Elimination of quotients in various localisations of premodels into models

The contribution of this article is quadruple. It (1) unifies various schemes of premodels/models including situations such as presheaves/sheaves, sheaves/flabby sheaves, prespectra/$\Omega$-spectra, simplicial topological spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors in categories/strong stacks and, to some extent, functors from a limit sketch to a model category versus the homotopical models for the limit sketch; (2) provides a general construction from the premodels to the models; (3) proposes technics that allows one to assess the nature of the universal properties associated with this construction; (4) shows that the obtained localisation admits a particular presentation, which organises the structural and relational information into bundles of data. This presentation is obtained via a process called an elimination of quotients and its aim is to facilitate the handling of the relational information appearing in the construction of higher dimensional objects such as weak $(\omega,n)$-categories, weak $\omega$-groupoids and higher moduli stacks.


Motivation 1.
There is an abundant literature on how to construct an algebraic object from one of its presentations [5,4,3,1,2]-this process will be referred to as a localisation. It is also well-known that the category of algebraic objects will satisfy strict universal properties if the objects themselves can be distinguished from their presentations by strict properties and, similarly, the category will usually satisfy weak universal properties if the objects can be distinguished from their presentations by weak properties, but little is known about how to derive strict universal properties for the category when the algebraic objects are only characterised by weak properties. One of the goals of the present paper is to address this lack.
If we think of an algebraic object as a model for a limit sketch [1], then algebraic objects can usually be distinguished from their presentations by lifting properties. Specifically, in the case of a limit sketch D, the presentations are given by the functors D → Set while the models are given by those presentations D → Set that preserve the chosen limits of D; as shown in [3], this type of property can be expressed in terms of a lifting property in the functor category Set D . On the other hand, the localisation of a presentation X into a model Q(X) is endowed with a reflection property, which equips X with a map i : X → Q(X) such that for every arrow f : X → M where M is a model, there exists an arrow f : Q(X) → M making the following diagram commute.
f the lifting properties characterising the models are strict, then one is able to show that the reflection is strict, that is to say that the arrow f : Q(X) → M is unique for any given f : X → M . For instance, in [3], one starts by characterising the models via strict lifting This research was carried out under an international Macquarie University Research Excellence Scholarship (iMQRES).
properties and the strictness of these is naturally carried over to the reflection property. This is the same idea in [4] where the author is able to construct a (strict) reflection from the strict lifting properties inherently associated with well-pointed endofunctors.
On the other hand, if the lifting properties are weak, then one is usually only able to show that the reflection is weak, in which case the arrow f : Q(X) → M is only proven to exist. For instance, in [6], the small object argument (recall that this argument comes from Homotopy Theory, which mostly, if not only, deals with weak lifting properties; see [7,8]) is used to construct weak reflections for subcategories of injective objects. Similarly, in Garner's framework [9,10], the small object argument is generalised to construct weak homomorphisms of ∞-categoriesà la Batanin [11] while the possibility to construct ∞-categories is assumed: the reason being that ∞-categories are objects that can be characterised by strict lifting properties [12,Corollary 1.19] while weak homomorphisms between these do not require such a strictness.
However, to the best of my knowledge, there has not been any published work explaining how to obtain strict reflection properties from weak lifting constructions such as the small object argument. In fact, it is not even clear how to obtain strict universal properties from weak characterisations in general. For instance, in [13], essential weak factorisation systems were introduced to study injective and projective hulls, which are meant to capture canonical envelops of injective and projective objects, with the goal of strengthening the lifting properties associated with the usual associated replacements (see intro. ibid.), but it is not said if these hulls can satisfy strict universal properties; in fact [14] gives a hint that this is unlikely and states that only an almost reflection property can be shown. The paper even emphasises the need of methods to pass from a weak setting to a strict one in its last section [14,Section 4], in which it is asked if it is possible to know when strict universal properties, such as naturality and functoriality, can be shown to be satisfied by a given weak reflection.
In an area of Mathematics in which the weakening of definitions and theories (e.g., ∞topos theory, univalent homotopy type theory, devired algebraic geometry, etc.) have now taken more and more importance, but whose language-Category Theory-also takes advantage of strict universal properties, it is, indeed, of interest to know if there are theorems that allow one to determine whether a set of weak lifting properties defining a type of algebraic object can provide the associated category with a strict universal property-at least stricter than the expected one.
The present paper is an effort to provide a set of technics and theorems showing that such a scheme is possible. Precisely, one of the main contributions of this paper is to propose a language (or context) in which it is possible to say if a category of algebraic objects that are characterised by weak lifting properties can be shown to possess a strict universal property (see Section 1.3). We will even show that the proposed argument is a generalisation of Quillen's small object argument (see Corollary 7.17) and will thus answer one of our earlier questions. The theorems given herein are meant to be generalised in future work (in which the boundary between strictness and weakness will become blurrier), the purpose being to pave the way for the construction of models taking their values in higher categorical structures. 1. 2. Motivation 2. The second matter that motivates the present paper is the so-called elimination of quotients mentioned in the title, which basically comes down to conclude that the way we encode an object is as important as its inherent properties. For instance, it is this same type of ideas that motivated the introduction of the elimination of imaginaries, in Shelah's Model Theory [15,16], in which quotients are eliminated in the form of definable quotient maps by using the various sorts available from the ambient (multi-sorted) theory; the development of the concept of covering space, in Algebraic Topology [17], that provides ways to blow up the quotients acting on a space and to bring out its homotopical properties by studying the automorphisms acting on the resulting quotient maps; the definition of stack, in Algebraic Geometry [18], due to the existence of non-trivial automorphisms that may occur because of the different ways a moduli space can be represented.
To really understand how the coding of objects, and, even that of sets, matters from the point of view of their algebraic structures, let us consider an example. Take a set X and consider the coproduct E := X + X encoded by the following logical specification.
If one takes R to be the binary relation on X + X that identifies (0, x) with (1, x) for every x ∈ X, then the quotient E/R is obviously isomorphic to X. However, in much the same way as it is fundamental to not confuse an isomorphism with an identity, it is, here, important to understand that E/R is not same as X. From the point of view of the present paper, the difference between X and E/R lies in the implicit algebraic structure with which E/R is equipped. This object can indeed be seen as a surjection p : E → X equipped with two sections s 0 , s 1 : X → E whose cospan structure defines a universal cocone, and this structure is noticeable even thought E/X is isomorphic to a mere set. In other words, the quotient E/R can be seen as living way beyond the category of sets, for the simple reason that isomorphisms are not the same as identities.
All this shows that the way we construct algebraic objects matters quite substantially, mainly because the algebraic properties coming along with their representations can turn out to be either very useful or extremely cumbersome (e.g., X versus E/R).
The goal of our so-called 'elimination of quotients' will be to eliminate the cumbersome quotients that may occur in the representation of algebraic objects and organise, in the form of quotient maps, the useful ones. Here, I feel important to mention that such a re-organisation is possible because our objects are characterised by weak lifting properties, which allow more freedom than strict ones.
If we look at how Kelly [4, Theorem 10.2] constructs algebraic objects, and to be more specific, models for some limit sketch (D, K), where K denotes the set of limit cones associated with D, we see that he isolates each cone c ∈ K and constructs, for each of these and every presentation X : D → Set, a well-pointed endofunctor i c : X → P c (X) where the object P c (X) completes the presentation X with operations required by the sub-theory (D, {c}) of (D, K). To complete X with respect to the operations required by the whole theory (D, K), he pushes out the wide span made of the arrows i c , for all c ∈ K, to obtain a well-pointed endofunctor i : X → P (X). In particular, each cone c ∈ K is equipped with a factorisation as follows.
Finally, the reflector X → Q(X) associated with the theory (D, K) is computed through a transfinite composition of the following form.
Isolating each cone c in K and proceeding to a pushout of the well-pointed endofunctors X → Q c (X) is a necessity if one wants to use the very neat and compact formalism of well-pointed endofunctors. However, this pushout procedure, as elegant as it may be, adds more cumbersome quotients than useful ones. Precisely, the wide pushout of the objects Q c (X) looks more like the type (X + X)/R ∼ = X because it mostly identifies all the copies of X living in each Q c (X) through the maps X → Q c (X).
As we can imagine, these cumbersome quotients become much more abundant when enriching our algebraic objects to other categories than Set and it would not be imaginable to be willing to do combinatorics with representations that repeat and contract the same information over and over. Not only do the results proposed in the present article avoid these cumbersome quotients, but they also bring out the hidden algebraic structure of the useful ones, where, here, the term 'algebraic structure' is used in the sense previously discussed for the quotient E/R.
In fact, our results go in the direction of Lawvere's work [5], in which the concept of congruence is used to construct a reflector from the category of presentations to that of models by showing how the quotients act on the free algebra functor applied on the presentations [5,Theorem 5.1]. It is worth noting that the concept of congruence has given rise to a very rich theory regarding the characterisation of congruence lattices for varieties of algebras [19,20]. Our results can therefore be seen as a refined extension of Lawvere's work. This refinement is presented in the form of a formal language that could be seen as suitable for a generalisation of Congruence Lattice Theory to more general objects than those proposed by Lawvere. 1. 3. Results for motivation 1. In the same fashion as there are categories of models for a theory [1], or categories of fibrants objects [21] or even systems of fibrant objects [22], it is, here, proposed the definition of system of premodels (see Definition 4.17), which gathers in the same structure a category of presentations together with maps along which the models are defined via weak lifting properties. An interesting feature of this structure is that it encompasses many examples that are meant to be captured operibus citatis; particular examples can also be found in [24,25,23,26]. There is also a novelty in the fact that the maps along which the weak lifting properties are defined are not maps in the category of values or that of presentations, but in a category whose level of definition allows one to verify whether the subcategory of the resulting models possesses a strict reflection property. For instance, this allows us to retrieve and explain the strict reflection property associated with the models for a limit sketch.
If we restrict ourselves to algebraic objects defined by limit-preserving functors, say valued in a category in which choices of colimits are obvious, a system of premodels is given by (1) a limit sketch (D, K); (2) a category C with enough limits and pushouts, if not all; (3) a subcategory P → C D ; (4) for every cone c ∈ K, a set V c of commutative squares in C, say as follows.
Before giving the definition of a model for this structure, we need to recall that a cone c in K is a natural transformation ∆ A (ou(c)) ⇒ in(c) where ou(c) is an object in D, A is a small category, ∆ A (ou(c)) is the obvious constant functor A → 1 → D picking out the object ou(c) in D and in(c) is some functor A → D. Now, a model for the previous structure is a functor D → C in P such that for every c ∈ K, the canonical arrow P (ou(c)) → lim P • in(c), for which we shall prefer the more compact notation P [c] := lim P • in(c), is orthogonal in the arrow category C 2 to every commutative square in V c (as shown below).
In the case of limits sketches, we retrieve the usual definition of model by taking, for every cone c ∈ K, the following pair of commutative squares in Set; the leftmost one encodes the surjectiveness of the map P (ou(c)) → P [c] while the other one encodes its injectiveness.
One of the very advantages of this language is to allow the specification of more general arrows than bijections such as weak equivalences (see characterisation in [27,Lemma 7.5.1]). This explains why this language is expected to be generalised to higher categorical structures in the future. Now, our main result, given in Theorems 8.18 and 8.21, can be simplified in terms of Theorem 1.1, in which items (i) and (ii) are in fact redundant. The statement makes use of the arrow β : S → D , which denotes, for every commutative square contained in V c and every c ∈ K, the universal arrow induced by the pair of arows β 1 and β 2 under the pushout (denoted by S ) of the arrows γ 1 and γ 2 .
Theorem 1. 1. Suppose that P → C D is an identity. For every object A in P, there exists an arrow i : A → Q(A) in P (Theorem 8.21) such that for every arrow f : A → X in P where X is a model for the system of premodels, if (i) the map β is an epimorphism for every square in V c and every c ∈ K; (ii) the arrow X(ou(c)) → X[c] is a monomorphism in C; (iii) the arrow β 1 is an epimorphism for every square in V c and every c ∈ K, then there exists a unique arrow g : Q(A) → X making the following diagram commute (Theorems 8.18 and 8.21).
g`À s one can see, the previous theorem explains, in the language of systems of premodels, why one can expect a strict reflection property in the case of set-valued models for a limit sketch.
In Theorem 1.1, the assumption that the inclusion P → C D is an identity will be replaced, in Theorems 8.18 and 8.21, with the notion of effectiveness, which translates a variation of the concept of definability in P (notice the parallelism with the concept of elimination of imaginaries given in Section 1.2). As will be shown in Theorem 8.13, this concept of definability becomes trivial if P is taken to be equal to C D . 1. 4. Results for Motivation 2. From the point of view of motivation 2, the present paper mainly focus on models for limit sketches in Set, so that we will mostly state our results from the perspective of these objects. This will nevertheless give an idea of what our theorems look like when generalised to other categories. The proof of the results stated below will be recapitulated in the conclusion of the present paper (Section 9).
We now consider a limit sketch (D, K), where, for simplicity only, K is supposed to be a finite set of finite-limit cones. The proposition given below states that it is possible to construct the reflector of any presentation in a very specific way, which is not visible from Kelly's construction [4].
For every presentation X in Set D and ordinal i ∈ ω, there exist a pair of objects E i (X) and B i (X) and an epimorphism p i : B i (X) + E i (X) → B i+1 (X) such that the reflector of X for the theory (D, K) is given by the transfinite composition of the following sequence of arrows in Set D .
In addition, the mappings X → E i (X) and X → B i (X) are functorial and the arrow Of course, one could argue that the map X → P (X) coming from Kelly's construction can be factorised into an epimorphism and a monomorphism X B(X) → P (X), so that we might recover the previous form, but it is not obvious whether P (X) can be decomposed into a functorial sum B(X) + E(X) in Set D , mainly because the quotients that acts on P (X) might prevent from doing so. In fact, there is a much stronger way to assess the difference between Kelly's construction and the previous one, which is given below. Proposition 1. 3. For every presentation X in Set D , there exist a sequence of epimorphism (p i : B i (X)+E i (X) → B i+1 (X)) i∈ω , as given in Proposition 1.2, for which there is a natural transformation of transfinite sequences for which α 0 is the identity on X and if there exists a pair of dashed arrows making the following triangle commute for n > 1, then the front arrow must factorise through the canonical map B n (X) → B n (X) + E n (X) and the object P n−1 (X) is a model for the limit sketch (D, {c}).
In other words, Kelly's construction has too many quotients to be non-trivially lifted to the elimination of quotients, and if a lift exists, then it cannot be in the free part E n (X), which means that, at rank n, the free operations added to satisfy the theory (D, {c}) are superfluous.
Even though the natural transformation α is to identify free operations between each other, note that it cannot identify too much information either as the universal property of a reflector implies that the transfinite colimit of α provides an isomorphism between the two underlying reflectors of X.
In fact, we will show that, in the case of models for a limit sketch, the so-called elimination of quotients takes the form given in Theorem 1.4, in which every cone c in K is again viewed as a natural transformation ρ : ∆ A (ou(c)) ⇒ in(c) where ou(c) is an object in D, A is a small category, ∆ A (ou(c)) is the obvious constant functor A → 1 → D picking out ou(c) in D and in(c) is some functor A → D. Theorem 1. 4. For every presentation X in Set D , there exist a sequence of epimorphisms as given in Proposition 1.3, for which we will denote the coproduct object B i (X) + E i (X) as a functor S i : D → Set, such that -B 0 (X) = X and E 0 (X) = ∅; -E i+1 (X) is the left Kan extension of the functor along the functor ou : K → D, where K is seen as a discrete small category; -the epimorphism p i is the quotient map S i → B i+1 (X) making the following identifications: (1) for every object d in D, it identifies a pair x, y ∈ S i (d) if there exists a cone c ∈ K and an arrow t : ou(c) → d in D for which the pushout of the canonical x and y to the same element; there exists a cone c ∈ K, an object z in the diagram A of c and a morphism t : in(c)(z) → d in D such that x and y can be lifted to a common element of the following composites.
Even though we have only discussed the finite-limit case, all of the previous propositions hold for non-finite limit-sketches. In this case, the ordinal ω becomes the cardinality of the limit-sketch (see the end of Section 4.1) and the transfinite sequence of arrows B i (X) + E i (X) → B i+1 (X) needs to be defined such that B α (X) is the transfinite colimits of all the arrows preceding the rank α. 1.5. Road Map. The main results of the paper start to be developed from Section 4, while Sections 2 and 3 give an account of various notations, conventions and technicalities. Specifically, Section 2 introduces a set of conventions meant to facilitate our notations while Section 7 focuses on a notion of smallness that will only be used in Section 7.
Even if Section 2 does not sound so attractive, the reader might want to skim through this section to get used to specific notations such as ι κ (Section 2.1); col D (Section 2.3); ξ i as well as col i (Section 2.5) and B D d ( ) (Section 2.14). Section 3 defines a notion of smallness that generalises the usual one. Recall that one usually says that an object D in some category C is small if for any functor 1 . defined from the ordinal category ω to C, say F : ω → C, the following canonical map is a bijection.
On the other hand, the smallness condition defined in Section 3 would be more of the following type. The property is now centred on the functor F and not on the object D any more; we then consider a set of objects G in C and say that a functor F : ω + 1 → C is G-convergent if the following canonical map is a bijection for every object D ∈ G.
The reason for this change is that the image F (ω) will not always be a colimit of the form col i∈ω F (i).
Then comes Section 4, in which is defined the notion of system of premodels. The difference with the simplified version given in Section 1.3 and that of Section 4 is that the canonical map Xou(c) → X[c] is now constructed from various parts of the system of premodel structure, so that it is now of the form Xou(c) → RX[c] where R is a right adjoint endofunctor on C. This right adjoint R will often be an identity functor in this paper, save for Ω-spectra, in which case it will be equal to the loop space functor Ω. In the future, the functor R will however take multiple forms.
Sections 6 and 5 work together to formalise the idea of algebraic structure associated with a quotient. Recall that completing a presentation with operations usually requires the adding of free operations along with certain quotients. In our case, the free structure will be added to the presentations, but the quotient structure will be resolved in a separate object q (see Section 6.7). The term resolved here refers to the concept of resolution developed in [28], which should be viewed as a way of passing from what looks like a set E/X to a higher dimensional structure, such as category or a quotient map E → X.
The purpose of Section 5, alone, is to give a theoritical generalisation of Quillen's small object argument [8] while Section 6 focuses on applying the formalism of Section 5 to systems of premodels.
The difference between our argument and Quillen's one is that one does no longer consider strict pushouts at every step and the lifts meant to be produced by these pushouts only commutes in the subsequent steps. These differences arise for two reasons. The first one is the desired elimination of quotients and the second one is due to the fact that the pushouts used in the usual argument do not necessarily commute with the right adjoints (including the limits) involved in the construction of the object RX[c].
To be able to formalise the previous ideas, we will introduce the concept of tome, whose goal is to gather all the squares that one would like to force to admit a lift through the small object argument. This will take the form of a functor ϕ : S → C 2 /h, where h is an object in the arrow category C 2 . Note that this tool will mainly find its use in the way the category S is encoded.
Specifically, in Section 6, this category S will be discrete and will take the form of a coproduct of what could look like two left Kan extensions.
The left-hand sum will allow us to parameterise all those squares that are to force the adding of the structural information to the presentations while the right-hand sum will allow us to handle all of the quotients that the adding of this information is supposed to generate. Note that the rightmost sum of S is only meant to quotient out what has been added at a previous step, leaving free the information added by the current leftmost sum and thus producing the elimination of quotients discussed in Section 1. 4. All the data needed to talk about an elimination of quotients such as J A , J Q , , χ, Λ A [ ], Λ Q [ ] (and some more) will be gathered into the notion of constructor (see Section 6.4). Remarks 6.13 and 6.15 might be helpful in seeing what all those left Kan extension-like constructions actually parameterise.
Finally, the small object argument is carried out in Section 7 where the smallness condition is used to prove the usual lifting properties. The universal property satisfied by our construction is discussed in Section 8 via Theorems 8.21 (existential part) and 8.18 (uniqueness). The latter mainly focus on the properties required to prove Theorem 1.4, whose proof is recapitulated in the conclusion (see Section 9.2). 1. 6. Acknowledgments. I would like to thank Steve Lack and the referees for comments that allowed the improvement of the earlier versions of this text. I would also like to thank the members of the Australian Category Seminar for various remarks regarding the content of this paper.

Background, Notations and Conventions
2.1. Ordinals. Any ordinal will be identified with the preorder category it induces. For every ordinal κ, the inclusion functor κ → κ + 1 will be denoted by ι κ . For convenience, the preorder category of one and two objects will be denoted by 1 and 2, respectively. We shall also use the notation ω to denote the least infinite ordinal.

Wide Subcategories.
Let C be a category. A subcategory A ⊆ C will be said to be wide if the inclusion functor A → C is surjective on objects. Put simply, this means that A contains all the objects of C.

Limits and Colimits.
For every category C and small category D, the obvious functor C 1 → C D mapping an object X : 1 → C to the pre-composition of X : 1 → C with the canonical functor D → 1 will be denoted by ∆ D . For convenience, the category C 1 will often be identified with the category C. If they exist, the left and right adjoints of ∆ D will be denoted by col D and lim D , respectively. Recall that the images of these two functors are understood as the colimits and limits of C over D, respectively. As usual, in the case where the functor lim D : C D → C 1 exists, the category C will be said to be complete over D. Similarly, the category C will be said to be cocomplete over D when the functor col D : C D → C 1 exists. Proposition 2.1. If a category C is complete (resp. cocomplete), then so is C D for any small category D where the limits (resp. colimits) are defined objectwise in C.

Proof.
Suppose that C is complete. For every object d in D, the restriction functor ∇ d : C D → C mapping X to X(d) has a right adjoint whose images are given by the Right Kan extensions along the functor 1 → D picking out d [29]. This implies that ∇ d commutes with limits. By duality, the other statement regarding colimits follows. 2.4. Cardinality. Let A be an object in Set. The cardinality of A is the least ordinal κ such that there is a bijection between A and κ. In ZFC, the axiom of choice ensures that the cardinality of a set A always exists, which will be denoted by |A|.
For any small category D, the cardinality of D is the cardinality of the following coproduct of sets, where Obj(D) is the set of objects of D.
The cardinality of D will be denoted by |D|. Below is given a well-known result on the commutativity of limits and colimits.
Proposition 2. 2. For every small category D and limit ordinal κ ≥ |D|, the canonical natural transformation col κ lim D ⇒ lim D col κ valued in Set over Set κ×D is an isomorphism. A. Similarly, for every complete category C and small category D, the functor ∆ D : C → C D commutes with colimits (see Proposition 2.1). In fact, it follows from Proposition 2.2 that the unit of the adjuncion ∆ D lim D commutes with colimits in Set as stated in the next proposition. Proposition 2. 3. For every small category D and limit ordinal κ ≥ |D|, denote by the letter η the units of the two adjunctions ∆ D lim D in Set and Set κ . The following diagram of canonical arrows in Set commutes for any functor F : κ → Set. A. 2.5. Universal Shiftings. Let i : T → S be a functor between small categories. The precomposition with i induces an obvious functor • i : C T → C S . Mostly for convenience, the composition of this functor with the colimit functor col S : C S → C will later be denoted by col i : C T → C. The obvious canonical natural transformation ξ i : col i ⇒ col S will be called the universal shifting along i. Similarly, the composition of the functor • i : C T → C S with the limit functor lim S : C S → C will be denoted by lim i : C T → C.

Proof. See Appendix
2. 6. Right Lifting Property. Let C be a category and A be a class of arrows in C. The class of arrows of C that have the right lifting property (abbrev. rlp) with respect to the arrows of A will be denoted by rlp(A).

Sequential Functors.
Let κ be some ordinal and C be a category. A functor F : κ + 1 → C will be said to be sequential if for any limit ordinal α in κ + 1, the object F (α) may be identified with the colimit of the functor F • ι α : α → C such that, for every ordinal β in α, the morphism F (β < α) : F (β) → F (α) corresponds to the arrow of the universal cocone of col α F • ι α associated with β.

Proof.
It is straightforward to show that if a morphism f has the rlp with respect to two composable arrows i and j, then it has the rlp with respect to the composition i • j. A direct generalisation to the transfinite case shows the proposition.

Limit Sketches.
A limit sketch is a small category S equipped with a subset Q of its cones 2 . The cones in Q will be said to be chosen. A model for a limit sketch S in a category C is a functor S → C that sends the chosen cones in Q to universal cones 3 in C. The models of a limit sketch S in C define the objects of a category Mod C (S) whose morphisms are natural transformations in C over S. For any limit sketch S, the category of models for S in Set will be denoted by Mod(S). Example 2.5 (Limit sketch for monoids). The category of monoids in Set may be defined as a category of models for a certain limit sketch Mon. The underlying small category of Mon is freely generated over a set of arrows and quotiented by commutativity relations. Specifically, the category Mon has four objects g 0 , g 1 , g 2 and g 3 , where g 1 is called the underlying object of the sketch, and a set of arrows as follows, where the identities have been forgotten.
Recall that these are, by definition, natural transformations of the form ∆ A (d) ⇒ U in S where A is a small category, U is a functor A → S and d an object in S, called the peak 3 'Universal' here means that the cone, say ∆ A (d) ⇒ U , defines a limit of the functor U : A → S The commutativity relations are given by the diagrams Example 2.8 (Limit sketch for rings). By definition, the subcategory of Ab(µ, η, δ) generated by p 1 , p 2 , p 12 , . . . , p 12 and ! is also included in Mon(µ , η ). The pushout of Ab(µ, η, δ) and Mon(µ , η ) along these underlying inclusions provides a certain limit sketch pRg(µ, µ , η, η ) that contains five objects and all the arrows and cones appearing in Ab(µ, η, δ) and Mon(µ , η ); the associated limit sketch combines the structure of a monoid with the structure of a commutative monoid. One thus recovers the theory of rings if one adds an object g 4 , a chosen cone g 2 q1 ←− g 4 q2 −→ g 2 and the following arrows and commutativity relations to pRg(µ, µ , η, η ).
The resulting limit sketch Rg(µ, µ , η, η ) then defines a sketch for which the models are rings. The limit sketch Rg(µ, µ , η, η ) to which the identity morphism 1 g1 : g 1 → g 1 is added to the set of chosen cones-when seen as a trivial cone-will later be denoted by Rg. 2. 10. Overcategories. Let C be a category and X be an object in C. The obvious functor C/X → C mapping an arrow f : A → X in C to the object A in C will be denoted by ∂.
Remark 2. 9. Let T be a small category. Any functor F : T → C/X may be seen as a natural transformation in C over T of the form h : ∂F ⇒ ∆ T (X). The converse is also true.
Let now G : A → C be a functor. It will come in handy to denote by C G the obvious functor on A satisfying the following mapping rule on the objects. 11. Covering Families. Let D be a small category and d be an object in D. A covering family on d is a collection S := {u i : d i → d} i∈A of arrows in D. For every morphism f : c → d in D, we shall speak of the pullback of S along f to refer to a collection of arrows f * S := {v i : c i → c} i∈A where the arrow v i is a pullback of u i along f . Also, note that every morphism g : d → c gives rise to a family g • S := {g • u i } i∈A . This last operation is used to define a more complex operation on S as follows. For every i ∈ A, take a covering family T i on d i . We will denote by S • {T i } i∈A the covering family on d obtained by the disjoint union of families u i • T i for every i ∈ A. (2) (Locality) for every i ∈ A and T i in J di , the covering family Such a collection will usually be denoted by J. A category D equipped with a Grothendieck pretopology J on D will be called a site.
Remark 2. 10. Every covering family S = {u i : d i → d} i∈A on an object d in J d may be seen as a functor A → D/d if A is seen as a discrete category. It follows from the stability and locality axioms that this functor extends to a product-preserving functor A → D/d where A is the completion of A under products. This functor will be called the stabilisation of S.

Families.
For any category C, the notation Fam(C) will be used to denote the category whose objects are pairs (S, F ) where S is a discrete category and F is a functor F : S → C and whose morphisms (S, F ) ⇒ (S , F ) are given by pairs (a, α) where a is a functor a : S → S and α is a natural transformation α : F ⇒ F a.

2.14.
Bounded Diagrams. Let D be a small category, d be an object in D and C be a category. We will denote by B D d C the category whose objects are triples (P, e, Q) where P and Q are functors D → C and e is an arrow P (s) → Q(s) in C and whose morphisms, say (P, e, Q) → (P , e , Q ), are given by pairs of natural transformations (α, α ) of respective forms P ⇒ P and Q ⇒ Q making the following square commute.
is the smallest subcategory of 2 × D consisting of the two copies of D and the arrow linking the two copies of d.

Convergent Functors
This section aims to define the notion of convergent functor, which is to replace the notion of 'small object' that is usually used in transfinite constructions. 3. 1. Emulations. Let S and T be two small categories and C be a category. A pair of functors g : C S → C T and h : Set S → Set T will be called an (S ↓ T)-emulation in C if it is equipped with a natural isomorphism as follows.
In terms of an equation, the previous diagram means that (g, h) is equipped with a natural isomorphism (in the variables X ∈ C, Y ∈ C S and t ∈ T) as follows.
Example 3. 1. Let T be a small category and C be a category. Take g to be the identity functor id : C T → C T and h to be the identity functor id : Set T → Set T . By definition, the pair (g, h) is a (T ↓ T)-emulation.
Example 3.2. Let U : T → S be a functor between small categories and C be a category. Take g to be the pre-composition functor C S → C T induced by U and h to be the equivalent version of g in Set.
It suffices a few lines of calculation to show that the following isomorphism holds, which implies that the pair (g, h) defines an (S ↓ T)-emulation.
Example 3. 3. Let T be a small category and C be a category. Take g to be the functor ∆ T : C 1 → C T and h to be the functor ∆ T : Set 1 → Set T . It follows from Example 3.2 that the pair (g, h) is a (1 ↓ T)-emulation.
Example 3. 4. Let S be a small category and C be a category complete over S. Take g to be the limit functor lim S : C S → C 1 and h to be the limit functor lim S : Set S → Set 1 . It is a well-known fact following from Yoneda's Lemma that the pair (g, h) is an (S ↓ 1)-emulation.
Example 3. 5. Let S be a small category and C be a category complete over S. We will denote by η the unit of the adjunction ∆ S lim S valued in any category. Now, take g to be the obvious functor C → C 2 mapping an object X in C to the arrow η X : X → lim S ∆ S (X) in C and h to be the equivalent version of g in the category Set (which is complete over S).
It follows from Yoneda's Lemma that the following diagram commutes, which implies that the pair (g, h) is an (S ↓ 2)-emulation.
Example 3. 6. Let S be a small category. For this example, we shall additionally need a small category A together a cone r : ∆ A (s) ⇒ U in S A . Let now C denote a complete category over A. The unit of the adjunction ∆ A lim A in C will be denoted by η. Now, to define our emulation, take g to be the obvious functor C S → C 2 mapping a functor P : S → C to the arrow in C and h to be the equivalent version of g in the category Set. It follows from Yoneda's Lemma that the pair (g, h) is an (S ↓ 2)-emulation. Specifically, the isomorphism associated with the pair (g, h) may be deduced from the isomorphisms involved in Examples 3.2, 3.4 and 3.5.
Example 3. 7. Let S be a small category. For this example, we shall need a small category A together a cone r : ∆ A (s) ⇒ U in S A . Let now C denote a complete category over A. The unit of the adjunction ∆ A lim A in C will be denoted by η. Now, to define our emulation, take g to be the obvious functor B S s C → C 2 mapping an object (P, e, Q) in B S s C to the arrow in C and h to be the equivalent version of g in the category Set. It follows from the isomorphisms involved in Examples 3.3, 3.4 and 3.6 that the pair (g, h) is an (2 s [S] ↓ 2)emulation.

Cocontinuous Emulations.
Let S and T be two small categories, C be a category and κ be a limit ordinal. An (S ↓ T)-emulation (g, h) in C will be said to be κ-cocontinuous, if for every object t ∈ T, the functor h : Set S → Set T preserves colimits over κ.
Example 3. 9. Consider the same context as that used in Example 3.2. Since Set is cocomplete over any small category D, the colimits of Set S are componentwise colimits, which means that for every functor F : D → Set S , the following isomorphism holds for every s ∈ S.
This directly implies that the functor h : Set S → Set T preserves colimits, which shows that the (S ↓ T)-emulation (g, h) is κ-cocontinuous for every limit ordinal κ.
Example 3. 11. Consider the same context as that used in Example 3.4 and suppose to be given a limit ordinal κ satisfying the inequality |T| ≤ κ. It directly follows from Proposition 2.2 that the functor h : Set S → Set T preserves colimits over κ. This shows that the (S ↓ 1)emulation (g, h) is κ-cocontinuous.
Example 3. 12. Consider the same context as that used in Example 3.5 and suppose to be given an limit ordinal κ satisfying the inequality |T| ≤ κ. It follows from Proposition 2.3 that the functor h : Set S → Set T preserves colimits over κ. This shows that the (S ↓ 2)-emulation (g, h) is κ-cocontinuous. Example 3. 13. By using the cocontinuity involved in Examples 3.11 and 3.12, we may show that the (S ↓ 2)-emulation (g, h) is κ-cocontinuous for any limit ordinal κ satisfying the inequality |A| ≤ κ.

Convergent Functors.
For any class G of objects of C, a functor F : κ + 1 → C will be said to be G-convergent in C if for every object D in G, the following canonical function (obtained by homing) is an isomorphism in Set.
If the class G turns out to be a singleton {D}, the functor will more explicitly be said to be D-convergent.
Remark 3. 15. One of the useful implications of the previous definition is that if a functor F : κ + 1 → C is G-convergent in C, then for every object D ∈ G and morphism f : D → F (κ) in C, there exist an ordinal α ∈ κ and a morphism f : D → F (α) making the following diagram commute in C.
Let now T and S denote two small categories and G : T → C be a functor. A functor F : κ + 1 → C S will be said to be unimorly G-convergent in C if for every object s in S and object t in T, the following canonical function is an isomorphism in Set.
In other words, the evaluation of F at an object s in S is {G(t) | t ∈ Obj(T)}-convergent.
Lemma 3. 16. Let T and S be two small categories such that |T| ≤ κ and C be a category. Let G : T → C be a functor and consider a uniformly G-convergent functor F : κ + 1 → C S in C. For every cocontinuous (S ↓ T)-emulation (g, h), the composite functor g • F : Proof. The following series of natural isomorphisms proves the statement.
This last isomorphism shows that g • F is G-convergent in C T .
Example 3. 17. Applying Lemma 3. 16 to the (T ↓ T)-emulation (g, h) of Example 3.1 implies that if a functor F : κ + 1 → C T is uniformly G-convergent in C and the inequality |T| ≤ κ holds, then the functor F : κ + 1 → C T is G-convergent in C T .
Example 3. 18. Applying Lemma 3. 16 to the (2 s [S] ↓ 2)-emulation (g, h) of Example 3.14 implies that if a functor (P, e, Q) : κ + 1 → B S s C S is uniformly G-convergent in C for some functor G : 2 → C and the inequality 2 ≤ κ holds, then the functor mapping an ordinal n in κ + 1 to the following composite arrow in C is G-convergent in C 2 . 19. It follows from Lemma 3.16 that if a functor F : κ + 1 → C is uniformly G-convergent in C, then F : κ + 1 → C is col T (G)-convergent in C. Specifically, this follows from the fact that ∆ T commutes with hom-sets (see Example 3.10) and the following series of isomorphisms.

Models for a Croquis
This section defines the notions of premodel and model for which we want to construct the localisation. We start with the type of theory on which the models are defined. 4. 1. Croquis. Let D be a small category. Recall that a cone in D over a small category A consists of two functors d 0 : 1 → D and d 1 : When such a cone is called c, the functor d 0 will be denoted by ou(c), the functor d 1 will be denoted by in(c) and the small category A will be referred to as the elementary shape of c and denoted by Es(c).
Definition 4.1. A croquis category (or croquis) in D consists of a set K of cones in D and a functor rou : K → D (where K is seen as a discrete category) called the regular output.
A croquis as above will be denoted by a triple (D, K, rou) and sometimes shortened to the pair (K, rou) when the ambient category D is obvious.

Convention 4.2.
For every croquis (D, K, rou), the operation ou( ) induces a function from K to Obj(D). Alternatively, this may be seen as a functor K → D. If the functor rou : K → D is equal to ou : K → D, then the croquis will be denoted by (D, K) or K and the functor rou will be said to be trivial.     4.7 (Flabby pretopologies). Let J denote a Grothendieck pretopology on a small (opposite) category D op . The croquis that will later give rise to flabby sheaves and the Godement resolution is the union of the two croquis (D, K J ) and Cr(D, id D ). Precisely, this croquis consists of the union of the two sets of cones K J and Mor(D) and the trivial regular output.
Example 4.8 (Segal croquis). Let ∆ denote the category of non-zero finite ordinals and preserving-order functions, which is known as the simplex category. Denote by ∆ + the wide subcategory of ∆ whose arrows are injective functions and, for every object r ∈ ∆, denote by ∂ r the composition of the functor ∂ : ∆/r → ∆ (see Section 2.10) with the obvious inclusion ∆ (2) s : 1 → 2 is the function with the mapping rule 0 → 0; Example 4.9 (Complete Segal croquis). Let ∆ be the simplex category. The complete Segal croquis of ∆ op is given by its Segal croquis (∆ op , K) to which is added the unique cone whose peak is the ordinal 1 and whose diagram in ∆ is given, below, underlying the cocone of dotted arrows, where, if one denotes 2 := {0, 1} and 4 := {0, 1, 2, 3}, (1) l : 2 → 4 is the function with the mapping rules 0 → 0 and 1 → 2; (2) r : 2 → 4 is the function with the mapping rules 0 → 1 and 1 → 3; The induced cone in ∆ op will be denoted by c iso as it is meant to describe the set of isomorphism structures relative to the natural categorical (or nerval) structure of ∆ op . The resulting croquis will be denoted by Cseg(∆ op ).
We shall speak of an elementary shape of a croquis (D, K, rou) to refer to the elementary shape of one of its cones. Because K is a small category, the class of elementary shapes of (D, K, rou) is a set, which will be denoted by Es(K). The cardinality of a croquis (D, K, rou) is then given by the cardinal of the coproduct of every small category in Es(K).
Premodels. Let (D, K, rou) be a croquis and C be a category. For any endofunctor R : C → C, denote by Pr C (K, rou, R) the category whose objects are triples (P, S, e) where (1) P is a functor D → C, (2) S is a functor 4 K → Set and (3) e denotes a collection of arrows e c,s : P rou(c) → RP ou(c) in C for every c ∈ K and s ∈ S(c) and whose morphisms, say of the form (P, S, e) ⇒ (P , S , e ), are pairs (f, a) where f and a are two natural transformations of respective forms P ⇒ P and S ⇒ S making the following diagram commute for every c ∈ K and s ∈ S(c).
The objects of Pr C (K, rou, R) will be called the R-premodels for (K, rou). For convenience, the category Pr C (K, rou, R) will sometimes be denoted as Pr C (K, R) when rou is trivial and as Pr C (K) when R is also an identity.
Example 4.10 (Premodels). The category of premodels for a sketch (D, K) to a category C corresponds to the full subcategory of Pr C (K) whose objects (P, S, e) are such that the images of S are equal to 1 and the morphism e c : P ou(c) → P ou(c) is an identity for every c ∈ Q. This subcategory is isomorphic to C D .
Example 4.11 (Presheaves). The category of presheaves over a site (D op , J) corresponds to the full subcategory of Pr Set (K J ) whose objects (P, S, e) are such that the images of S are equal to 1 and the morphism e c : P ou(c) → P ou(c) is an identity for every c ∈ K J . This subcategory is isomorphic to Set D .

Example 4.12 (Prespectra).
If Ω : pTop → pTop denotes the loop space functor on the category of pointed topological spaces and pred denotes the predecessor operation n → n−1 on N * , then the category of prespectra is the full subcategory of Pr Top (Cr(N, pred), Ω) whose objects (P, S, e) are such that the images of S are equal to 1. This subcategory will be denoted by PrSpc. Example 4.13 (Pre-localised rings). Let Set denote the category of sets and Rg be the limit sketch defined in Example 2. 8. The category of 'pre-localised rings' is defined as the full subcategory of the category Pr Set (Rg) whose objects (P, S, e) are such that (1) P : Rg → Set is a model for Rg; (2) the image of S : K → Set above the cone 1 g1 : g 1 → g 1 is equal to a subset of P (g 1 ) while its images above all the other cones are equal to 1 and (3) the morphism e c,s : P (g 1 ) → P (g 1 ) is given by -the right multiplication map x → P (µ )(x, s) for every s ∈ S(c) if c = 1 g1 ; -the identity morphism e c : P ou(c) → P ou(c) otherwise.
This subcategory will be denoted by PrLocRg. Example 4.14 (Pre-Segal spaces). Let Top denote the category of topological spaces and continuous functions. The category of pre-Segal spaces is the category of simplical topological spaces; it is given as the full subcategory of Pr Top (Seg(∆ op )) whose objects (P, S, e) are such that the images of the functor S are equal to 1 and the morphism e c : P ou(c) ⇒ P ou(c) is an identity for every c ∈ Seg(∆ op ). Thecategory of pre-complete Segal spaces is defined similarly by replacing Seg(∆ op ) with Cseg(∆ op ). 15. Let D be a small category and C be a category. For any given endofunctor R : C → C, a category of R-premodels is a subcategory of the category Pr C (K, rou, R).
Example 4. 16. Premodels for a sketch, presheaves on a site, prespectra, pre-localised rings and pre-Segal spaces are examples of such categories (see the previous examples). 4. 3. Models. Let D be a small category, (K, rou) be a croquis in D and C be a complete category over the elementary shapes of K. Suppose to be given a right adjoint R : C → C.
The first goal of this section is to define a functor G K c : Pr C (K, rou, R) → Fam(C 2 ) for every cone c ∈ K. In this respect, for every cone c in K of the form t : ∆ A (d 0 ) ⇒ d 1 , for which we shorten the notation rou(c) to the symbol r, the functor G K c maps any premodel (P, S, e) to the family taking any s ∈ S(c) to the following composite arrow in C.
For every morphism of R-premodel of the form (f, a) : (P, S, e) ⇒ (P , S , e ), the image morphism G K c (f, a) : G K c (P, S, e) ⇒ G K c (P , S , e ) is given, for every s ∈ S(c), by the following morphism in C 2 .
Definition 4.17 (System of premodels). A system of R-premodels consists of (1) a croquis (D, K, rou); (2) a category C that is complete on the elementary shapes of K and admits a terminal object; (3) a category of R-premodels P → Pr C (K, rou, R) where R is a right adjoint and (4), for every cone c ∈ K, a set V c of commutative squares in C, called the diskads (see left diagram, below) equipped with a pushout in C (see right diagram, below).
The collection consisting of all the sets V c will usually be denoted by V. A system of R-premodels will be denoted as a 4-tuple (K, rou, P, V) and said to be defined over D in C. The diagrams used in Definition 4.17 can more efficiently be described as a colimit sketch in C (i.e. diagram equipped with colimits) of the following form.
This type of colimit sketch will be called a vertebra and denoted by the symbols γ 2 , γ 1 ·β. For such a vertebra, it will come in handy to refer to the arrows γ 2 , γ 1 , β and β • δ 1 as the seed, coseed, stem and trivial stem, respectively. Finally, the left adjoint of R : C → C will conventionally be denoted by L.
Definition 4.18 (Model). An R-premodel (P, S, e) in a system of R-premodels (K, rou, P, V) will be said to be an R-model if, for every cone c ∈ K, every component of the arrow G K c (P, S, e) ⇒ 1 in Fam(C 2 ) has the right lifting property with respect to all the diskads of V c when these are seen as arrows γ 1 ⇒ β 1 in C 2 with respect to the notations of Equation (4.1). 4.19 (Models for a sketch). For every limit sketch (D, K), define the system of premodels consiting of the croquis K (see Example 4.5); the associated category of premodels Set D → Pr Set (K) and, for every cone c in K, the set made of the following vertebrae in Set.
The id Set -models of such a system correspond to the models for the sketch (D, K).    4.11); the associated category of premodels Cat D → Pr Cat (K J ) and, for every cone c in K J , the set made of the following vertebrae for the obvious choices of morphisms, where (1) 1 is a terminal category; (2) iso is the free living isomorphism category (i.e., two objects, one isomorphism); (3) 2 is the free living arrow category (i.e., two objects, one arrow); (4) 2 ⊕ 2 is category made of two objects and two parallel arrows between them.
The id Cat -models of such a system correspond to those 'sheaves' D → Cat for which the sheaf condition is not a bijection but an equivalence of categories. only. The id Cat -models of such a system correspond to the strong stack (see [23]). The strong stacks completion constructed in ibid corresponds to a special case of the general construction given in this paper. only. The id Cat -models of such a system may be identified to the strong stacks of [23] up to the notion of homotopy defined thereof. , which is included in Pr Top (Seg(∆ op )) and (i) for every cone c in K 0 ⊆ Seg(∆ op ), the set of obvious vertebrae induced by the diskads given in Equation (4.4), where -n runs over the natural numbers; -the object D n is the topological n-disc; -the map ι n : D n → D n+1 is the obvious hemisphere inclusion; , the set of Vertebrae (4.5), where n runs over the positive integers and -the object S n−1 is the topological (n − 1)-sphere; -the maps between the different objects are induced by the obvious inclusions; The id Top -models of such a system correspond to the Segal spaces in Top (see [25] for a definition enriched in simplicial sets). (2) for the cone c iso (see Example 4.9), the set of vertebrae of the form (4.5) for every positive integer n. The id Top -models of such a system correspond to the complete Segal spaces in Top (see [25] for a definition enriched in simplicial sets).  (Cr(N, pred), Ω) and, for every cone c in Cr(N, pred), the set of vertebrae of pointed spaces defined in Diagram (4.6), where n is a positive number and -where the object S n−1 /∂ is the quotient of the (n − 1)-sphere by itself (i.e., a point); -where the object D n /∂ is the quotient of the n-disc by its boundary; -where the object S n /p is the quotient of the n-sphere by its equator; -where the object D n+1 /p is the quotient of the (n + 1)-disc by its equator; -where the object D n+1 /h is the quotient of the (n+1)-disc by one of its hemispheres; -where the object D n+1 /∂ is the quotient of the (n + 1)-disc by its boundary; -where the maps between the different objects are the obvious inclusions. (4.6) The Ω-models of such a system correspond to the Ω-spectra.   [30] or [31,Proposition 8]) that, in some nice model category C, the notion of weak equivalence may be characterised via the type of right lifting property expressed in Example 4. 18. For instance, Examples 4.25 and 4.26 on Segal spaces could have been extended to any nice cofibrantly generated model category, which need not be simplicial (contrary to usual practice). In fact, it is worth noting that the type of localisation described in the present article is an alternative to the usual simplicial Bousfield Localisation process (see [7]). On could also look at the type of localisation discussed in [32,Corollary 8.8], which could be comprised in a more technical generalisation of the present work. Future work will also aim at generalising Example 4.22 to weaker functors in order to charactise the notions of (∞, n)-stack and strong (∞, n)-stack.

Narratives and the Small Object Argument
s This section aims to introduce the small object argument that will be used for the construction of the localisation. The difference from that given below and the one defined by Quillen [8] is the notion of 'degree' coming along with the concept of narrative (see the table below). The degree is the key ingredient that allows us to obtain our so-called elimination of quotients.

Notions Descriptions
Tome A collection of commutative squares whose rightmost vertical arrows are all equal: this can be visualised as a 'book' whose pages are glued along a spine. The pages can satisfy certain compatibility relations.
Morphisms Regular : relates the spine and the pages of two 'books' together. of tomes Loose: only relates the spines.

Oeuvre
An ordered collection of tomes related via loose morphisms; the theme is the common object towards which the spines of the books go to.

Narrative of degree δ
An oeuvre that is equipped with sub-diagrams of its tomes, called the events, and choices of lifts for these sub-diagrams, called the viewpoints These lifts only 'commute' from the k-th book to the (k + δ)-th book. 5. 1. Numbered Categories and Compatibility. In the sequel, the term numbered category will denominate any pair (C, κ) where C is a category and κ is a limit ordinal. A small category T will be said to be compatible with (C, κ) if (1) the category C admits colimits over T and (2) the inequality |T| ≤ κ holds. By extension, a functor i : T → A will be said to be compatible with a numbered category (C, κ) if its domain T is compatible with (C, κ).

Lifting Systems.
Let us now define in formal terms what will later be seen as a set of generating cofibrations for our small object argument. Let (C, κ) be an numbered category.
A lifting system in (C, κ) is a set J of objects of Cat/C 2 that are compatible with (C, κ) as functors. 5. 3. Right Lifting Property. Let (C, κ) be an numbered category and J be a lifting system in (C, κ). For every functor ϕ : T → C 2 in J, the image of an object s in T via ϕ will usually be denoted by ϕ(s) : A(s) → B(s). A morphism f : X → Y in C will be said to have the right lifting property with respect to the system J if for any functor i : T → S in J, the morphism f : X → Y has the rlp with respect to the arrow col T ϕ : In the sequel, the class of morphisms of C that have the right lifting property with respect to a lifting system J will be denoted by rlp(J).
Example 5. 1. If J is a set of functors of the form 1 → C 2 picking out some objects of C 2 , then the preceding right lifting property corresponds to the usual one. 5. 4. Tomes. Let C be a category. A tome in C is a triple consisting of a morhism h : X → Y in C, a small category S on which C admits all colimits and a functor ϕ : S → C 2 /h. According to Remark 2.9 applied to the arrow category C 2 , a way of seeing a tome in C is in the form of a cocone (u, v) : ∂ϕ ⇒ ∆ S (h) in C 2 over the functor ∂ϕ : S → C 2 . Because C has all colimits over S, the earlier cocone provides an arrow col S ∂ϕ ⇒ h in C 2 after applying the adjunction property of col S ∆ S on it. This latest arrow will be referred to as the content of (S, ϕ, h). Note that for any functor i : T → S, we may pre-compose the universal shifting induced by i (see section 2.5) with the content of (h, S, ϕ) as follows.
The resulting arrow col i ∂ϕ ⇒ h will later play a central role and be referred to as the content of (f, S, ϕ) along i : T → S. 5. 5. Morphisms of Tomes. Let C be a category. A loose morphism of tomes from T 0 := (h 0 , S 0 , ϕ 0 ) to T 1 := (h 1 , S 1 , ϕ 1 ) is given by a morphism (x, y) : h 0 ⇒ h 1 in C 2 . A regular morphism of tomes T 0 ⇒ T 1 is given by a morphism (x, y) : h 0 ⇒ h 1 in C 2 and a functor σ : S 0 → S 1 making the next right diagram commute.
The arrow symbol associated with loose morphisms will be denoted as T 0 ⇒ T 1 . The category whose objects are tomes in C and whose arrows are regular (resp. loose) morphisms of tomes will be denoted by Tome(C) (resp. Ltom(C)). For a fixed object Q in C, the wide subcategory of Ltom(C) that is restricted to the loose morphisms (x, y) : T 0 ⇒ T 1 whose components y : Y 0 → Y 1 are identities on Q will be denoted by Ltom(Q, C). 5. 6. Oeuvres and Narratives. Let (C, κ) be a numbered category and Q be an object in Convention 5. 2. In the sequel, the image of an inequality k < l in κ + 1 via an oeuvre O will be denoted by ( For convenience, when l is successor of k in κ + 1, the notations χ l k will be shortened to χ k . For every object k in κ + 1, the morphism h k will be denoted as an arrow G k → Q while the image of the composite functor ∂ϕ k : S k → C 2 at an object s in S k will be denoted as ∂ϕ k (s) : For every finite ordinal δ ∈ ω, a narrative of theme Q and degree δ in (C, κ) is an oeuvre (1) (events) for every ordinal k ∈ κ, a set J k , called the set of events at rank k, consisting of objects of Cat/S k that are compatible with (C, κ) as functors; (2) (viewpoint) for every functor i : T → S k in the set J k , a lift for the commutative square (living in C) resulting from the pre-composition of the content of (h k , S k , ϕ k ) along i : T → S k with the arrow χ k+δ k : h k ⇒ h k+δ ; the square is therefore of the form col T ∂ϕ k ⇒ h k+δ in C 2 . The lift will later be referred to as the viewpoint at rank k along i. Remark 5. 3. It follows from Convention 5.2 that the viewpoint at rank k along i mentioned in item (2) must be of the form col i B k → G k+δ .
Convention 5. 4. The functor κ + 1 → C induced by the sequence of arrows χ l k : G k → G l for every inequality k < l in κ + 1 will be denoted by G and called the context functor.
Observe that any oeuvre and, a fortiori, any narrative as defined above provides a factorisation in C as given below. This factorisation is that used for our small object argument.
Also, notice that the set of events J k induces an obvious lifting system {∂ϕ k • i | i ∈ J k }, which will be denoted by E k (O). 5. 7. Small Object Argument. Let (C, κ) be a numbered category, Q be an object in C and O : κ + 1 → Ltom(C) be a narrative of theme Q and degree δ. A lifting system J in (C, κ) will be said to agree with the narrative O if for every ordinal k ∈ κ and functor ϕ : T → C 2 in J admiting a lift ψ : T → C 2 /h k of ϕ along ∂ (see left diagram below), there exists a functor i : T → S k in J k whose composite with ϕ k gives the lift ψ (see right diagram below).
Proof. The goal of the proof is to show that the morphism h κ : G κ → Q is in rlp(J). To do so, let ϕ : T → C 2 be a functor in J and consider any arrow (x, y) : col T ϕ ⇒ h κ . The proposition will be proven if the commutative square encoded by this arrow admits a lift. By assumption, the functor G : κ + 1 → C is uniformly (dom • ϕ)-convergent in C. It follows from Remark 3.19, taken from the viewpoint of Remark 3.15, and the fact that κ is limit 6 that there exist an ordinal k ∈ κ and an arrow (x , y) : col T ϕ ⇒ h k factorising (x, y) : col T ϕ ⇒ h κ as follows.
Note that an application of the universal property of the adjunction col T ∆ T on the leftmost arrow of Equation (5.3) provides an arrow in C T as follows (where the leftmost arrow, given below, is the unit of col T ∆ T ).
According to Remark 2.9, Arrow (5.4) induces a functor ψ : T → C 2 /h k , which makes the leftmost diagram of Equation (5.2) commute. Because the lifting system J agrees with the narrative O, there must exist a functor i : T → S k making the right diagram of Equation (5.2) commute. This means, after re-applying the adjunction col T ∆ T , that Equation (5.3) is in fact of the following form, where the leftmost arrow is precisely the content of the tome O k along i : T → S k .
It follows from the viewpoint axiom (see Section 5.6) satisfied by O that the Composite col i ∂ϕ k ⇒ h k ⇒ h k+δ admits a lift. This implies that the whole composite (5.5) admits a lift, which, a fortiori, implies that the arrow (x, y) : col T ϕ ⇒ h κ admits a lift. 5. 8. Strict Narratives. Let (C, κ) be a numbered category and Q be an object in C. For any narrative O : κ + 1 → Ltom(C) of theme Q, recall that the set of events J k gives a collection of functors that induces a cocone under the category S k (see Section 5.6). A narrative O : κ + 1 → Ltom(C) of theme Q and degree δ will be said to be strict in C if (1) for every ordinal k ∈ κ, the cocone induced by the elements of J k is universal in Cat.
(2) it is equipped with a morphism π k : col S k B k → G k+δ factorising the content of O k into a pushout as follows; Recall that if κ is limit, then for every ordinal k ∈ κ, the successor k + δ is also in κ for every δ ∈ ω (3) for every functor i : T → S k in J k , the viewpoint π i k : col i B k → N k along i is equal to the pre-composition of π k with the universal shifting along i as follows; Proposition 5. 6.
) (see end of Section 5.6) for every k ∈ κ, then it has the rlp with respect to the arrow χ κ 0 : G 0 → G κ (see Diagram (5.1)). Proof. Let f : X → Y be a morphism that has the rlp with respect to the lifting system E k (O) for every k ∈ κ. For any k ∈ κ, this means that it has the rlp with respect to the following arrow in C, for every functor i : It directly follows that f has the rlp with respect to the coproduct of these arrows over the set J k (seen as a discrete category), which may be identified to the arrow col S k (∂ϕ k ) up to isomorphism as shown below.
It follows from classical facts that, since f has the rlp with respect to col S k ∂ϕ k , it has the rlp with respect to any of its pushouts, and hence with respect to χ k : G k → G k+δ for any k ∈ κ. It finally follows from Proposition 2.4 and the fact that the context functor G : κ + 1 → C is sequential that f has the rlp with respect to the arrow χ κ 0 : G 0 → G κ in C. 5. 9. Morphisms of Oeuvres. Let (C, κ) be a numbered category. For every pair of oeuvres O : κ+1 → Ltom(C) and O : κ+1 → Ltom(C), of respective themes Q and Q , a morphism of oeuvres from O to O consists, for every ordinal k ∈ κ, of a regular morphism of tomes such that the underlying loose morphisms (x k , y k ) : in the functor category Ltom(C) κ+1 (see Remark 5.7). The category whose objects are oeuvres for the numbered category (C, κ) and whose arrows are morphisms of oeuvres will be denoted by Oeuv(C, κ).
Remark 5. 7. The previous definition implies that all the arrows y k are equal to the same morphism y : Q → Q for every k ∈ κ+1. In addition, it forces the equality χ k •x k = x k+1 •χ k to hold in C for every k ∈ κ.

Constructors and Their Tomes
This section introduces the notion of constructor that allows one to associate systems of premodels with tomes. Constructors contain all the necessary information that permits the 'elimination of quotients'. We will see that their definition already brings out what is meant to be analytic (or structural) and what is meant to be quotiented out. Even if they appear to comprise many components, the main goal of the items defined in Sections 6.2 and 6.4 is to be able to define two sums whose forms look like the following type. The hom-sets Hom(e 0 (ϑ), ) -which are defined in Section 6.4-are meant to ensure a certain functoriality (i.e., they are the monomials for a certain type of species [33]) while the hom-sets Hom(e 1 (ϑ), Φ(ϑ, s))-which are defined in Section 6.2-are meant to contain the 'squares' that will enable us to perform our small object argument. In the sequel, I shall therefore try to give evoking names to the different parameters used to define these sums. In particular, one sum is to encode the structural data of our elimination of quotients while the other one is to encode the quotient acting on this data. To make the reader more confident with the items of Sections 6.2 and 6.4, here is a preluding summary of the different notations used therein.
The following conventions are meant to ease the combinatorial description of a constructor and its associated tomes, which will be defined in Section 6.6.
Convention 6.1 (Vertebrae). The diskad of a vertebra v := γ 2 , γ 1 · β will be denoted by disk(v) and seen as an arrow γ 2 ⇒ βδ 1 in C 2 . The other arrow γ 1 ⇒ βδ 2 in C 2 , which is induced by the 'dual' vertebra v rv := γ 1 , γ 2 · β, will be denoted by disk(v rv ) and called the codiskad of v. Finally, the stem β and seed γ 2 of v will be referred to by the notations stem(v) and seed(v), respectively.

Convention 6.2 (Domains and codomains). Let
A and C be two categories and F : A → C 2 be a functor. In order to avoid too many notations in our reasonings, the image F (X) of an object X of A in the arrow category C 2 will be denoted as F (X) : F • (X) → F • (X). This implies that every morphism f : X → Y in A gives a commutative diagram as follows.
Similarly, for every functor H : A → C 2×2 , we will denote by H • : A → C 2 and H • : A → C 2 the 'source' and 'target' arrows of the squares involved in the image of H. Example 6. 3. For every vertebra v in C as displayed in Equation (4.1), the arrow disk(v) • is equal to seed(v). Thus, when the reader reads α • ( ) in Section 6.4, where α is a functor I → C 2×2 mapping any element in I to the diskad of a certain vertebra in C, they should think of the seed of the so-called vertebra. Convention 6.4 (Closedness). Let A, B and C be three categories. The image of any functor of the form G : A × B → C will later be denoted as F a (b) for any pair of objects (a, b) in A × B -instead of the usual notation F (a, b). Convention 6.5 (Families). Let C be a category. In the sequel, we will denote by I the obvious functor Set × C → Fam(C) mapping a pair (S, X) to the functor ∆ S (X) : S → C. Also, mainly for convenience, the images of any object F : S → C in Fam(C) at some s ∈ S will be denoted by F s . This means that the equation I S (X) s = X holds for every s ∈ S. Convention 6.6 (Families of arrows). Convention 6.2 will be extended to Fam(C 2 ) in the obvious way: for every functor Φ : A → Fam(C 2 ), we shall denote by Φ • and Φ • the obvious functors A → Fam(C 2 ) mapping any object X ∈ A to the families s → Φ(X) • s and s → Φ(X) • s , respectively.
Convention 6.7. Later on, I shall often identify a set with a discrete category and identify many functions with functors. The reason for this is that we shall pre-compose these functions with functors going from discrete categories to non-trivial categories, which, for their parts, should really be seen as functors. This convention should thus ease the back and forth between set theory and category theory.

Preconstructors.
This section introduces the concept of preconstructor. This notion tries to capture what it takes to specify the data of a localisation. For instance, in Modern Algebra, localising a ring (R, +, ·) requires one to specify: the underlying set that one wants to act on, which is here the set R; the subset S ⊆ R by which one wants to localise the ring; the operation that one wants to inverse, which is here given by the S-indexed family of group morphisms e s : R → R defined by the mappings x → x · s; the type of inversion one wants to see happening on the maps e s . Regarding this last item, the inversion would, for instance, be expressed in terms of a bijection for the type of localisation used in Classical Algebraic Geometry, but it would be expressed in terms of a quasi-isomorphism in the category of unbounded chain complexes in Derived Algebraic Geometry.
To pass from the earlier description to the formalism of preconstructors, one can try to describe what a preconstructor would be for the previous list of items, so that we could make the following associations (also, see the structure below): the data ρ would specify the object R while the data λ would give the subset S; the data Φ, Υ and Ψ would enumerate the maps e s with theirs domains R and codomains R (which would be required to be independent of the indices in S); and the data α and ω would specify the type of inversion one wants to see happening. We now give a formal definition.
Let The previous string diagrams amount to saying that the following equations hold in the functor 'category' [B, Fam(C)] for every θ ∈ I; c) two functors α : I → C 2×2 and ω : I → C 2 , called the analysor and the quotientor, such that the image α(θ) encodes the diskad of a vertebra of stem ω(θ) for every θ ∈ I; As mentioned in the preamble of Section 6, a preconstructor contains all the information that is necessary to define the parametrising 'squares' on which we will run the smallobject-argument algorithm. These so-called parameters will be presented either as families (see Definition 6.8) or as formal sums (see Definition 6.10) -both presentations being useful. Γ Q ( ){ } : B 2 × I → Fam(Set) whose images are determined, for every arrow f : X → Y in B and object θ ∈ I, by the following mappings (or families) over λ θ (X).
Remark 6.9 (Concept of vertebra). The relationship between the analytic family and the quotient family is established in item c) via the concept of vertebra. At this stage, this should suggest to the reader that the notion of vertebra subtly encompass both the idea of quotient-or coherence-via its stem and the idea of cellular structure-or ana-lysis-via its diskad.
Preconstructor of a System of Premodels. Let (K, rou, P, V) be a system of Rpremodels over a small category D in a category C. The goal of this section is to associate any such system with a preconstructor of type D[P, C]. In this respect, define the set I to be the following leftmost disjoint sum.
Remark 6.11 (Encoding). Any element θ in I may be presented as a pair (c 0 , v) where c 0 is a cone in K and v is a vertebra in V c0 .
By keeping the notational convention suggested by Remark 6.11, one defines the data of the preconstructor for the system of premodels (K, rou, P, V) as follows: (1) the regulator is given by the mapping ρ : θ → rou(c 0 ); (2) the localisor is given by the evaluation λ : (θ, (P, S, e)) → S(c 0 ); (3) the analysor is given by the mapping α : θ → disk(v); (4) the quotientor is given by the mapping ω : θ → stem(v); and because both equations G K c0 (P, S, e) • s = P (rou(c 0 )) and G K c0 (P, S, e) • s = limRP in(c 0 ) hold for every s ∈ S(c 0 ), one may define the functor Φ : I × P → Fam(C 2 ) as the obvious functor satisfying the mapping (θ, (P, S, e)) → G K c0 (P, S, e) on objects, so that the two associated functors Υ and Ψ are defined as follows. where: c 0 is a cone in K; v is a vertebra in V c0 ; s is an element in S(c 0 ) and c is a commutative square in C 2 of the form given below, on the left, for the notation θ := (c 0 , v), which may also be seen as the right commutative cube in C when viewed from the bottom-left corner.
Similarly, the image of the quotient functor Γ K Q (f )[θ] contains the tuples (c 0 , v, s , s) where: c 0 is a cone in K; v is a vertebra in V c0 ; s is an element in S(c 0 ) and s is an arrow stem(v) ⇒ Ψ θ (f ) in C 2 for the notation θ := (c 0 , v). 6.4. Constructors. This section introduces the concept of constructor. In comparison to the informal introduction of Section 6.2, a constructor should be seen as a structure giving all the data that we need to describe the localisation of the ring R by a subset S in terms of freely-added tuples and relations acting on these.
Specifically, one usually constructs the localisation S −1 R by freely adding tuples of the form (x, s), for every x ∈ R and s ∈ S, to the set R. These tuples are often denoted as quotients x/s. Because S has not been supposed to be a multiplicative set, one would also need to specify tuples of the form (x, s 1 , s 2 , s 3 , . . . , s n ) for every x ∈ R and s i ∈ S where 1 ≤ i ≤ n. The equivalence relations defined on the pairs (x, s) are quite well-known: two pairs (x, s) and (x , s ) are equivalent if there exists u ∈ R for which the following relation holds.
u · (xs − x s) = 0 In the case of the elements of the form (x, s 1 , s 2 , s 3 , . . . , s n ), it is less obvious how this should be done. A constructor can help us with this as it contains all the required structure for this type of general description without involving the need of focusing on the encoding.
In terms of the notations given below, in the definition of constructor, the data would specify the set of elements that are to be paired with elements in S; the data χ would specify the set of elements that are to be subject to relations of the form given earlier; the data ι and δ, which are used for coherence purposes, would be identities; the data µ and ν would specify the types of quotients one would like to see happening: they provide the seeds and the stems of the vertebrae given by the data α coming from the preconstructor structure; the maps denoted by ϑ would map every element x ∈ R to x · s (for the analytic links) and every pair (x, x ) where x := (x/s ) · s to a pair (x · s, x · s) (for the quotient links); and the data  would specify how the set R injects into the localisation S −1 R. With respect to the definition given below, all of these data would be associated with the canonical ring morphism f : R → 1. (2) two functors ι : J A → I and δ : J Q → I called the analytic and quotient indicators; (3) a functor µ : J A → C 2 called the transitive analysor and, for every ϑ ∈ J A , a function ϑ , called the analytic link, of the following form; a functor ν : J Q → C 2 called the transitive quotientor and, for every ϑ ∈ J Q , a function ϑ , called the quotient link, of the following form; ϑ : C 2 (ωδ(ϑ), Ψ δ(ϑ) (f )) → C 2 (ν(ϑ), Υ χ(ϑ) (f )) (5) a functor  : I → J A , called the analytic section, satisfying the equalities ι •  = id I , •  = ρ and µ •  = α • so that the analytic link (θ) is an identity for every θ ∈ I; For such a constructor, we define, for every object f ∈ B 2 , an analytic functor Γ A (f ) : D → Set and a quotient functor Γ Q (f ) : D → Set whose images Γ A (f )(d) and Γ Q (f )(d) are given by the following formulae, respectively. 6. 5. Constructor of a System of Premodels. Let (K, rou, P, V) be a system of Rpremodels over a small category D in a category C. The goal of this section is to associate any such system with a constructor of type D[P, C]. We shall, of course, use the preconstructor structure defined in Section 6. 3. To define the supplementary structure, let us now define the following set (where Obj(Es(c 0 )) denotes the set of objects of the elementary shape of c 0 ∈ K) and let us associate every arrow (f, a) : (X, S, e) ⇒ (Y, S , e ) in P with two sets J A and J Q as follows where the set J(d) is defined for every d ∈ D as the following sum, in which c denotes a tuple of the form (c 1 , . . . , c n ) in K n and S(c) stands for the products of sets S(c 1 ) × · · · × S(c n ).
The initial section  : I → J A is taken to be the canonical monomorphism. respectively, where c 0 ∈ K, v ∈ V c0 , z ∈ Obj(Es(c 0 )) and, obviously, n ≥ 1.
Now, if one denotes by θ, θ , ϑ A and ϑ Q any tuple of I, I , J A and J Q as displayed in Remark 6.13, one defines the mappings , χ, ι, δ, µ and ν associated with the constructor structure of (K, rou, P, V) as follows: Finally, one produces a constructor of type D[P, C] by defining the analytic link ϑ A as an identity map when ϑ A ∈ I, and, otherwise, as a compositional iteration of the form ( 6.2) l cn,sn,tn • l cn−1,sn−1,tn−1 • · · · • l c1,s1,t1 ( ) where the triples (c 1 , s 1 , t 1 ), . . . , (c n , s n , t n ) are made out of the obvious components of ϑ A and the functor l ci,si,ti maps any commutative square as given below, on the left, to the commutative trapezoid given on the right, where ε denotes the counit of the adjunction L R and the component t i is, here, seen as an arrow of the form ou For its part, the quotient link ϑ Q , which is defined for every ϑ Q ∈ J Q , is given by a first application of the functor l z that maps any commutative square as given below, on the left, to the commutative trapezoid given on the right, where ς z is the universal projection of the adjunction ∆ lim at z, and, in the case where ϑ Q is not in I , followed by successive iterations of the functor l ci,si,ti over the triples (c i , s i , t i ) made out of the obvious components of ϑ Q (see Formula 6.2). It is easy to check that the initial section  : I → J A satisfies the axioms of item 5) of Section 6. 4. The constructor associated with (K, rou, P, V) will later be referred to as Γ K .
Remark 6.14. In the case where the associated maps P rou(c) → RP ou(c) of our premodels are identities, the functors R and rou are trivial and the associated sets S are all equal to a fixed one, the set J(d) can be set empty for every d ∈ {rou(c 0 ), in(c 0 )(z)} and c 0 ∈ K so that  can be defined as an identity. In this case, the validity of our results still holds for Examples 4.10 and 4.11, but not for Examples 4.12 and 4.13, which require J(d) to be as above. See Remark 6.28 and the proof of Theorem 6.29 for more insight. Φ θ (f ) s is an arrow in C 2×2 as displayed in Equation (6.1) for the notation θ := (c 0 , v).
Similarly, the image of the quotient functor Γ K Q (f )(d) contains the tuples (c 0 , v, z, t, s , s) and the tuples (c 0 , v, n, c, s, t, z, t, s , s) where: c 0 is a cone in K; v is a vertebra in V c0 ; n is a natural number; c, s and t are the tuples defined in Remark 6.13 and used to define the quotient link; z is an object of Es(c 0 ); t is an arrow in D of the form in(c 0 )(z) → d for the first type of tuple and an arrow ou(c n ) → d otherwise; s is an element in S(c 0 ) and s is an arrow stem(v) ⇒ Ψ θ (f ) in C 2 for the notation θ := (c 0 , v). Remark 6.16 (Encoding). It is not hard to see from Remark 6.15 that any type of tuple in Γ K A (f )(d) may be written as a tuple of the form (ϑ, t, s , c) where the encoding of the parameter ϑ may vary. Similarly, it follows from Remark 6.15 that any tuple in Γ K Q (f )(d) may be written as a tuple (ϑ, t, s , s) where the encoding of the parameter ϑ may vary. 6. 6. Tomes of a Constructor. Let Γ denote a constructor of type D[B, C] as defined in Section 6. 4. This section shows that Γ may be associated with a variety of canonical tomes, each of them being used for specific purposes. The first one, called the operadic tome, is meant to be used in the small object argument (see Section 5) and is constructed out of the preconstructor structure of Γ as follows: For every object θ ∈ I, arrow f : X → Y in B and s ∈ λ θ (X), it is given by the functor s θ : and is defined on each term of Γ A (f )(d)-which denoted as t ϑ,s below-as follows.
Explicitly, the functor maps any tuple (ϑ, t, s, c) in Γ K A (f )(d) (see Remark 6.16) to the composite arrow given, below, by Equation (6.3) in C 2 .
A third tome, called the quotient tome, is given by a functor ϕ Q : Γ Q (f )(d) → C 2 /Υ d (f ) and is defined on each term of Γ Q (f )(d) -which denoted as t ϑ,s below -as follows.
Explicitly, the quotient tome ϕ Q maps any tuple (ϑ, t, s, s) in Γ K Q (f )(d) (see Remark 6.16) to the composite arrow given, below, by Equation (6.4) in C 2 .
The proofs of the following propositions follow from the previous definitions.
Proposition 6.17. The operadic tome s θ : This amounts to saying that the mapping f → (Φ θ (f ) s , Γ A (f )[θ], s θ ) induces a functor T op θ,s : B 2 → Tome(C). Proposition 6. 18. The analytic tome ϕ A : This amounts to saying that the mapping d → (Υ d (f ), Γ A (f )(d), ϕ A ) induces a functor T an f : D → Tome(C). Proposition 6. 19. The quotient tome ϕ Q : This amounts to saying that the mapping d → (Υ d (f ), Γ Q (f )(d), ϕ Q ) induces a functor T qu f : D → Tome(C). 6. 7. Quotiented Arrows. Let Γ denote a constructor of type D[B, C] as defined in Section 6. 4. This section defines the concept of 'quotient' whose essential idea is to restrict the quotient family of Γ to certain parametrising 'squares' only. In this respect, a Γ -quotient for a morphism f : X → Y in B consists of a collection of discrete categories, as given below, on the left, as well as a collection of functors as given on the right . We may associate any such Γ -quotient q with a functor q[ ] : I → Set defined as follows for every θ ∈ I.
This functor will be called the species of q. In much the same fashion as the quotient species of Γ was used to define its quotient functor, we use the species of q to define a third functor q : D → Set given by the following formula.
This functor will be referred to as the quotienting functor of q.
Convention 6. 21. The natural transformation of Proposition 6.20 may be composed with the quotient tome ϕ Q of Γ to give a natural transformation ϕ Q : q ⇒ C 2 Υ(f ). Because this arrow lives in the functor category Set D , it may be factorised into an epimonomorphism followed by a monomorphism as follows (this is an image factorisation).
For every object d ∈ D, the imageq(d) will be thought of as the set q(d), but quotiented by the obvious binary relation. In any case, the elements ofq(d) and q(d) will be denoted as tuples (ϑ, t, s, υ, s) where t is an arrow of the form χ(ϑ) → d; s is an element in λ δ(ϑ) (X); υ is an element in E s (δ(ϑ)) and s is an element in q s θ {υ} → Γ Q (f ){θ} s . Remark 6.22 (In preparation for Theorem 6.29). Let f : X → Y be a morphism in B as above. For every object d ∈ D, denote by D(J Q , d) the following sum of sets, which is defined with respect to the structure of f provided by the constructor Γ .
The definition of Γ -quotient for f : X → Y implies that any function of the form h : D(J Q , d) → D(J Q , d ) that maps a pair (ϑ, t) in D(J Q , d) to a pair (ϑ , t ) in D(J Q , d ) so that the equality δ(ϑ) = δ(ϑ ) is satisfied lifts to a function h : q(d) → q(d ) mapping any tuple (ϑ, t, s, υ, s) in q(d) to the tuple (h(ϑ, t), s, υ, s) in q(d ).
Example 6.23 (In preparation for Theorem 6.29). In the case of a constructor Γ K associated with a system of R-premodels (K, T, P, V) over a small category D in a category C, the disjoint sum D(J Q , d) associated with a morphism (f, a) : (X, S, e) ⇒ (Y, S , e ) in P contains two types of tuples, which are of the form (c 0 , v, z, t) and (c 0 , v, n, c, s, t, z, t) with respect to the same notations given in Remark 6. 15. For every c ∈ K and s ∈ S(c), if one takes r to be rou(c) and d 0 to be ou(c), then it is possible to define a function h c,s : D(J Q , r) → D(J Q , d 0 ) with the following mapping rules, where cc stands for (c 1 , . . . , c n , c), ss stands for (s 0 , . . . , s n−1 , s) and tt stands for (t 0 , . . . , t n−1 , t). h c,s : → (c 0 , v, 1, c, s, t, z, id d0 ) (c 0 , v, n, c, s, t, z, t) → (c 0 , v, n + 1, cc, ss, tt, z, id d0 ) Because the following equations hold, it follows from Remark 6.22 that the function h c,s : D(J Q , r) → D(J Q , d 0 ) extends to a function q(r) → q(d 0 ).
In fact, the function h c,s : q(r) → q(d 0 ) also restricts to a function h c,s :q(r) →q(d 0 ). To see this, take two tuples x * := (ϑ * , t * , s * , υ * , s * ) and x † := (ϑ † , t † , s † , υ † , s † ) in q(r) that are equivalent inq(r), that is to say that have the same image under ϕ Q (see below, according to Formula (6.4)). Υ It follows that their images via h c,s :q(r) →q(d 0 ) are also equivalent inq(d 0 ). This comes from the fact that the previous equation gives rise to the following one, after some obvious compositional operations on it (see the definitions for l c,s,t * and l c,s,t † in Section 6.5). Υ However, this last equation also amounts to saying that the images of h c,s (x * ) and h c,s (x † ) via ϕ Q are the same, and thus shows that h c,s restricts to a functionq(r) →q(d 0 ).

Definition 6.24 (Quotiented arrows)
. From now on, we shall speak of a Γ -quotiented arrow in B to refer to any arrow f : X → Y in B that is equipped with a Γ -quotient q for f .
A Γ -quotiented arrow as defined above will be denoted either as a pair (f, q) or as a paired arrow (f, q) : X → Y . A morphism of Γ -quotiented arrows, denoted as an arrow (f, q) ⇒ (g, p), will be understood as a morphism f ⇒ g in B 2 . The category of Γ -quotiented arrows and their morphisms will be denoted by Γ B 2 . 6. 8. Merolytic Functors and Their Tomes. Let Γ denote a constructor of type D[B, C] as defined in Section 6.4 where C has coproducts. For every Γ -quotiented arrow (f, q) : X → Y , define the merolytic functor of (f, q) as the coproduct of functors given below.
Then, define the merolytic tome of (f, q) : X → Y as the coproduct ϕ q : Γ f,q (d) → C 2 /Υ d (f ) of the following cospan whose right leg is given by the rightmost arrow of Equation ( 6.5). 25. For every (f, q) ∈ Γ B 2 , the merolytic tome ϕ q : This amounts to saying that the mapping rule d → (Υ d (f ), Γ f,q (d), ϕ q ) induces a functor T f,q : D → Tome(C).
Proof. Follows from Propositions 6.18, 6.19 and 6.20. Because the tome T f,q (d) is functorial in d ∈ D, so is its content col∂ϕ q ⇒ Υ d (f ) (see Section 5.4). In other words, the content gives us a commutative diagram in C D as follows.
The previous diagram will be referred to as the functorial content of T f,q . 6. 9. Effectiveness of Quotiented Arrows. The goal of this section is to introduce what logicians could see as a concept of definability. The concept of effectiveness will allow us to designate those arrows that can be equipped with well-defined pushout factorisations in the category associated with a constructor. We prepare the notion of effectiveness by introducing the (almost-trivial) concept of realisability. Let Γ denote a constructor of type D[B, C] as defined in Section 6.4 where C has coproducts. A Γ -quotiented arrow (f, q) : X → Y in B will be said to be Γ -realised if one may form a componentwise pushout square inside the functorial content of its merolytic tome as shown below.
The functor d → [f, q](d) will then be called the Γ -realisation of (f, q) while the pair of arrows (p q f , h q f ) will be referred to as the Γ -prefactorisation of (f, q). Definition 6.27 (Effectiveness). Let Γ denote a constructor of type D[B, C] as defined in Section 6.4. A Γ -quotiented arrow (f, q) : X → Y in B will be said to be effective if it is Γ -realised and its Γ -prefactorisation in C D lifts to a factorisation of f : X → Y in B, as shown in Equation (6.7), such that the arrow λ θ ({f } q ) : λ θ (X) → λ θ ([f /q]) is an identity for every θ ∈ I.
The leftmost factorisation of Equation (6.7) will be called the Γ -factorisation of (f, q).
Remark 6. 28. Let S 0 be a given set and Γ K be the constructor of a system of R-premodels (K, T, P, V) over a small category D in a category C where every object (X, S, e) in P is such that S is equal to S 0 and e is made of identities only. In this case, the underlying functor Υ : P → C D is fully faithful and it follows that if C has pushouts, then every Γ K -quotiented arrow in P is effective. This means that the theorem given below becomes trivial, which explains why the set J(d) mentioned in Remark 6.14 may be set empty since it is not really needed anywhere else in the paper except for Theorem 6.29 (and Theorem 8.13, which is a copy of it). See Example 6.32 in the case where J(d) is defined as in Section 6.5.
Theorem 6. 29. Let (K, rou, P, V) be a system of R-premodels over a small category D in a category C. If C has pushouts and the inclusion P → Pr C (K, rou, R) is an identity, then every Γ K -quotiented arrow in P is effective.
Proof. For convenience, the symbol Γ K will be shortened to Γ . Since C has pushouts, every Γ -quotiented arrow is Γ -realised by definition. Let (f, a, q) : (X, S, e) ⇒ (Y, S , e ) be an Γ -quotiented arrow in B. We are going to prove that the Γ -realisation of (f, q) has an R-premodel structure of the form ([f, q], S, e q ) and that this structure lifts the γprefactorisation of f : X ⇒ Y in C D to another one in P. In this respect, fix c ∈ K and s ∈ S(c) and denote rou(c) and ou(c) by r and d 0 , respectively. For simplicity, we will denote by e c,s (f ) the obvious morphism f (r) ⇒ Rf (d 0 ) in C 2 whose components are given by the pair of arrows e c,s and e c,ac(s) in C.
We are now going to show that the following diagram commutes.
On the set Γ A (r), the calculation on a tuple x = (c 0 , v, t, s , c) goes as follows.
On the setq(r), the calculation for x = (c 0 , v, z, t, υ, s , s) goes as follows.
On the set Γ A (r), the calculation on a tuple x = (c 0 , v, n, c, s, t, t, s , c) goes as follows.
On the setq(r), the calculation for x = (c 0 , v, n, c, s, t, z, t, υ, s , s) goes as follows.
Now, the equation ξ c,s ϕ q r = ϕ q d0 ζ c,s tells us that the content of the tome T f,q (d 0 ) along ζ c,s is equal to the content of T f,q (r) after applying the functor ξ c,s on it. More specifically, the equation means that the respective composites of Equations ( 6.9) and (6.10) are equal. (6.9) col ζc,s ∂ϕ q Lec,s(f ) If one denotes by η the unit of the adjunction L R, the definition of adjunction implies that the function R( ) • η is inverse of ε • L( ). Since the content col∂ϕ q d0 ⇒ f (d 0 ) appearing in Equation ( 6.9) may be factorised as in Diagram (6.6) on d 0 , an application of the inverse function of ε • L( ) on the arrow represented by Equations ( 6.9) and (6.10) provides the following commutative diagram, where Equation (6.9) provides the inside while Equation (6.10) provides the outside.
Now, because the top left corner of the previous diagram corresponds to the top left corner of the commutative square defining the Γ -realisation of (f, a, q) when evaluated at r, it follows that there exists a natural transformation e q c,s : The previous diagram provides a morphism (p q f , id) : (X, S, e) ⇒ ([f, q], S, e q ) in the category of R-premodels Pr C (K, rou, R). The universality of [f, q] also provides a morphism (h q f , a) : ([f, q], S, e q ) ⇒ (Y, S e ) in Pr C (K, rou, R). These two morphisms obviously define a factorisation of the morphism (f, a) : (X, S, e) ⇒ (Y, S , e ) in Pr C (K, rou, R). Finally, since the second component of the morphism (p q f , id) is the identity on S, its image via the functor λ θ is an identity for every θ ∈ I (see Section 6.5). In other words, the arrow λ θ (X, S, e) → λ θ ([f, q], S, e q ) mentioned in Definition 6.27 is indeed an identity. Definition 6.30 (Fibered). A system of R-premodels (K, rou, P, V) over a small category D in a category C will be said to be fibered if the category C has pushouts and the Γ -factorisation of any Γ -quotiented arrow (obtained in Theorem 6.29) lifts to P. Example 6. 31. By Theorem 6.29, any system of R-premodels (K, rou, P, V) where C has pushouts and P is identified with the category Pr C (K, T, R) is fibered.
Example 6. 32. In the proof of Theorem 6.29, note that if the objects (X, S, e) and (Y, S , e ) are such that the associated arrows e c,s and e c,ac(s) are identities, then so is e q c,s . This implies that any system of R-premodels (K, rou, P, V) where C has pushouts and P may be identified with the functor category C D is fibered (e.g., Examples 4.19-4.26) Example 6. 33. In the proof of Theorem 6.29, note that if the objects (X, S, e) and (Y, S , e ) are such that the images of S and S are equal to 1, then so is the Γ K -realisation ([f, q], S, e q ). This implies that the system of Ω-premodels given in Example 4.27 is fibered.
Remark 6.34. A system of R-premodels (K, rou, P, V) is not always fibered (e.g., Example 4.28), which is often due to a too strong restriction of the premodels via the inclusion P → Pr C (K, rou, R). However, Theorem 6.29 shows that if P is too strong, we might want to stay in Pr C (K, rou, R) to process most of our calculations. The idea would then be that it is possible to go back to P at the very end of a transfinite calculation. Example 6. 35. This example discusses the form that the Γ -realisation takes when considering categories of models for a limit sketch. Let (D, Q) be a limit sketch seen as a croquis.
Consider the system of premodels defined in Example 4.19 for the category Set D . Recall that the vertebrae associated with any cone c ∈ Q were of the following form.
It follows from the definition of the transitive analysor and quotientor that, for any Γ Kquotiented arrow (f, q) : X → 1, the Γ K -realisation of (f, q) evaluated at an object d ∈ D is defined over the following types of span.
The contribution of the left span to the construction of the Γ K -realisation [f, q](d) is to add an element to X(d) while the contribution of the right span to the construction of the Γ K -realisation [f, q](d) is to quotient a pair of elements in X(d). After unravelling the indices that parametrise the two types of span, we may deduce that the colimit [f, q](d) is of the following form, where Γ K A,0 (f )(d) and After further unravelling the parameterisation of the rightmost summand, we may show that the colimit [f /q](d) may be expressed as follows, where R is a binary relation on X in Set D .
Concretely, the set C 2×2 (disk(v 0 ), Φ ι(ϑ A ) (f )) is nothing but the set X[c 0 ] := limXin(c 0 ) with respect to the notations of ϑ A given in Remark 6.13 while the object (ϑ A ) is given by ou(c n ) for the same notations.
Recall that, according to Remarks 6.14 and 6.28, the set J A could in fact be given by the set I itself in the present situation (i.e. premodels for a sketch). In this case, the expression of Equation ( 6.11) turns out to be as follows. 10. Rectification of Effective Quotiented Arrows. Let Γ denote a constructor of type D[B, C] as defined in Section 6.9, with the usual notations, and (f, q) : X → Y be an effective Γ -quotiented arrow in B. Usually, effectiveness does not mean that the quotiented arrow is as we would like it to be. It is in fact necessary to rectify its defaults via a second quotient. The goal of this section is to define the 'rectification' of (f, q), which is nothing but a Γ -quotient u for the arrow To do so, let us define, for every element θ ∈ I and s ∈ λ θ (X) = λ θ ( f q ), the associated functor of the following form.
First, define the discrete category E s (θ) to be the set Γ A (f ){θ} s . By definition, an element υ ∈ E s (θ) may be identified with an element c ∈ C 2×2 (α(θ), Φ θ (f ) s ), which may be sent to the arrow The arrow encoded by c • may be identified with an element in the image of the analytic tome of ϕ A : Γ A (f )(ρ(θ)) → C 2 /Υ ρ(θ) (f ) as follows (see Formula ( 6.3) and the assumption of the initial section  : This therefore defines a function i : E s (θ) → Γ A (f )(ρ(θ)) mapping any element c ∈ E s (θ) to the tuple ((θ), id ρ(θ) , s, c) whose image via the merolytic tome ϕ q : This being said, denote by r the element ρ(θ) and, for every c ∈ E s (θ), denote by i c the function 1 → Γ A (f )(r) that picks out the element i(c). From the point of view of these notations, we have showed that the image of the composite ϕ q • i c corresponds to the commutative square c • . However, this also means that the content of the merolytic tome of (f, q) along i c : 1 → E s (θ) is equal to the commutative Square ( 6.12) in C as illustrated below.
Because the left arrow col∂ϕ q r ⇒ Υ r (f ) (i.e., the content) may be factorised as shown in Diagram ( 6.6), it follows that the commutative square encoding c • factorises as shown below, on the left.
The diagram displayed above, on the right, is for its part the image of the Γ -factorisation of (f, q) in B via the functor Φ θ : B → C. The definitions of the diagrams involved in Equation ( 6.13) imply that the commutative square c ∈ C 2×2 (α(θ), Φ θ (f ) s ) factorises as follows, where the image α(θ) is replaced with the diskad of a vertebra γ 2 , γ 1 · β for which β = ω(θ) by definition.
Notice that the previous commutative cube provides the following left commutative square.
By using the structure of the vertebra γ 2 , γ 1 · β, we may form a pushout S inside so that we obtain a canonical arrow w : S → Φ • θ ([f /q]) s making the preceding right diagram commute. It is not hard to deduce from the universality of this pushout that both arrows s are solutions for a same universal problem over S (Diagram (6.14) might come in handy to visualise this fact). In particular, this means that the following diagram must commute.
Because β corresponds to the image ω(θ), we have defined a functor u θ,s { } : E s (θ) → Set mapping a commutative cube c ∈ E s (θ) to the subset of C 2 (ω(θ), Φ • θ ( f q ) s ) consisting of Diagram (6.15) only. Thus, the images of u θ,s { } are sets (or singletons) included in Γ Q ( f q ){θ} s so that the collection of functors given below, denoted by u, defines a Γquotient for the arrow Definition 6.36 (Rectification). The Γ -rectification of the Γ -quotiented arrow (f, q) : X → Y is the Γ -quotiented arrow ( f q , u), which will sometimes be denoted by Rec(f, q).
Later on, the diagram obtained in Equation ( 6.15), which is entirely determined by the image of the Γ -rectification of (f, q) above a cube c ∈ E s (θ) at the parameters θ ∈ I and s ∈ λ θ ([f /q]), will be referred to as the obstruction square of (f, q) for c at (θ, s). Definition 6.37 (Ideal). A Γ -quotiented arrow (f, q) : X → Y will be said to be ideal if it is effective, its Γ -rectification ( f q , u) is effective and for every θ ∈ I, s ∈ λ θ ([f /q]) and c ∈ E s (θ), there exists an arrow π 1 (θ, s) : D → Φ • θ ([ f q /u]) s factorising the obstruction square of (f, q) for c at (θ, s) as follows.
Remark 6.38 (Structure of narrative of degree 2). Consider an ideal Γ -quotiented arrow (f, q) : X → Y and a commutative cube c in C 2×2 (α(θ), Φ θ (f ) s ). According to the previous discussion, this cube c may be factorised as in Diagram ( 6.14). Merging this factorisation of c with: (1) the factorisation of the obstruction square of (f, q) for c at (θ, s) on its front face and (2) the Γ -factorisation of the Γ -rectification of (f, q) on its back face leads to the following factorisation of c (where the top front corner has been forgotten and r = ρ(θ)).
This means that the composite arrow given in Equation ( 6.17), whose the leftmost arrow is given by the content of the operadic tome s θ : This last fact will later imply that we may construct a narrative of degree 2 out of the operadic tome.
Remark 6.39 (About π 0 ). This section discusses the encoding of the arrow that we have denoted π 0 . We shall use the same notations as that introduced at the beginning of the section. Recall that we defined the element i c = ((θ), id (θ) , s, c), which we used to shift the merolytic tome of (f, q) and obtain the leftmost diagram of Equation (6.13). Therefore, we have the following formula if we use the notation of Diagram (6.6).
If we now denote i c = ((θ), t, s, c) for some arrow t : (θ) → d, the functionality of π q f and the construction of the merolytic tome of (f, q) gives the following formula.
This formula will later come in handy in the proof of Theorem 8. 21.
Theorem 6.40. Let (K, rou, P, V) be a system of R-premodels over a small category D in a category C. If C admits pushouts and the inclusion P → Pr C (K, rou, R) is an identity, then every Γ K -quotiented arrow is ideal. Proof. For convenience, the symbol Γ K will be shortened to Γ . The present proof uses the construction made in the proof of Theorem 6. 29. In particular, we shall use the notations defined thereof, such as h q f and p q f . Let (f, a, q) : (X, S, e) ⇒ (Y, S , e ) be an Γ -quotiented arrow in B. By Theorem 6.29, it is effective and so is its Γ -rectification (h q f , a, u) : ([f, q], S, e q ) ⇒ (Y, S , e ). There now remains to show the existence of an arrow π 1 (θ, s) : D → Φ • θ ([ f q /u]) s factorising the obstruction square of (f, q) for any cube c ∈ E s (θ) at any parameter θ ∈ I and s ∈ λ θ ([f /q]) (see Diagram (6.16)).
First, recall that, for every θ ∈ I, s ∈ λ θ ([f /q]) and cube c ∈ E s (θ), the obstruction square of (f, q) for c at (θ, s) is given by an arrow in C 2 of the following form.
By using the notations of Section 6.5 and the adjointness properties of R and lim z , the preceding righthand arrow may be turned into the following arrow in C 2 for every z ∈ Es(c 0 ).
Now, observe from the definitions of Section 6.5 that, for every θ = (c 0 , v) ∈ I and z ∈ Es(c 0 ), we may define an object θ z := (θ, z) in J Q , which precisely lands in the component I of J Q . From the notations of Section 6.5, the arrow given in Equation ( 6.18) may in fact be rewritten as follows 8 .
In order to avoid overloading the next diagrams, denote by d : Es(c 0 ) → D the functorial mapping z → χ(θ z ) and, for every s ∈ u θ,s {υ}, denote by i s,z the function 1 → Γ f q,u (d(z)) that picks out Tuple ( 6.19) inũ(d(z)) for every z ∈ Es(c 0 ). Now, to resume, the previous discussion showed that the image of the composite ϕ u d(z) • i s,z corresponds to the arrow ς z • ε • L(s). However, this is equivalent to saying that the content of the merolytic tome of ( f q , u) along i s,z : 1 → Γ f q ,u d(z) is equal to the arrow ς z • ε • L(s) as illustrated below.
Because the rightmost arrow col∂ϕ u d(z) ⇒ Υ d(z) ( f q ) may be factorised as shown in Diagram ( 6.6), it follows that the commutative square encoded by ς z • ε • L(s) factorises as follows. LS The idea is now to obtain a factorisation of the form given in Equation (6.16) by reconstructing the obstruction square s (from which the previous diagram is derived) without losing the factorisation.
First, note that, by definition of the quotient acting onũ (see Convention 6.21), the collection of arrows {i s,z } z∈Es(c0) is natural in z ∈ Es(c 0 ) since the following tuples have the same images via the functor ϕ Q for every arrow t : z → z in Es(c 0 ).
The functoriality of Diagram (6.6) over D and the naturality of i s,z : 1 → Γ f q,u (d(z)) in z ∈ Es(c 0 ) then implies that the earlier commutative diagram is natural over z ∈ Es(c 0 ). Forming the limit of that diagram over Es(c 0 ) and then applying the inverse of the function ε • L( ) (which is given by the function R( ) • η if η denotes the unit of L R) provides a factorisation of the original obstruction square s as follows.
Example 6.41. This example continues the discussion started in Example 6.35 (we shall use the same notations as those used thereof) in order to describe, in more details, the binary relation R(d) acting on X(d) (see Formula (6.11)) in the case where f is taken to be the canonical map ! X : X → 1. Recall that the quotient X(d)/R(d) was meant to simplify the following expression.
) Also, recall that, by definition, the binary relations contained in Γ K A,1 (! X )(d) (see Remark 6.15 for the encoding of Γ K A ) are those pairs (x, y) : 1 + 1 → X(d) that may be related to commutative diagrams as follows.
Precisely: The above diagram says that two elements x, y ∈ X(d) will be identified if there exist a cone c ∈ K, a morphism t : ou(c) → d and two elements x and y in Xou(c) such that the identities X(t)(x ) = x and X(t)(y ) = y hold and the elements x and y have the same image via the canonical map Xou(c) → limXin(c).
On the other hand, the binary relations contained inq(d) were given as part of our assumptions. However, in the sequel, the idea will be to defineq(d) either as the empty binary relation or as we defined the setũ(d) in Section 6. 10. In the latter case, in order to make sense ofq(d), we need to suppose that the image X(d) takes the form given below for some functor Y : D → Set and binary relation R : D → Set.
The quotient Y (d)/R (d), which will later be shortened as Y (d), is supposed to identify pairs of elements coming from a previous Γ K -quotientp(d). In this case, the pairs contained in the relationq(d) = Rec(! X ,p) are those pairs (x, y) : 1 + 1 → X(d) that are the top parts of commutative diagrams of the form displayed below, where the leftmost commutative square is one of those obstruction squares constructed in Section 6. 10.
Precisely: After unravelling the details of the construction of the corresponding obstruction square, the above diagram says that two elements will be identified if there exist a cone c ∈ K, say encoded by a natural transformation ρ : ∆ou(c) ⇒ in(c), an element z ∈ Es(c), a morphism t : in(c)(z) → d and two elements x and y living in X(in(c)(z)) of the form such that the following relations hold.
We can clearly see that the role of two binary relationsq(d) and Γ K A,1 (! X )(d) is to turn the canonical arrow X(ou(c)) → X[c] into a surjection and an injection, respectively. Example 6.42 (Comparison with Kelly's construction). Let us compare the quotients acting on the pushout object [! X /q], as described in Examples 6.41 and 6.35 (where ! X denotes the canonical arrow X → 1), with those acting on the pushout object of Kelly's construction [4]. Recall that, for each cone c ∈ K, the latter is given by a well-pointed endofunctor id ⇒ P c in Set D . More specifically, if we take c to be a cone of the usual the form ρ : ∆ Es(c) (ou(c)) ⇒ in(c) in D, then for every functor X : D → Set, the object P c (X) can be computed in Set D as the pushout object of the following span [4, diag. (10.1), p. 31], whose components are further detailed below, while the natural transformation id ⇒ P c is the bottom arrow of the resulting pushout square.
X( ) For every object d ∈ D, we can decompose the previous span in four parts as follows: (1) The arrow given below, part of the vertical leg, maps every pair (t, x), where t is an arrow ou(c) → d and x ∈ X(ou(c)), to the element X(t)(x) in X(d); D(ou(c), d) × X(ou(c)) → X(d) (2) The arrow given below, also part of the vertical leg, maps every pair (t, (x z ) z∈Es(c) ), where t is an arrow in(c)(z) → d in the colimit col z D(in(c)(z), d) and (x z ) z∈Es(c) is a tuple in X[c] = lim X • in(c), to the element X(t)(x z ) in X(d); 3) The arrow given below, part of the horizontal leg, is induced by the canonical arrow X(ou(c)) → X[c] and maps every pair (t, x) to the pair (t, x ), where x is the tuple (ρ z (x)) z∈Es(c) in the limit object X[c]; (4) The arrow given below, also part of the horizontal leg, is induced by the canonical arrow col z D(in(c)(z), d) → D(ou(c), d) and maps every pair (t, x), where t is an arrow in(c)(z) → d for some object z ∈ Es(c) and x ∈ X[c], to the pair (t • ρ z , x); It takes a few lines of calculations to see that the pushout P c (X)(d) of the previous span evaluated at d can be described as a quotiented sum of the form (6.20) where: and y ∈ X(d), such that there exist a ∈ X(ou(c)) and an arrow t : ou(c) → d for which the following identities hold.
x = (t, (ρ z (a)) z∈Es(c) ) y = X(t)(a) and y ∈ X(d), such that there exist z ∈ Es(c) and an arrow t : in(c)(z) → d for which the following identities hold.
We can see that the definition of the relation R 1 exactly matches that of the relatioñ q(d) = Rec(! X ,p) given in Example 6.41. On the other hand, we can check that for every relation (x 1 , x 2 ) ∈ Γ K A,1 (! X )(d), as described in Example 6.41, there is an (obvious) element y for which both relations x 1 R 0 y and x 2 R 0 y are satisfied.
However, a relation of the form xR 0 y cannot be retrieved from the union of the relations Γ K A,1 (! X )(d) andq(d), given in Example 6.41. It can only be retrieved if one allows a use of these relations up to quotients. Indeed, the reader can check that the identification of the second line, below, cannot be made unless the one given in the fist line has already occured.
first identify (ρ z , (ρ z (a)) z∈Es(c) )q(d) X(ρ z (a)) which then allows us to identify (t, (ρ z (a)) z∈Es(c) ) Γ K A,1 (! X )(d) X(t)(a) As mentioned in Section 1.4, Kelly's construction is pursued by pushing out all the maps X ⇒ P c (X) to give a natural transformation X ⇒ P (X) where P (X) identifies each component X appearing in the expression of the objects P c (X) for every c ∈ K. We therefore obtain an expression as follows, for very object d ∈ D.
This expression should be compared with the (similar) expression of the Γ K -realisation [! X /∅] obtained in Example 6.41, whose sum over K is, here, quotient-free.
Because the relations contained in Γ K A,1 (! X )(d) can be written as a zigzag of relations in R 0 , we can construct an obvious arrow from [! X /∅] to P (X) matching all the components D(ou(c), d) × X[c] together (here, the symbol ∼ stands for the obvious relation).
In fact, our earlier discussion showed that, if we denote X 1 = [! X /∅] and X n+1 = [! Xn /u n ] where u 1 = Rec(! X , ∅) and u n+1 = Rec(! Xn , u n ), then we can continue this process iteratively, by matching the components of the sum over K, so that we have arrows as follows.
[! X1 /u 1 ] ⇒ P (P (X)) One can check that all these arrows are compatible, in an obvious way, with the arrows X n ⇒ [! Xn /u n ] and P n (X) ⇒ P n+1 (X). However, one of our previous remarks on the fact that R 0 can only be retrieved from the relations Γ K A,1 (! X )(d) andq(d) up to quotients indicates us that if there exists a pair of dashed arrows making the following diagram commute then the front arrow must factorise through the following canonical arrow (see the reason below).
Indeed, otherwise we could derive a contradiction from the elements of the form which must be identified with the elements x z in P c P n (X)( ) via the relation R 1 , but must be left free in the expression of [! Xn /u n ]( ). The empty case X n [c] = ∅ obviously leads to the same conclusion.
If we now look at Formula ( 6.20), this factorisation means that that all the elements in the component D(ou(c), d) × P n (X)[c] of P c P n (X)(d) must be identified with elements in the other component P n (X)(d). From the point of view of the relation R 0 at d = ou(c) where t is taken to be the identity on ou(c), this means that the canonical arrow P n (X)(ou(c)) → P n (X)[c] must be a surjection.
Finally, observe that, when n > 0, the arrow P n (X)(ou(c)) → P n (X)[c] is also an injection because the images of P n (X) are quotiented by the relations R 0 and hence the relation Γ K A,1 (! P n−1 (X) )(d), which precisely characterises its injectiveness (see Example 6.41). In other words, the canonical arrow P n (X)(ou(c)) → P n (X)[c] is a bijection, which makes the object P n (X) a model for (D, {c}).

Combinatorial Categories and Their Oeuvres
The notion of combinatorial category encompasses all the assumptions that are necessary to the application of the small object argument in the case of systems of premodels. , where C has coproducts, together with a limit ordinal κ such that the category B admits colimits over every limit ordinal λ ∈ κ + 1 when seen as a preorder category. Such a structure will be denoted as a pair (Γ , κ) where Γ will be equipped with its usual notational conventions. 7.2. Factorisable Morphisms. Let (Γ , κ) be a numbered constructor of type D[B, C]. A morphism f : X → Y in B will be said to be (Γ , κ)-factorisable if it is equipped with a sequence (f n , u n ) n∈κ+1 of ideal Γ -quotiented arrows in B satisfying the following conditions: initial case: f 0 = f ; successor cases: Rec(f n , u n ) = (f n+1 , u n+1 ); limit cases: for any (infinite) limit ordinal λ ∈ κ + 1, the arrow f λ is the colimit col n∈λ f n in B of the following diagram over the category λ. By induction, we may show that the arrows χ λ n (f ) and {f n } un define a sequential functor G(f ) : κ + 1 → B with the following mapping rules.
The functor G(f ) : κ+1 → B turns the mapping n → f n into an obvious functor G (f ) : κ + 1 → B 2 , which also lifts to the category Γ B 2 via the mapping n → (f n , u n ) (see Diagram (7.1)).
Theorem 7.3. Let κ denote a limit ordinal and (K, T, P, V) be a system of R-premodels over a small category D in a category C. If C is cocomplete, R preserves colimits over every limit ordinal λ ∈ κ + 1 and the inclusion P → Pr C (K, T, R) is an identity, then every morphism in P may be equipped with the structure of a (Γ K , κ)-factorisable morphism.
Proof. First, the assumption that C is cocomplete and R preserves colimits over every limit ordinal λ ∈ κ+1 implies that Pr C (K, T, R) admits colimits over every limit ordinal λ ∈ κ+1.
We are now going to show that every morphism (f, a) : (X, S, e) ⇒ (Y, S , e ) of the category Pr C (K, T, R) may be equipped with the structure of a (Γ K , κ)-factorisable morphism by induction. Let us define the sequence of Γ K -quotiented arrow (f n , a n , u n ) n∈κ+1 as follows: For the initial case, take (f 0 , a 0 ) to be the morphism (f, a) : (X, S, e) ⇒ (Y, S , e ) and u 0 to be given by the collection of empty functors {∅ : 1 → Set} θ∈I,s∈λ θ (X) ; By Theorem 6.40, the Γ K -quotiented arrow (f n , a n , u n ) is ideal and we can take the next Γ K -quotiented arrow (f n+1 , a n+1 , u n+1 ) to be Rec(f n , a n , u n ); For any (infinite) limit ordinal λ ∈ κ + 1, the arrow (f λ , a λ ) is given by the colimit col n∈λ (f n , a n ) in Pr C (K, T, R) of Diagram (7.2) over the category λ while u λ is given by the collection of empty functors {∅ : 1 → Set} θ∈I,s∈λ θ (X) Corollary 7.4. Let κ denote a limit ordinal and (K, T, P, V) be a fibered system of Rpremodels over a small category D in a category C. If C is cocomplete and R preserves colimits over every limit ordinal λ ∈ κ + 1, then every morphism in P may be equipped with the structure of a (Γ K , κ)-factorisable morphism. Proof. Follows from fiberedness and Theorem 7.3.
Example 7.5 (Systems of premodels). Let κ denote a limit ordinal and (K, T, P, V) be a system of R-premodels over a small category D in a category C where P may be identified with the category of R-premodels C D → Pr C (K, T, R) -hence R is an identity. It follows from Example 6.32 and Corollary 7.4 that the morphisms of P are all (Γ K , κ)-factorisable. (1) for every n ∈ κ + 1, the set of events J n d contains all the functors 1 → Γ fn,un (d); (2) for every n ∈ κ + 1 and functor i : 1 → Γ fn,un (d) in J n d , the viewpoint associated with the arrow is given by the Γ -realisation of (f n , u n ) (see Diagram (6.6)) that may be inserted in the content col∂ϕ un d ⇒ Υ d (f n ), so that we obtain a lift π 0 for the previous composite that makes the following diagram commute.
Note that the object [f n−1 /u n−1 ] must stand for X when n = 0.
By definition (see Section 5.8), the previous narrative is strict.
For every object θ ∈ I and s ∈ λ θ (X), the mapping n → T op θ,s (f n ) induces an oeuvre O θ,s [f ] : κ + 1 → Ltom(C 2 ) of theme Φ θ (Y ) that is equipped with the structure of a narrative of degree 2.
Proof. The fact that the mapping n → T op θ,s (f n ) induces an oeuvre follows from Proposition 6.17 and Remark 7.2. One thus obtains an oeuvre O f (θ) : κ + 1 → Ltom(C 2 ) of theme Φ θ (Y ). The narrative structure is defined as follows: (1) for every n ∈ κ + 1, the set of events J n θ,s contains all the functors 1 → Γ A (f n ){θ} s ; (2) for every n ∈ κ + 1 and functor i : 1 → Γ A (f n ){θ} s in J θ n , the viewpoint is given by the pair (π 0 , π 1 (θ, s)) defined in Section 6.10 if one replaces the functor i c with i and the Γ -quotiented arrow (f, q) with (f n , u n ). As noticed in Remark 6.38, the version of Diagram (6.14) for these parameters provides the wanted lift.
For every limit ordinal κ, a category B will be said to be κ-combinatorial in a category C if it is equipped with a numbered constructor (Γ , κ) of type D[B, C] such that (1) every morphism in B is (Γ , κ)-factorisable; (2) for every object f in B 2 and object θ in I, the functor Φ θ,s • G(f ) : κ + 1 → C 2 , which is the context functor of the oeuvre O θ,s [f ], is Cosd(Γ )-convergent.
Remark 7. 10. In practice, it is easy to prove that for every morphism f : X → Y in B and object d in D, the context functor is Gen(Γ )-convergent. This is generally due to the fact that the context functor Υ d • G(f ) is sequential and the vertebrae {v α (θ)} θ∈I are rather 'small'.
Example 7. 11. The following discussion continues the discussion began in Examples 6.35 and 6.41. In this respect, let (D, K) be a limit sketch seen as a croquis and consider the system of premodels defined in Example 4.19 for the category Set D . If one numbers the constructor Γ K with an ordinal κ ≥ ω, then for every morphism f : X → Y in Set D and object d in D, the context functor is U -convergent for any finite set U . This comes from the fact that any sequential functor of the form κ + 1 → Set where κ ≥ ω is convergent with respect to finite sets. Now, in the case of the constructor Γ K , the set Gen(Γ K ) is made of the finite sets ∅, 1 and 2 = 1 + 1, so the functor Υ d • G(f ) : κ + 1 → Set is Gen(Γ K )-convergent.
Example 7. 12. Let CW denote the wide subcategory of Top restricted to inclusions A → B defining relative CW-complex structures (see [17]). It is well-known that any sequential functor of the form κ + 1 → CW, where κ ≥ ω, is convergent with respect to compact topological spaces (see Appendix of [17]). Since topological spheres and discs are compact, it follows that the functor Υ d •G(f ) : κ+1 → Top associated with the constructors of the systems of Ω-premodels defined in Examples 4.25 and 4.26 is Gen(Γ K )-convergent when K is taken to be equal to Seg(∆ op ) and Cseg(∆ op ), respectively.
Example 7.13 (Systems of premodels). Let (K, T, P, V) be a fibered system of R-premodels over a small category D in a category C. In addition, suppose that C is cocomplete and R preserves colimits over every limit ordinal λ ∈ κ+1. Corollary 7.4 shows that every morphism in P is (Γ K , κ)-factorisable for any limit ordinal κ. Let us prove that the category P becomes κ-combinatorial if -κ is a well-chosen limit ordinal; -the statement of Remark 7.10 holds.
As specified by Remark 7.10, for every morphism f : (X, S, e) ⇒ (Y, S , e ) in P and object d in D, the context functor G(f )(d) : κ + 1 → C of the oeuvre O[f ](d) is generally Gen(Γ K )-convergent. Recall that this functor lifts to a functor landing in P as follows.
Let c denote a cone of the form t : Let also g denote the functor (C D ) 2 → C 2 defined in Example 3.7 where the cone 'r' used thereof is replaced with the natural transformation t : ∆ A (d) ⇒ d 1 . By definition, the following equations hold for every ordinal n ∈ κ + 1, cone c ∈ K, vertebra v ∈ V and element s ∈ S(c).
if n is succ. g(col n∈λ e un c,s ) if n is limit. In the case where the inequalities 2 ≤ κ and |A| ≤ κ hold, Example 3.18 says that the composite of the functor G(f ) : κ + 1 → P with the functor Φ (c,v),s : P → C 2 is Cosd(V)convergent. In other words, this shows that if κ is greater than or equal to the cardinality |(K, T )| and 2, then the context functor of the oeuvre O f (θ) is Cosd(V)-convergent.
Definition 7.14 (Lifting system). Let B be a combinatorial category as defined above. For every morphism f : X → Y in B, every θ ∈ K and s ∈ λ θ (X), denote by J soa θ,s the lifting system consisting of the functors 1 → (C 2 ) 2 picking out the codiskad disk(v α (θ) rv ) : Proposition 7. 15. For every morphism f : X → Y in B, every θ ∈ K and s ∈ λ θ (X), the lifting system J soa θ,s agrees with the narrative O θ,s [f ] : κ + 1 → Ltom(C 2 ) in the numbered category (C, κ).
By definition, the functor ψ picks out an element in C 2×2 (α(θ), Φ θ (f n ) s ) which is therefore an element of Γ A (f n ){θ} s . This means that we found a functor i : 1 → Γ A (f n ){θ} s in the set of events J n θ,s whose composite with s θ gives the lift ψ as follows.
This exactly shows the statement of the proposition.
Theorem 7. 16. Let κ be a limit ordinal and B be a κ-combinatorial category as defined above. Every morphism f : X → Y may be factorised into two arrows such that, for every θ ∈ I and s ∈ λ θ (X), the arrow Φ θ (f κ ) s : Φ θ (G(f )(κ)) s → Φ θ (Y ) s in C 2 has the rlp with respect to the codiskad of v α (θ) and, for every object d in D, the ) has the llp with respect to every morphism in rlp κ (E n (O[f ](d))) (see end of Section 5.6) for every n ∈ κ + 1.
Proof. The factorisation is given by the image of the arrow 0 → κ via the functor G (f ) : Let κ be a limit ordinal. A category C will be said to be trivially κ-combinatorial over a set G ⊆ Obj(C 2 ) if it is κ-combinatorial when equipped with the numbered constructor (Γ 1 , κ) associated with the obvious category of id C -premodels C 1 → Pr(1, id, id) whose set of vertebrae consists of the following degenerate vertebrae for every arrow δ ∈ G.
Corollary 7.17 (Quillen's small object argument). Let C be a trivially κ-combinatorial category over a set of arrows G in C. Every morphism f : X → Y in C may be factorised into two arrows where the arrow f κ is in the class rlp(G) and the arrow χ κ 0 (f ) is in the class llp(rlp(G)). Proof. Theorem 7. 16 implies that every morphism f : X → Y in C may be factorised into two arrows χ κ 0 (f ) : X → G(f )(κ) and f κ : where the arrow f κ is in the class rlp(G) and the arrow χ κ 0 (f ) has the llp with respect to every morphism in rlp(E n (O[f ](d))) for every n ∈ κ+1. However, because of the triviality of our data, it follows that the equality rlp(E n (O[f ](d))) = rlp(G) holds for every n ∈ κ + 1.
Remark 7.18. For every system of R-premodels (K, T, P, V) where: C is cocomplete; R : C → C preserves colimits over every limit ordinal λ ∈ κ + 1 and P → Pr C (K, T, R) is combinatorial (see Example 7.13), Theorem 7.16 provides every arrow ! : (X, S, e) ⇒ 1 in P with a factorisation (X, S, e) χ(f ) + 3 G(X, S, e) ! + 3 1 where G(X, S, e) is an R-model and the arrow χ(f ) satisfies nice lifting properties. In the case of a category of premodels for a sketch, Example 6.35 shows that the 'localisation' (X, S, e) ⇒ G(X, S, e) admits a presentation as given in Theorem 1. 4. There now remains to show that the arrow (X, S, e) ⇒ G(X, S, e) is universal. This is the goal of the next and last section.

Universal Property
This section discusses the universal properties of the factorisations provided by Theorem 7. 16. To do so, we shall require our constructor to be 'normal' (see Section 8.1). An existential resut is given in Theorem 8.21 while a universal one is given in Theorem 8. 18 (2) for every θ ∈ I and s ∈ λ θ (X), the functor Φ θ,s : B → C 2 preserves 1.
(3) the mappings f → J A and f → J Q (see Section 6.4) induce functors from B 2 to Set that extend the mappings f → , f → χ, f → ι and f → δ into obvious functors from B 2 to Cat/D, Cat/D, Cat/I and Cat/I, respectively; Example 8. 1. The constructor associated with a system of R-premodels (K, T, P, V) over a small category D in a category C that possesses a terminal object 1 is normal. Item (1) is straightforward and item (2) follows from the fact that R : C → C preserves any terminal object by adjointness. The functoriality of the sets J A and J Q is induced by the action of a morphism (f, a) : (X, S, e) ⇒ (X, S , e ) on the sets S and S (see Section 6.5) while the functoriality of the functors , χ, ι and δ is straightforward .

8.2.
Quasi-Models and Models. Let Γ be a normal constructor of type D[B, C]. For every object X in B, there is an obvious morphism in B 2 given by the following commutative square.
Applying the functor Γ A ( )( ) : B 2 ×D → Set (see Remark 8.2) on this morphism provides the following natural transformation in Set over D, which is natural in X ∈ B.
An object X in B will be said to be a quasi-model of Γ if for every object d in D, the function ℘ A (X, d) : Such a structure will be denoted as a pair (X, σ). Remark 8. 3. It follows from the definition of a surjection that an object X ∈ B is a quasimodel of Γ if and only if for every object d ∈ D and tuple (ϑ, t, s, c) in Γ A (! X )(d), every commutative cube c ∈ Γ A (! X ){ι(ϑ)} s admits a lift as follows (where θ stands for ι(ϑ)).
Remark 8.4. The difference between a quasi-model and a model is that the lifts are chosen. Specifically, any model (X, σ) of Γ is determined by a collection of lifts (h, h) chosen for every object ϑ ∈ J A , element s ∈ λ ι(ϑ) (X) as follows.
Indeed, if one denotes the previous commutative cube by c and its upper commutative part seen as a degenerate commutative cube by lift(ϑ, s, c), the section σ( ) : Γ A (! X )( ) → Γ A (id X )( ) is determined by the following mapping rules.
(ϑ, t, s, c) → (ϑ, t, s, lift(ϑ, s, c)) The fact that such a mapping defines a natural section of the natural surjection ℘ A (X, ) : Conversely, if a natural section ℘ A (X, ) was not of this form, we could find two arrows t : (θ) → d and t : (θ) → d such that the elements (ϑ, t, s, c) and (ϑ, t , s, c) would be sent to elements of the following form via the section σ.
(ϑ, t, s, lift(ϑ, t, s, c)) (ϑ, t , s, c, lift(ϑ, t , s, c)) However, the naturality of σ( ) above the arrows t and t also implies the equalities which show that the section has to be of the form previously given in the remark.
Example 8.5 (System of premodels). Let (K, T, P, V) be a system of R-premodels as in Example 8. 1. The R-models are exactly given by the quasi-models of Γ K . By Remark 8.4, it is always possible to turn a quasi-model X into a model (X, σ) by using the axiom of choice on the different possible lifts.
Let now A be an object in B. An A-quasi-model for the constructor Γ consists of a morphism f : A → X in B where X is equipped with the structure of a quasi-model X.
Similarly, an A-model for the constructor Γ consists of a morphism f : A → X in B where X is equipped with the structure of a model (X, σ). The latter structure will be denoted as a triple (f, X, σ).  → Set} θ∈I,s∈λ θ (A) whose component at the parameters θ ∈ I and s ∈ λ θ (A) is given by the following image factorisation for every υ ∈ E s (θ).
Then, we may define another collection of functors of the form The diagram given on the left of Remark 8.10 induces a commutative diagram in C D of the form given below. This diagram will be referred to as the functorial content of T mod f,u . 8. 5. Effectiveness of Quotiented Models. Let Γ be a normal constructor of type D[B, C] where C has coproducts and A be an object in B. A Γ -quotiented A-model f : (A, u) → (X, σ, ß) will be said to be Γ -realised if one may form a pushout square inside the functorial content of its merolytic tome as shown below.
By Remark 8.10, the pushout square may be supposed to be exactly the same as that defined for the Γ -realisation of (! A , u). In particular, the following result holds.
Theorem 8. 13. Let (K, rou, P, V) be a system of R-premodels over a small category D in a category C. If C has pushouts and the inclusion P → Pr C (K, rou, R) is an identity, then every Γ -quotiented relative model is effective.  29. In fact, the proof of the present theorem is very similar to that of Theorem 6.29, except that it uses Diagram (8.7) instead of Diagram (6.8) for every c ∈ K and s ∈ S(c). As in the proof of Theorem 6.29, the symbols r and d 0 stand for the objects rou(c) and ou(c) in D, respectively.
The proof that Diagram (8.7) commutes goes as in the proof of Theorem 6.29 by using Formula (8.4). Then, Diagram (8.7) may be used to show that the following diagram commutes.
e c,ac (s) The substantial information given by the previous diagram is the inner bottom commutative trapezoid, which shows that the lift ([! A , u], S, e u ) ⇒ (X, S , e ) exists; the desired factorisation is deduced by universality.
Definition 8.14 (Strongly Fibered). A system of R-premodels (K, rou, P, V) over a small category D in a category C will be said to be strongly fibered if it is fibered and, for every Γ -quotiented arrow (! A , u) : (A, S, e) → 1, the Γ -factorisation of any corresponding Γ -quotiented (A, S, e)-model (obtained in Theorem 6.29) lifts to P.
(θ, s, c) gives a commutative diagram as follows (where s = λ θ (g)(s)). (8.11) S Example 8. 19. The vertebrae of Examples 4.19 and 4.20 satisfy assumptions (i), (iii) and (iv) of Theorem 8. 18. Similarly, the id Set -models generated by these examples, which are quasi-models of the associated constructor by Remark 8.3, or, in fact, actuals models, by Remark 8.4 and the axiom of choice, satisfy condition (ii). Finally, it follows from Remark 6.14 that condition (v) can also be satisfied in the case of these examples. 8. 6. Factorisable Models. Let (Γ , κ) be a normal numbered constructor of type D[B, C] where C has coproducts, A be an object in B and (X, σ) be a model of Γ . An A-model f : A → (X, σ) will be said to be (Γ , κ)-factorisable if the morphism ! A : A → 1 is equipped with the structure of a (Γ , κ)-factorisable arrow, say (! An , u n ), together with a sequence {f n : (A n , u n ) → (X, σ, ß n )} n∈κ+1 of effective Γ -quotiented relative models satisfying the following conditions: initial case: f = f 0 ; successor cases: f n+1 is given by the arrow f n un : [! An /u n ] → X; limit cases: for any (infinite) limit ordinal λ ∈ κ + 1, the arrow f λ is the colimit col n∈λ f n in B of the following diagram over the category λ.
Later on, the arrow χ κ 0 (! A ) : A → G(! A )(κ) will be denoted as ρ A : A → G(A) and called the localisation of f : A → (X, σ). According to Theorem 7.16 and Remark 8.3, if the category B is equipped with the structure of a κ-combinatorial category for the constructor Γ , then the object G(A) must be a quasi-model of Γ .
Theorem 8.21 (Weak localisation). Let κ denote a limit ordinal and (K, rou, P, V) be a system of R-premodels over a small category D in a category C. If C is cocomplete, R preserves colimits over every limit ordinal λ ∈ κ + 1 and the inclusion P → Pr C (K, rou, R) is an identity, then every relative model of Γ K may be equipped with the structure of a (Γ K , κ)-factorisable relative model. Proof. This corollary is an obvious generalisation of Theorem 8.21 that takes advantage of the notion of strong fiberedness (see Definition 8.14). 8. 7. Elimination of Quotients. A normal numbered constructor (Γ , κ) of type D[B, C] will be said to eliminate quotients if the category B is a κ-combinatorial category for the constructor Γ and every canonical arrow A → 1 is equipped with the structure of a (Γ , κ)factorisable morphism such that every A-model f : A → (X, σ) is (Γ , κ)-factorisable for this structure.
Remark 8. 23. For every object A in B, all A-models f : A → (X, σ) are equipped with the same localisation ρ A : is a quasi-model (see Remark 8.20). The way in which this arrow has been defined from the data of Γ is the key of the so-called 'elimination of quotients'.   This means that these examples eliminate quotients and are equipped with a localisation of the form given in Theorem 8. 24. In particular, this localisation tends to organise the different sorts of data appearing in the diskads of the systems in the form of bundles-this was explicited in Examples 6.35 and 6.41 in the case of the models for a limit sketch.
Remark 8. 26. Under the conditions of Theorem 8.18, the factorisation of Theorem 8.24 may be shown to be unique by using an obvious transfinite induction. 9. Concusions 9.1. Conclusions for Motivation 1. In Section 1.3, one of our main goals was to provide a language that would allow us to show strict universal properties from weak definitions. In this paper, we address this question in the form of Theorem 8. 18. This theorem shows us what the main ingredients that are responsible for universal properties look like and most of them pertain to the sets of vertebrae associated with our systems of premodels (see conditions (i), (ii) and (iii)).
In fact, many sections and concepts were introduced in this paper because of these vertebrae. The need for each of these sections can be explained by the following storyline. At the centre of things is Section 4.3, which introduces the concept of system of premodels. This structure is a formal way to present the lifting problems associated with our vertebrae. To handle these lifting problems, we have to introduce the analytic and quotient species given in Definition 6. 10. However, because these species need some formal setting, the concept of constructor is introduced in Section 6.4, which a fortiori motivates the introduction of preconstructors in Section 6.2. Note that the main purpose of the latter is to allow the handling of the premodel structure (e.g., the maps e c,s : P rou(c) → RP ou(c) defined in Section 4.2) while the purpose of the former is to allow the handling of the vertebrae associated with systems of premodels. The way one handles the species is formalised via the tools of Section 5, in which is expressed our small object argument (Proposition 5.5). This section really allows us to see the big picture without introducing too much detail. On the other hand, from Section 6.6 to the end of Section 6, we give all the details of this big picture in the case of systems of premodels. We also use Section 7 to explain what it takes, in terms of required assumptions, to be able to apply the small object argument of Section 5. The need for Section 8 naturally presents itself if one is interested to know more about the universal properties satisfied by the models living in systems of premodels. As one is able to see there, this section heavily relies on the concept of species introduced in Section 6.4 and hence the concept of vertebra.
The fact that this last section relies so heavily on the vertebrae is not so surprising when one knows that vertebrae are meant to encode some sort of homotopical information and that, on the other hand, Homotopy Theory is all about coherence property. In fact, this idea of coherence-and universal property-coming from vertebrae is already discussed in my thesis [34] and this is exactly the spirit in which Theorem 8.18 should be regarded. In this respect, I will use the rest of this conclusion to explain why the formalism of systems of premodels is something that one might want to consider if one wants to solve higher coherence problem.
A way to put it would be to ask what happens if one starts changing the assumptions of Theorem 1.1 (see Section 1) in terms of homotopical properties. The notion of epimorphism used thereof could be replaced with a notion of epimorphism up to homotopy. For instance, the arrow β : S → D could be called a weak epimorphism if for every pair of arrows f, g : D → X for which the equation f • β = g • β holds, we can form the pushout S of β with itself (see below) so that a given arrow β : S → D factorises the universal arrow induced by f and g under S as follows.
A quick look at the beginning of the proof of Theorem 8.18, in which β should be viewed as the transitive quotientor ν(ϑ), shows that such a notion of weak epimorphism would imply that the universal solution of the reflection would be unique up to a homotopy relation as defined in Equation (9.1). However, this type of statement would only hold if the vertebra satisfies some nice compositional properties, and, more specifically, compositional properties of the type defined in [34]. In other words, our vertebra would need to satisfy axioms of the same type as those usually considered in the case of (co)limit sketches -the compositional properties would try to recapture the idea of composition of cells in (Higher) Category Theory.
Interestingly, these axioms would also mingle different vertebrae together. For instance, it is interesting to note that our current discussion has made us consider two vertebrae: one for which β is a stem (as usual) and one for which β is both a seed and a coseed, given in Equation (9.2). This pair of vertebrae can be arranged in the form of the following diagram.
Such a commutative diagram defines what is called a spine (of degree 1) in [34]. There, spines are shown to be essential in the understanding of higher coherence results of the type mentioned above and one can see that these structures arise very naturally once one starts talking about universal properties. The degree of a spine hides a dimensional nature and it is interesting to note that this dimensional aspect already arises among the examples discussed in [14,Section 4] when it is asked whether weak reflections can possess strict universal property such as functoriality and naturality.
In conclusion, the idea of universal property and coherence fits the language of systems of premodels nicely, so that these structures appear to be the right setting to address the question whether a class of algebraic objects defined via weak lifting properties can satisfy strict (or at least stricter than expected) universal properties-and an important part of the work to be done in this direction can already be perceived in [34]. 9.2. Conclusions for Motivation 2. In Section 1.4, our other main goal was to prove Theorem 1.4, along with Propositions 1.2 and 1. 3. These results were proven in different sections of the present paper. Before addressing the usefulness of these results, we briefly recapitulate their proof below.
Let (D, K) be a limit sketch, seen as a croquis, and consider the system of premodels defined in Example 4.19 for the inclusion Set D → Pr Set (K). First, Example 8.25 tells us that the reflector ρ A : A → G(A) associated with a premodel 9 A in Set D is given by Theorem 8. 24. Its strict universal property then follows from Remark 8.26, where one needs to look at Example 8.19 in order to use Theorem 8. 18. The functoriality of the reflection A → G(A) and the naturality of the reflector ρ A obviously follows from this (strict) universal property. Now, if one consider the transfinite construction of the reflector ρ A : A → G(A) given in Section 7.2, one may see that the transfinite sequence that gives rise to the reflector ρ A is of the desired form -for Proposition 1.2 by using Example 6.35; -for Proposition 1.3 by using Example 6.42; -for Theorem 1.4 by using Examples 6.35-6.42, for which one needs to realise that a sum of the form is the same thing as a left Kan extension E(X) of the form given in Equation (9.3), where K must regarded as a discrete category.

G G Set
The question that now remains to be answered is: what is the combinatorial presentation given by Theorem 1.4 useful for? Recall that, according to Theorem 1.4, the reflector associated with a presentation X in Set D is the transfinite composition of a sequence of arrows as follows where, for every i ≥ 0, the object E i+1 (X) is the left Kan extension of the functor where, here, the functor S i : D → Set denotes the sum E i (X) + B i (X).

G G Set
The restriction of the quotient map p i+1 (see Section 1.4) to the object E i+1 (X) gives us a way to organise the data of E i+1 (X) with respect to its fibres. Of course, this organisation is also present in Kelly's construction via the quotients acting on E i+1 (X) (see Example 6.42), but this organisation is also unlikely to be the one that one wants to consider if one decides to study the combinatorial properties of the models. In fact, while Kelly's construction forces us to consider an actual quotient of the object E i+1 (X), the elimination of quotients leaves the object E i+1 (X) free of quotients, so that one can now use any other type of relations on E i+1 (X) without being forced to deal with the relations of the localisation. Furthermore, the formalism of quotient maps (formalised in terms of quotiented arrows in Section 6.7) makes compatibility and distributivity questions between potential new relations and those forced by the localisation much easier to study.
For instance, one could want to study the colimits of the category of models for (D, K). Recall that colimits in this category are given by the images of the reflection G on the corresponding colimits in the category of premodels Set D , as shown below.
col F = G(col F ) In addition, recall that a colimit of the form col F in Set D can be seen as a quotiented sum.
x F (x) / ∼ The relations ∼ acting on the sum x F (x) usually generates the type of identifications that one wants to study. Specifically, one usually wants to understand how these propagate through the transfinite constructions building the models. However, their propagation is usually non-obvious and requires some more-or-less non-trivial case-by-case analysis, depending on how complicated the theory (D, K) is. This case-by-case analysis might not even depend on the quotients implied by the localisation and might instead depend on the properties of the objects F (x). In order to be efficient and clear, this case-by-case analysis needs to be processed in a quotient-free environment separated from the quotients generated by the localisation, but what is better than a quotient map whose domain is a quotient-free left Kan extension of the form given in Equation (9.3) to make such a separation? Interestingly, the construction of the quotient maps p i+1 has motivated the formalisation of the concept of quotient (in Section 6.7), so that our results open the door to the development of a new language to talk about quotients living in algebraic objects in general.

Appendix A.
Recall that the category of sets Set is complete and cocomplete. The limit lim D F of a functor F : D → Set for some small category D is given by the set (A.1) {(x d ) d∈Obj(D) | x d ∈ F (d) and for any t : d → d in D : F (t)(x d ) = x d } while the colimit col D F of a functor F : D → Set for some small category D is given by the quotient set {(d, x) | d ∈ Obj(D); x ∈ F (d)}/ ∼ where ∼ denotes the binary relation whose relations (d, x) ∼ (d , x ) are defined when there exists an object e and two arrow t : d → e and t : d → e in D such that the equation F (t)(x) = F (t )(x ) holds. Note that in the case where D is a preorder category κ for some ordinal κ, the binary relation ∼ is an equivalence relation.
Proof of Proposition 2.2. A proof may be found in [35,Corollaire 9.8]. For the sake of selfcontainedness, the proof is recalled in this appendix. Let F ( ) : κ × D → Set be a functor. An equivalence class for the equivalence relation ∼ will be denoted into brackets, i.e. [(k, x)]. The notation (x d ) F d∈Obj(D) will be used to mean that the collection (x d ) d∈Obj(D) is compatible with the action of the functor F in the appropriate way (see Equation (A.1)). By definition, the following equations hold.
The natural transformation col κ lim D ⇒ lim D col κ ( ) is given by the following mapping.
Let us prove its surjectiveness. Consider an element in lim D col κ F of the following form.
By definition of the compatibility with the action of F , for any arrow t : d → d in D, there exist arrows s d : k d → e t and s d : k d → e t in κ such that the next equation holds.
Since κ is a limit ordinal greater than or equal to |D|, we may define the following supremum in κ. .
In For every object d in D, the arrows g d • s d : k → e are equal in κ. The same is true for g d • s d : k → e . It follows that the equation Proof of Proposition 2. 3. We keep the convention set in the proof of Proposition 2.2. We only need to check that the diagram of the statement commutes. For any set X, the unit η X : X → lim D ∆ D (X) maps an element of x ∈ X to the constant collection (x) d∈Obj(D) . Similarly, for any functor X : κ → Set, the unit η X( ) : X( ) → lim D ∆ D (X( )) maps an element of x ∈ X(k) to the constant collection (x) d∈Obj(D) in lim D ∆ D (X(k)). The diagram of the statement is therefore encoded by the following mapping rules. [ In particular, this shows that the diagram commutes.