A result of Krasner in categorial form

In 1957 M.\ Krasner described a complete valued field $(K,v)$ via the projective limit of a system of certain structures, called hyperfields, associated to $(K,v)$. We put this result in purely category-theoretic terms by translating into a limit construction in certain slice categories of the category of valued hyperfields and their homomorphisms.


Introduction
If one considers the operations of classical algebraic structures (such as groups, rings, fields,. . . ) by looking at their graphs, then one sees that they satisfy two fundamental assumptions: they are left total and functional.In other words, these properties can be spelled out as operations being everywhere defined (i.e., the operation can be applied to any two elements to obtain at least one result) and single-valued (i.e., an application of the operation to any two elements yields at most one result).A hyperfield is a field-like structure where the latter property is relaxed for the additive operation.In the literature, such structures appear perhaps more than one would expect: hyperfields are of interest e.g. in tropical geometry [36,16,12], symmetrization [11,15,33], projective geometry [5], valuation theory [20,21,26,8] and ordered algebra [7,9,17].There are even reasons to believe that their theory generalises field theory in ways that can be used to tackle deep problems such as the description of F 1 , the "field of characteristic one" (cf.[35,5]).
More generally, since the pioneer papers [28,29,30] of F. Marty, structures with multivalued operations (also called hyperstructures) generalising classical singlevalued structures have been the object of several research projects.For example, modules with a multivalued operation and a scalar multiplication over Krasner hyperrings (the old brothers of hyperfields) have been studied e.g. in [32,3,2].In addition, hypergroups are mentioned in a journal of theoretic physics, in [13].
In this paper, we focus on valued hyperfields, generalising valued fields: the object of study of classical valuation theory.To any valued field pK, vq, Krasner's quotient construction associates a projective system H of valued hyperfields, indexed by the non-negative elements of the value group.The projective limit of the system H is a valued field isomorphic (as a valued field) to the valuation-theoretic completion of pK, vq.This result was proved by Krasner in [18] to achieve an approximation result for complete valued fields of positive characteristic by means of other fields of characteristic 0. His proof, makes heavy use of the ultrametric structure of the hyperfields in H induced by the canonical one of pK, vq.On the other hand, it can be noted that, while valuation maps generalise readily to the setting of hyperfields, a valuation on a hyperfield needs not to induce an ultrametric distance as in the singlevalued case.Krasner's broader aim led him to prefer the approach of declaring "valued" only those hyperfields which admit a valuation map inducing (in a prescribed way) an ultrametric on the underlying set.However, examples of hyperfields with interesting valuation maps which do not induce an ultrametric as required by Krasner are now known (see e.g.[20,Example 4.3]).Among these, generalised tropical hyperfields [25,Example 2.14] happen to encode ordered abelian groups as valued hyperfields.
In addition, without the ultrametric condition, valuations on (hyper)fields become nothing but homomorphisms with a specific target, making the category vHyp of valued hyperfields and their homomorphisms a natural framework (also) for classical valuation theory.
In this article, we propose the question whether the metric properties of the members and of the projections of the system associated to valued fields (such as H), can somehow be captured in the just mentioned category vHyp, while respecting the principle of equivalence.
We show in fact that the metric properties of the systems H are local in vHyp, in the sense that they can be deduced by seeing H as a diagram in the slice categories of vHyp by (suitable) generalised tropical hyperfields.There, Krasner's result goes through as a clean limit construction and valuation-theoretic completions of valued fields are hence described as vertices of such limit cones.
We organised the paper by first explaining briefly the necessary category-theoretic backgroung and the notation that we adopt to treat it.Then we survey the algebraic theory of hyperfields as well as Krasner's quotient construction associating to a valued field the projective system H of valued hyperfields mentioned above.We define the category vHyp of valued hyperfields and explain how ordered abelian groups (OAGs) can be interpreted as objects in it.Then, by passing to slice categories over OAG-extensions of the value group of a valued field pK, vq, we show how its associated system forms a diagram in those slice categories and that the limit of this diagram is isomorphic as a valued field to the completion of pK, vq.

Category Theoretic Preliminaries and Terminology
Many references for category theory may be cited, which cover the necessary background for the scope of this paper.The classic [27] is certainly one of them and we found particularly useful the following books as well [1,34,22].
Since we believe that one way to better appreciate our contribution is to be as precise as possible with terminology and notation, we shall assume only some familiarity with the concepts of category, small category, and functor during this preliminary section.
As for basic notations, for a category C, we write A P ObpCq to mean that A is an object in C and, for any ordered pair pA, Bq of objects in C, we denote the set of arrows f : A ÝÑ B in C by CpA, Bq.For the composition of arrows the symbol ˝will be employed and the identity C-arrow of A P ObpCq will be written as If C is an object in C, then the slice category C{C has C-arrows f : A ÝÑ C, with A P ObpCq, as objects, while a C-arrow a : A ÝÑ B is an arrow in C{C if and only if the following triangular diagram: is commutative in C, i.e., g ˝a " f .The concept of limit in a category is fundamental and, partly due to its generality, admits many equivalent definitions.The terminology we adopt for limits is similar to that of [22], which seemed to be the most appropriate in this case.Let us go through a brief recap.for any cone on D as in (1), there exists a unique arrow h : V ÝÑ L such that p I ˝h " s I holds, for all I P ObpSq.By a limit of a diagram D : S ÝÑ C we mean the vertex of a limit cone over D.
An isomorphism between objects A, B in a category C is a C-arrow f : A ÝÑ B with the property that a C-arrow f ´1 : B ÝÑ A exists such that f ´1 ˝f " 1 A and f ˝f ´1 " 1 B .The notation f ´1 used for this arrow suggests that from the mere existence, uniqueness follows too, which is in fact well-known to be the case.
The uniqueness of the arrows whose existence is guaranteed by the universal property of limit cones, implies that, when they exist, limit cones are unique up to (a unique) isomorphism (of cones).In particular, limits in a category C are unique up to (a unique) C-isomorphism.It is up to this isomorphism that we speak of the limit of a diagram.
The sides of limit cones are often called projections.This name comes from the analogy with the limit of diagrams of shape 2, that is, the category consisting of 2 objects with their identity arrows solely.The latter specially shaped limits are called (binary) products.In fact, in the category Set of sets and functions, their vertex is the familiar cartesian product of sets, while their sides are nothing but the projections onto its components.For the product pA ˆB, p 1 , p 2 q of two objects A 1 , A 2 P ObpCq, where p i : A 1 ˆA2 ÝÑ A i denote the projections (i " 1, 2), the universal property of limits has the following form: for any object B in C admitting two arrows f i : B ÝÑ A i (i " 1, 2) in C, there exists a unique arrow f 1 ˆf2 : B ÝÑ A 1 ˆA2 such that the following diagram: is commutative.
Another specially shaped limit, which is named terminal object, is defined in a category C as the limit cone of the unique diagram H ÝÑ C, where H denotes the category with no objects and, consequently, no arrows (the empty category).If T is a terminal object in C, then the universal property of limits has the following form: for any object C in C, there exists a unique arrow !C : C ÝÑ T .
When limit cones on diagrams of a certain shape S exist in a category C, then one says that C has limits of shape S. One then usually simplifies the terminology further in case the particular shape has been given a name.For instance, phrases like "C has products" or "C has a terminal object" mean that C has limits of shape 2 and H, respectively.Remark 2.1.It is important to keep in mind that uniqueness up to isomorphism does not necessarily mean absolute uniqueness1 .For example, in the category Set of sets and functions, where isomorphisms are bijections, all singleton sets are terminal objects.

Valued Fields and Hyperfields
Let pK, `, ¨, 0, 1q be a field and pΓ, ď, `, 0q a linearly ordered abelian group (always denoted additively)2 .That is, Γ is an abelian group equipped with a linear order relation ď and an abelian group structure whose operation `is compatible with ď, i.e., the following implication: , where 8 is a symbol such that γ `8 " 8 `γ " 8 ą γ for all γ P Γ, is called a (Krull) valuation on K if and only if it satisfies all of the following three properties: (VAL1) vpxq " 8 if and only if x " 0, for all x P K. (VAL2) vpxyq " vpxq `vpyq, for all x, y P K.
If a valuation v on a field K is given, then pK, vq is called a valued field, while the image of v in Γ, denoted by vK, is called the value group of pK, vq.The value vpxq of x P K will be written as vx whenever no risk of confusion arises.If pK, vq is a valued field, then is a subring of K, called the valuation ring of pK, vq.It determines the valuation map v up to valuation-equivalence, i.e., up to composition with an order preserving isomorphism of the value group.The prime ideals of the valuation ring O v are linearly ordered by set-inclusion and have the following form: to the trivial convex subgroup t0u of vK, is the unique maximal ideal of O v .The field Kv, defined as the quotient ring O v {m v , is called the residue field of pK, vq.
A homomorphism of valued fields from pK, vq to pL, wq can be defined as a homomorphism of fields σ : K ÝÑ L such that σpO v q Ď O w .The latter condition is sometimes phrased as "σ preserves the valuation".Since homomorphisms of valued fields are in particular homomorphisms of fields, they are automatically injective and will thus sometimes be called embeddings.We say that pL, wq is a valued field extension of pK, vq if K Ď L and the inclusion map is an embedding of valued fields.In this way, valued fields and their homomorphisms form a category vFld, which is a subcategory of Set.By an isomorphism of valued fields we mean an isomorphism in vFld, namely, a bijective homomorphism of valued fields whose inverse (as a function) is an arrow in vFld too.This can be spelled out further as follows: a function σ : K ÝÑ L is an isomorphism of valued fields pK, vq » pL, wq if and only if it is an isomorphism of fields K » L such that σpO v q " O w .
Fix now a valued field pK, vq.There is a smallest ordinal3 κ serving as the index set of a sequence pγ ν q νăκ that is cofinal in vK, i.e., such that for each δ P vK there exists ν ă κ such that δ ă γ ν .We say that a sequence px ν q νăκ of elements of K is a Cauchy sequence if and only if for every γ P vK there exists ν 0 ă κ such that if ν 0 ď ν, µ ă κ, then vpx ν ´xµ q ą γ.
A sequence px ν q νăκ is, instead, said to be convergent to an element x belonging to some valued field extension pL, wq of pK, vq if and only if for every γ P wL there exists ν 0 ă κ such that if ν 0 ď ν ă κ, then wpx ´xν q ą γ.
If the latter property happens to hold, then we also say that the sequence px ν q νăκ converges in L. If pL, wq is a valued field extension of pK, vq, then we say that K lies dense in L if every Cauchy sequence in K converges in L, while pK, vq is called complete if and only if every Cauchy sequence in K converges in K. Fact 3.1 (Theorem 2.4.3 in [6]).Every valued field pK, vq admits one and (up to isomorphism of valued fields) only one valued field extension pK c , v c q -called the completion of pK, vq -which is complete and in which K lies dense.
An important consequence of the fact that K lies dense in K c is that the value group v c K c and the residue field K c v c of pK c , v c q are (canonically) isomorphic to vK and Kv, respectively (cf.[6, Proposition 2.4.4]).
3.1.Krasner hyperfields.Hyperfields first appeared in [18].In introducing them, Krasner was motivated by his interest for certain structures obtained from valued fields by means of the "factor (or quotient) construction", which he himself described for the first time in the same article and later in [19].For the algebraic definition of hyperfields we refer to [25, Definition 2.7] and references therein.A more categorial treatment of these structures within the category of sets and (total) relations can be found in [23].
The following definition of homomorphism for hyperfields has become standard in the literature.
Let us recall Krasner's factor construction which yields a projective system of hyperfields associated to any valued field (considered also in [21,26]).All hyperfields considered in this paper can be obtained via this construction (see [25,Proposition 2.17]).Nevertheless, the statement "all hyperfields are quotient" is not valid in full generality [31].Whether non-trivially valued hyperfields are automatically quotient or not is not known, as discussed in [26].
For a valued field pK, vq and an element γ P vK such that γ ě 0, one considers the subgroup of the 1-units of level γ in K ˆ: U γ v :" tu P K | vpu ´1q ą γu.It can be easily verified that vu " 0 for all u P U γ v , so that the valuation map v on K factors through the quotient group K γ :" K ˆ{U γ v and yields a map v γ : K γ ÝÑ vK Y t8u, where K γ :" K γ Y tr0s γ u.In this paper, we follow the notation of [21] and we denote the multiplicative coset xU γ v of x P K in K γ as rxs γ (in particular, r0s γ " t0u) and call the valued γ-hyperfield associated to pK, vq the hyperfield pK γ , ', ¨, r0s γ , r1s γ q, where rxs γ ' rys γ :" trx `yus γ | u P U γ v u and rxs γ ¨rys γ :" rxys γ .The use of the term "valued" is motivated once one observes that the map v γ satisfies (VAL1), (VAL2) and the following property analogous to (VAL3): (VAL3*) v γ rzs γ ě mintv γ rxs γ , vrxs γ u, for all rxs γ , rys γ P K γ and all rzs γ P rxs γ ' rys γ see also [20,24,25].More generally, we shall call pH, vq a valued hyperfield whenever v is a map from the hyperfield H to an ordered abelian group Γ (with the addition of 8) satisfying (VAL1), (VAL2) and (VAL3*).These requirements are equivalent (cf.e.g.[25,Lemma 3.4]) to v being a homomorphism of hyperfields H ÝÑ T pΓq, where T pΓq denotes the generalised tropical hyperfield associated to Γ: [25]).Let 8 be a symbol such that γ `8 " 8 `γ " 8 ą γ for all γ P Γ.For γ, δ P Γ Y t8u we denote by rγ, δs the closed interval containing all ε P Γ Y t8u satisfying γ ď ε ď δ.Then by setting γ ' 8 " 8 ' γ " tγu for all γ P T pΓq and: It is not difficult to check that pT pΓq, ', `, 8, 0q is a hyperfield, called generalised tropical hyperfield associated to Γ.Note that, conversely, the order of Γ can be recovered as follows: In [25,Section 3] the author shows that, as in the case of fields, the set O v of the elements in a valued hyperfield pH, vq with non-negative value under v determines the valuation map (up to valuation-equivalence).As a consequence, homomorphisms of valued hyperfields are defined analogously to arrows in vFld and a category vHyp is thus obtained.A field K with additive operation `can be regarded as a hyperfield with additive operation ' defined as x ' y :" tx `yu.Conversely, any hyperfield with a singlevalued additive operation, i.e., such that x ' y is a singleton for all x, y P H, can be regarded as a field.It is in this spirit that we view vFld as a subcategory of vHyp.One can observe that the identity 1 T pΓq : T pΓq ÝÑ T pΓq of the hyperfield T pΓq in vHyp is a valuation on T pΓq.Thus, a valuation v : H ÝÑ Γ Y t8u on a hyperfield H is, equivalently, a vHyp-arrow pH, vq ÝÑ pT pΓq, 1 T pΓq q.
Let us fix a valued field pK, vq.The following statement contains a number of fundamental properties of the valued γ-hyperfields associated to pK, vq, where 0 ď γ P vK.Lemma 3.4 (Lee Lemma).Take x, y, x 0 , . . ., x k P K for some positive integer k and let γ P vK be such that γ ě 0. The following assertions hold: piq If x ‰ 0, then rxs γ " ty P K | vpx ´yq ą γ `vxu.piiq If x and y are not both 0, then ď prxs γ ' rys γ q " tz P K | vpz ´px `yqq ą γ `mintvx, vyuu.
The above proposition permits to induce from v γ an ultrametric distance on K γ which we denote by d γ (see e.g.[25, Definition 4.1]).Proposition 3.6.Let pK γ , v γ q be the valued γ-hyperfield of a valued field pK, vq, where 0 ď γ P vK.If x and y are not both 0, then rxs γ ' rys γ is the open ultrametric ball of radius δ :" γ `mintvx, vyu around rx `ys γ P K γ with respect to d γ .

Main results
For the rest of the paper: ‚ pK, vq denotes a valued field.‚ pΓ, ď, `, 0q denotes an ordered abelian group containing pvK, ď, `, 0q as a substructure in the language tď, `, 0u of ordered groups.In the first result of this final section, we highlight another consequence of the fact that a valued field lies dense in its completion.As usual, if a homomorphism of hyperfields σ is bijective and its inverse is a homomorphism of hyperfields as well, then σ is called an isomorphism of hyperfields.Lemma 4.1.Let pK c , v c q be the completion of pK, vq and identify K and vK with the subsets of K c and v c K c to which they are canonically isomorphic.Then for all γ P vK such that γ ě 0, there is an isomorphism of hyperfields σ γ : K γ ÝÑ K c γ such that v γ pσ γ paqq " v c γ paq holds, for all a P K c γ .Proof.Fix γ P vK such that γ ě 0. Just for this proof, we will denote by rxs c the class of x P K c in K c γ and by rys the class of y P K in K γ .It follows from Lemma 3.4 piq that, for all nonzero x P K c , we have that rxs c , as a subset of K c , is an open ultrametric ball (with respect to the ultrametric induced on K c by v c ).Since K lies dense in K c , there is y P K such that y P rxs c .On the other hand, rxs c is an equivalence class in K c with respect to an equivalence relation whose restriction to K has rys among its equivalence classes.Consequently, rys c " rxs c as subsets of K c and if y 1 P K satisfies y 1 P rxs c as well, then rys c " rxs c " ry 1 s c and thus rys " ry 1 s must hold.Since all x P K belong to the class rxs c in K c γ , this proves that the assignment rxs Þ Ñ rxs c defines a bijective function σ γ : K γ ÝÑ K c γ .From the definitions and the inclusion U γ v Ď U γ v c it easily follows at this point that σ γ is an isomorphism of hyperfields satisfying the assertion of the lemma.
Proof.First we show that ρ δ,γ is well-defined for all γ, δ as in the statement.To this end, assume that rxs δ " rys δ .Then there exist t P U δ v such that x " yt.Since γ ď δ we have that U δ v Ď U γ v , so we obtain that x " yt for some t P U γ v and thus ρ δ,γ prxs δ q " rxs γ " rys γ " ρ δ,γ prys δ q.
The assignment x Þ Ñ rxs γ defines a function ρ γ : K ÝÑ K γ , for all non-negative γ P vK.It follows from the definitions that these functions are homomorphisms of valued hyperfields such that vx " v γ rxs γ for all x P K, i.e., the following triangular diagrams: Therefore, the functions ρ γ are arrows in the slice category vHyp{T pΓq.Moreover, they respect the functions ρ δ,γ in the sense that, for all non-negative γ, δ P vK we have that the following diagram: commutes in vHyp.The above discussion shows that pK, vq is the vertex of a cone over the diagram (3) in vHyp{T pΓq, i.e., the following diagram: is commutative in vHyp.Now consider the completion pK c , v c q of pK, vq.If, as before, we identify vK with the subset of v c K c to which it is canonically isomorphic, then from Lemma 4.1 we deduce that K c too is the vertex of a cone over the same diagram (3).In addition, K embeds as a valued field in K c by Fact 3.1 and such an embedding can be seen to be an arrow in vHyp{T pΓq.Before moving forward let us prove the following useful lemma, which states that (with the right choice of Γ) all cones in vHyp{T pΓq over the diagram (3) are (valued) fields.
Lemma 4.3.Let pH, wq be a valued hyperfield such that there are order-preserving group-embeddings vK ãÑ wH ãÑ Γ and assume that pH, wq is the vertex of a cone in vHyp{T pΓq over the diagram (3).Then H is a field, i.e., for all x, y P H we have that x ' y is a singleton, where ' denotes the additive operation of the hyperfield H.
Proof.We work up to the given embeddings of ordered abelian groups.Fix γ P vK such that γ ě 0. We denote by f γ : H ÝÑ K γ the sides of the given cone in vHyp{T pΓq.Pick x, y P H ând z, z 1 P x ' y.We claim that 0 P z 1 ' z ´, where z ´.Since f γ is an an arrow in the slice category vHyp{T pΓq, we obtain that f γ pzq, f γ pz 1 q P f γ pxq ' γ f γ pyq holds in K γ , where ' γ denotes the additive operation of the hyperfield K γ .In addition, we obtain that the equalities wx " v γ f γ pxq and wy " v γ f γ pyq hold in Γ.Thus, by Proposition 3.6, we have that f γ pxq' γ f γ pyq is an open ultrametric ball in K γ of radius γ `mintwx, wyu.Let now pγ ν q νăκ be an increasing and cofinal sequence of non-negative elements of vK and take an arbitrary δ P vK.We consider some ν ă κ which is large enough so that γ ν `mintwx, wyu ą δ holds in Γ.If we suppose that r0s γν R f γν pz 1 q ' γν f γν pzq ´, then by Proposition 3.5 and since f γν is a homomorphism of hyperfields, we obtain that for any a P z 1 ´z´, the value wa " v γν f γν paq P vK Y t8u is larger than δ.Since δ is arbitrary in vK, this implies that wa " 8 and so a " 0. We deduce that r0s γν " f γν paq P f γν pz 1 q ' γν f γν pzq ´.
This contradiction shows that r0s γν P f γν pz 1 q ' γν f γν pzq ´must hold in K γν and, as a consequence, we obtain that f γν pz 1 q " f γν pzq.Furthermore, since f γν is an arrow in vHyp{T pΓq, we have that wz 1 " wz and by enlarging ν (if necessary) we can ensure that γ ν `wz ą δ as well.Now, for all a P z 1 ' z ´we obtain that f γν paq P f γν pz 1 q ' γν f γν pzq ´and, again by Proposition 3.6, f γν pz 1 q ' γν f γν pzq ´is an open ultrametric ball of radius γ ν `wz and center r0s γν .Therefore, wa " v γν f γν paq P vK Y t8u will be larger than δ and since δ is arbitrary in vK, it follows that a " 0 anyway.At this point our claim is proved and z 1 " z follows.The proof is complete.
By the following theorem the valued field extensions of a valued field which embed in its completion are characterised in terms of the diagram (3).Theorem 4.4.Let pL, wq be a valued field extension of pK, vq such that there is an order-preserving group-embedding wL ãÑ Γ.Then the following statements are equivalent: piq pL, wq embeds as a valued field into pK c , v c q. piiq For all γ P vK such that γ ě 0, there is an isomorphism in vHyp{T pΓq: σ γ : pL γ , w γ q pK γ , v γ q.
Proof.We begin by proving that piq implies piiq.Since L contains K which lies dense in K c , it follows that L lies dense in K c too.Hence piiq follows as in the proof of Lemma 4.1.
For the implication from piiq to piq, a little more effort is needed.First, we need to fix an increasing and cofinal sequence of non-negative elements pγ ν q νăκ in the value group vK, as we have done in the proof of Lemma 4.3.Then, for any x P L ˆand all ν ă κ, we set y ν P K ˆto be a representative for the class σ γν prxs γν q P K γν .By the assumption piiq and the definition of the hyperfield valuations v γν and w γν , we deduce that vy ν " wx holds in Γ for all ν ă κ.In addition, since pγ ν q νăκ is increasing, Lemma 3.4 piq yields that vpy ν ´yµ q ą γ ν `wx holds in Γ, for all ν ă µ ă κ.Now, by the cofinality of pγ ν q νăκ in vK, the latter inequality implies that py ν q νăκ is a Cauchy sequence in K which then converges to some y P K c .
We claim that y does not depend on the choice of the representatives y ν P K, but only on the class σprxs γν q P K γ .For let y 1 ν P K be such that ry 1 ν s γν " σ γν prxs γν q be another choice.As above, py 1 ν q νăκ is a Cauchy sequence in K and we denote by y 1 its limit in K c .If δ P vK and ν ă κ are such that γ ν `wx ą δ and v c py ´yν q, v c py 1 ν ´y1 q ą δ hold in Γ, then, by Lemma 3.4 piq and since vy 1 ν " wx " vy ν , we obtain that vpy ν ´y1 ν q ą γ ν `wx ą δ and then v c py ´y1 q " v c py ´yν `yν ´y1 ν `y1 ν ´y1 q ě mintv c py ´yν q, v c py ν ´y1 ν q, v c py 1 ν ´y1 qu ą δ hold in Γ.Since δ is arbitrary in vK " vK c , we may conclude that v c py ´y1 q " 8, i.e., y 1 " y holds in K c .Our next claim is that the assignment x Þ Ñ y defines an embedding of valued fields σ : pL, wq pK c , v c q.
To see why this holds, most of the efforts are devoted to the verification that σ preserves the additive operations.For take x, y P L and assume without loss of generality that wx ď wy.If z ν , a ν , b ν P K are such that rz ν s γν " rx `ys γν , ra ν s γν " rxs γν and rb ν s γν " rys γν for all ν ă κ, then, as before, these elements form Cauchy sequences in K. Let us then denote by z, a and b their limits in K c , respectively.By definition of σ, we have that σpx `yq " z, σpxq " a and σpyq " b.Our aim now will be to prove that z " a `b holds in K c .We first obtain from Lemma 3.4 pivq that z ν P ď pra ν s γν `rb ν s γν q holds, for all ν ă κ.Therefore, if we fix any δ P vK and let ν ă κ be large enough so that γ ν `wx ą δ and v c pa ´aν q, v c pb ´bν q, v c pz ´zν q ą δ hold in Γ, then an application of Lemma 3.4 piiq yields that vpa ν `yν ´zν q ą γ ν `wx ą δ holds in Γ, where we used the fact that va ν " wx for all ν ă κ.Thus, we obtain that v c pa ν `bν ´zq " v c pa ν `bν ´zν `zν ´zq ě mintv c pa ν `bν ´zν q, v c pz ν ´zqu ą δ and, consequently, v c pa `b ´zq " v c pa ´aν `b ´bν `aν `bν ´zq ě mintv c pa ´aν q, v c pb ´bν q, v c pa ν `bν ´zqu ą δ hold in Γ.Since δ is arbitrary in vK and v c pa `b ´zq P vK Y t8u, we deduce that v c pa `b ´zq " 8 and, consequently, z " a `b, as contended.Now, it suffices to recall that r´xs γ " ´rxs γ and that if y ‰ 0, then rxy ´1s γ " rxs γ rys ´1 γ hold in L γ to immediately deduce that σp´xq " σpxq ´and that σpab ´1q " σpaqσpbq ´1 must hold in K c γ .We have proved that σ is a homomorphism of fields and, as such, an embedding.
Finally, since vy ν " wx holds in Γ, for all x P L ˆand all ν ă κ (as we have already shown above) it can be easily verified from the definition of convergent sequences, that the element σpxq P K c , to which the Cauchy sequence py ν q νăκ in K converges, satisfies v c pσpxqq " vy ν , for all ν ă κ.We conclude that vpσpxqq " wx holds, for all x P L. In particular, we have that σ is an embedding of valued fields.
In the proof of the implication from piiq to piq in the above theorem, we have used the assumption that pL, wq is an extension of pK, vq only for identifying vK and v c K c with a canonical subset of wL.However, in the final analysis, this identification is not necessary and is performed only for a smoother exposition of the reasonings in the proof.Scholium 4.5 (4 ).If pL, wq is any valued field such that there is an order-preserving groupembedding wL ãÑ Γ and, up to this embedding, condition piiq of Theorem 4.4 holds, then there is a vHyp{T pΓq-arrow pL, wq ÝÑ pK c , v c q. Now, under the assumptions of the above result, for a valued hyperfield pL, wq, we deduce, first, that L is a field by Lemma 4.3.Then, we denote by f γ : L Ñ L γ » K γ and by ργ : K c Ñ K c γ » K γ the projections onto the valued γ-hyperfields associated to pK, vq of pL, wq and pK c , v c q, respectively.It is so straightforward to verify that the embedding σ that we have constructed in the proof of Theorem 4.4 is unique with the property that f γ " ργ ˝σ for all γ P vK such that γ ě 0. Indeed, this conclusion follows from the fact that for any x P L ˆ, the classes prxs γ q γPvKě0 form a chain of ultrametric balls in L of increasing radii and, moreover, the set of this radii is cofinal in vK.We have now fully proved Krasner's result [18, § 4] in a purely categorial language: Theorem 4.6.The completion pK c , v c q of pK, vq is (vHyp-isomorphic to) the limit of the diagram (3) in vHyp{T pΓq, for any ordered abelian group extension Γ of vK.
We note moreover, that in the above setting the assumption Γ " vK " wL is a posteriori not restrictive.
Corollary 4.7.pK, vq is complete if and only if it is the source of the limit of the diagram (3), with Γ " vK, in vHyp{T pΓq.

Conclusion and future work
It is possible (cf.[26,Proposition 1.17]) to associate a diagram of the form (3) as above also in case the additive operation of pK, vq is not assumed to be singlevalued, i.e., starting with any valued hyperfield pK, vq.As a corollary to our approach, we deduce that if among the cones over the diagram (3) there is one whose vertex is a valued field, then the limit exists in vHyp and is up to isomorphism the valuation-theoretic completion of the latter valued field.In particular, this happens under weaker assumptions on pK, vq than that K is a field, e.g., by [26,Proposition 1.27], it suffices that v induces an ultrametric on K in the way originally required by Krasner.The problem whether the just mentioned assumption on pK, vq is also necessary or can be weakened further is outside the scope of this work and left open for future investigations.
Finally, a characterisation of generalised tropical hyperfields in purely category-theoretic terms becomes of interest.A characterisation is given in [25,Theorem 5.2] which relates the solution to the latter problem to a categorial description of the broader class of stringent hyperfields (in the sense of [4]).

2. 1 .
Limits.Fix a category C. If S is a small category, then a functor D : S ÝÑ C is called a diagram in C of shape S. A cone on a diagram D : S ÝÑ C consists of an object V in C, called the vertex of the cone, together with a family, indexed by the collection of objects in S, (1) `V DpIq ˘IPObpSq sI of C-arrows, called the sides of the cone such that the following triangular diagram: for all arrows f P SpI, Jq.A cone over a diagram D : S ÝÑ C, `L DpIq ˘IPObpSq pI is called a limit cone if it satisfies the following universal property: