In this paper, we introduce the notion of n-tilting (resp. n-cotilting) objects in functor categories and give some characterizations of n-tilting objects and n-tilting classes (resp. n-cotilting objects and n-cotilting classes).
The article is devoted to non-associative algebras with metagroup relations and modules over them. Their functors are studied. Satellites of functors are scrutinized. An exactness of satellite sequences and diagrams is investigated.
This paper develops extension theory for Brown functors in directed graph homotopy theory. We establish a systematic method for extending Brown functors from finite directed graphs to arbitrary directed graphs using inverse limits over finite subdigr...
The categorical treatment of fuzzy modules presents some problems, due to the well known fact that the category of fuzzy modules is not abelian, and even not normal. Our aim is to give a representation of the category of fuzzy modules inside a genera...
Various types of topological and closure operators are significantly used in fuzzy theory and applications. Although they are different operators, in some cases it is possible to transform an operator of one type into another. This in turn makes it p...
Mallios and Zafiris emphasize that adjoint functors, or adjunctions, are not only “ubiquitous” in category theory but also characterize the naturality of their approach to physical geometry. Hence, in this paper, the history and theory of...
This paper is devoted to the investigation of cardinal invariants such as the hereditary density, hereditary weak density, and hereditary Lindelöf number. The relation between the spread and the extent of the space SP2(R,τ(A)) of permutation...
This work establishes definitive conditions for the inheritance of categorical completeness and cocompleteness by categories of internal group objects. We prove that while the completeness of Grp(C) follows unconditionally from the completeness of th...
This paper establishes a comprehensive framework for the hereditary transfer of categorical completeness and cocompleteness to categories of internal group objects in D-modules. We prove that while completeness of Grp(D-Mod) follows unconditionally f...
This paper introduces and investigates the fundamental properties of L-primals, a generalization of the primal concept within the framework of L-fuzzy sets and complete lattices. Building upon the established theories of L-topological spaces and L-pr...
We study the relationship between the category of R-modules (CR-M) and the category of intuitionistic fuzzy modules (CR−IFM). We construct a category CLat(R−IFM) of complete lattices corresponding to every object in CR−M and then sh...
We construct a 2-categorical extension of the relative entropy functor of Baez and Fritz, and show that our construction is functorial with respect to vertical morphisms. Moreover, we show such a ‘2-relative entropy’ satisfies natural 2-c...
Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunction seeming to be the primary lens. Our topic is a theory showing “where adjoints...
In this paper, we study the space of G-permutation degree of some classes of topological spaces and the properties of the functor SPGn of G-permutation degree. In particular, we prove: (a) If a topological space X is developable, then so is SPGnX; (b...
This paper is devoted to the study of the interrelationships among LF-grills, LF-ideals, LF-neighborhoods, LF-topologies, and LF-co-topologies. We establish a categorical framework that demonstrates the interconnections among these concepts. In addit...
In this paper, we investigate the Koszul behavior of finitely semi-graded algebras by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincaré series, and the Yoneda algebra are defined for this class of algebr...
The natural transformation constitutes one of the most important entity of category theory and it introduces a piece of sophisticated dynamism to the categorial structures. Each natural transformation forms a unique mapping between the so-called func...
Many of the new MV-valued fuzzy structures, including intuitionistic, neutrosophic, or fuzzy soft sets, can be transformed into so-called almost MV-valued fuzzy sets, or, equivalently, fuzzy sets with values in dual pair of semirings (in symbols, (R,...
In this paper we, prove that if the product Xn of a space X has certain tightness-type properties, then the space of permutation degree SPnX has these properties as well. It is proven that the set tightness (T-tightness) of the space of permutation d...
Since its introduction, researching malware has had two main goals. On the one hand, malware writers have been focused on developing software that can cause more damage to a targeted host for as long as possible. On the other hand, malware analysts h...
We present a categorical formalization of a variant of first-order logic. Unlike other texts on this topic, the goal of this paper is to give a very transparent and self-contained account without requiring more background than basic logic and set the...
Two categories of lower and upper lattice-valued F-transforms with fuzzy relations as morphisms are introduced, as generalisations of standard categories of F-transforms with maps as morphisms. Although F-transforms are defined using special structur...
The study of the interactions among intentional agents, with rationality being the main source of intentional behavior, requires mathematical tools capable of capturing systemic effects. Here, we choose an alternative toolbox based on Category Theory...
The Müller-Wichards model (MW) is an algebraic method that quantitatively estimates the performance of sequential and/or parallel computer applications. Because of category theory’s expressive power and mathematical precision, a category t...
We provide a stochastic extension of the Baez–Fritz–Leinster characterization of the Shannon information loss associated with a measure-preserving function. This recovers the conditional entropy and a closely related information-theoretic measure tha...
The general theory of information (GTI) is a synthetic approach, which reveals the essence of information, organizing and encompassing all main directions in information theory. On the methodological level, it is formulated as system of principles ex...
IN MEMORIAM OF ALEXANDER GROTHENDIECK. THE MAN.
The growing complexity of modern practical problems puts high demand on mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice is becoming particularly import...
Consciousness is a central issue in neuroscience, however, we still lack a formal framework that can address the nature of the relationship between consciousness and its physical substrates. In this review, we provide a novel mathematical framework o...
The concept of distribution is a concept within traditional logic that has been fundamental for the syntactic development of Sommers and Englebretsen’s term functor logic, a logic that recovers the term syntax of traditional logic. The issue he...
Causal discovery involves searching intractably large spaces. Decomposing the search space into classes of observationally equivalent causal models is a well-studied avenue to making discovery tractable. This paper studies the topological structure u...
Universal Causality is a mathematical framework based on higher-order category theory, which generalizes previous approaches based on directed graphs and regular categories. We present a hierarchical framework called UCLA (Universal Causality Layered...
With his research on Aristotle’s syllogistic, Jan Łukasiewicz initiates the branch of logic known as the calculus of names. This field deals with axiomatic systems that analyse various fragments of the logic of names, i.e., that branch of...
These notes are an informal overview of techniques related to deformation theory in the context of physics. Beginning from motivation for the concept of a sheaf, they build up through derived functors, resolutions, and the functor of points to the no...
In this work, we use the concept of G-weak graded rings and G-weak graded modules, which are based on grading by a set G of left coset representatives for the left action of a subgroup H of a finite X on X, to define the conjugation action of the set...
In this paper, we construct a quotient structure on digroups. This construction yields a new functor from the category of digroups to the category of groups. We obtain a modular property for digroups and use it to prove an analogue of the Zassenhaus...
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/ Ω -spectra, simplicial topological spaces/(complete) Segal spac...
In this work, we analyze the combinatorial properties of the category of augmented semi-simplicial sets. We consider various monoidal structures induced by the co-product, the product, and the join operator in this category. In addition, we also cons...
Using a free simphicial group with given a CW-basis, we give an alternative description of the top group of the free square complex and 2-crossed complex of groups on 2-construction data in terms of tensors and coproducts of crossed module of groups....
We study the duality between corings and ring extensions. We construct a new category with a self-dual functor acting on it, which extends that duality. This construction can be seen as the non-commutative case of another duality extension: the duali...
The starting point is Peano’s expression of the axiomatics of natural numbers in the framework of Leśniewski’s elementary ontology. The author enriches elementary ontology with the so-called Frege’s predication scheme and goes on to...
We apply the geometric quantization procedure via symplectic groupoids to the setting of epistemically-restricted toy theories formalized by Spekkens (Spekkens, 2016). In the continuous degrees of freedom, this produces the algebraic structure of qua...
Recently, Gorenstein dimensions relative to a semidualizing module have been the subject of numerous studies with interesting extensions of the classical homological dimensions. Although all these studies share the same direction, a common basis, and...
In this article, we investigate the fundamental properties of coalgebras with coalgebra comultiplications, counits, and coalgebra homomorphisms of coalgebras over a commutative ring R with identity 1R based on digital images with adjacency relations....
The definition of Azumaya algebras over commutative rings \(R\) requires the tensor product of modules over \(R\) and the twist map for the tensor product of any two \(R\)-modules. Similar constructions are available in braided monoidal categories, a...
Let (C,E,s) be an extriangulated category with a proper class ξ of E-triangles and X a resolving subcategory of C. In this paper, we introduce the notion of X-resolution dimension relative to the subcategory X in C, and then give some descriptions of...
If M is an algebra in a semidegenerate congruence-modular variety V, then the set Con(M) of congruences of M is an integral complete l-groupoid (= icl-groupoid). For any morphism f:M→N of V, consider the map f•:Con(M)→Con(N), where, fo...
Let CAn=C[S2≀S2≀⋯≀S2] be the group algebra of an n-step iterated wreath product. We prove some structural properties of An such as their centers, centralizers, and right and double cosets. We apply these results to explicitly...
In this discussion paper we argue that category theory may play a useful role in formulating, and perhaps proving, results in ergodic theory, topogical dynamics and open systems theory (control theory). As examples, we show how to characterize Kolmog...
In this paper, we consider the digital cohomology modules of a digital image consisting of a bounded and finite subset of Zn and an adjacency relation. We construct a contravariant functor from the category of digital images and digital continuous fu...
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