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The aim of this paper is to introduce a tensor structure for the Serre quotient category of an abelian monoidal category with biexact tensor product to make the canonical functor a monoidal functor. In this tensor product, the Serre quotient category of a multiring category (resp. a multitensor category) by a two-sided Serre tensor-ideal is still a multiring category (resp. a multitensor category). Besides, a two-sided Serre tensor-ideal of a tensor category is always trivial. This result can be generalized to any tensor product. If the canonical functor is a monoidal functor, then the corresponding Serre subcategory of the tensor category is trivial.
As categorical dimension increases, classical categorical concepts often become too rigid and must be replaced by appropriately lax analogues. A basic manifestation of this principle is that higher-categorical versions of familiar geometric constructions -- such as cylinders, cones, and suspensions -- are not functorial in the classical sense. To remedy this situation, we extend the theory of higher categories to a framework of so-called oriented and antioriented categories, which provides a unified formalism for lax and oplax phenomena and reveals the geometric nature inherent in higher category theory. We characterize (anti)oriented categories as deformations of higher categories in which the various compositions only commute up to coherent (anti)oriented interchange law, and provide a geometric presentation of (anti)oriented categories as sheaves on a suitable thickening of the simplex category. We also construct an embedding of the category of $(\infty,\infty)$-categories into the category of (anti)oriented categories and characterize the image as those $(\infty,\infty)$-categories satisfying a strict interchange law.
We construct a strict pivotal monoidal category $\mathcal{D}_{\mathrm{DNA}}$ whose objects are DNA sequences (words over $\{A,C,G,T\}$) and whose morphisms are isotopy classes of typed noncrossing planar matchings, composed of through-strands and Watson-Crick-typed arcs, in a rectangle with source and target boundaries. The dual of a sequence is its reverse complement, evaluation and coevaluation are canonical duplex pairings, and the snake identities hold by planar isotopy. A bending correspondence identifies each morphism $x \to y$ with a secondary structure on the combined word $x{}^{\vee} y$; in particular, the generalized elements $\varepsilon \to w$ are exactly the non-pseudoknotted secondary structures on $w$. Composition, viewed in this straightened picture, is computed by a zip-and-transfer operation on complementary interfaces, a combinatorial rearrangement of base-pair connectivity of which toehold-mediated strand displacement is a kinetically specific instance. Because $\mathcal{D}_{\mathrm{DNA}}$ is rigid monoidal, it shares the categorical backbone of pregroup grammars and the DisCoCat framework for compositional semantics: a strong monoidal functor from a grammatical
We prove a category-theoretic independence theorem for four fundamental notions: meaning, object, name, and existence. Working in a Lawvere-style categorical semantics and in particular in toposes, we show that these notions occupy distinct structural levels (object, morphism, element, and internal logical level) and are not uniformly recoverable from one another. The key separation arises between internal existence and global naming. Using a concrete example in the topos $\mathbf{Sh}(S^1)$-the sheaf of local sections of a nontrivial covering-we exhibit an object that is internally inhabited but admits no global element. These results provide a precise structural basis for treating geometric universes as foundational frameworks for information networks.
This article introduces Hilbert $*$-categories: an abstraction of categories with similar algebraic and analytic properties to the categories of real, complex, and quaternionic Hilbert spaces and bounded linear maps. Other examples include categories of Hilbert W*-modules and of unitary group-representations. Hilbert $*$-categories are "analytically" complete in two ways: every bounded increasing sequence of Hermitian endomorphisms has a supremum, and every suitably bounded orthogonal family of parallel morphisms is summable. These "analytic" completeness properties are not assumed outright; rather, they are derived, respectively, from two new universal constructions: codirected $\ell^2$-limits of contractions and $\ell^2$-products. In turn, these are built from directed colimits in the wide subcategory of isometries.
Recommender systems have made significant strides in various industries, primarily driven by extensive efforts to enhance recommendation accuracy. However, this pursuit of accuracy has inadvertently given rise to echo chamber/filter bubble effects. Especially in industry, it could impair user's experiences and prevent user from accessing a wider range of items. One of the solutions is to take diversity into account. However, most of existing works focus on user's explicit preferences, while rarely exploring user's non-interaction preferences. These neglected non-interaction preferences are especially important for broadening user's interests in alleviating echo chamber/filter bubble effects.Therefore, in this paper, we first define diversity as two distinct definitions, i.e., user-explicit diversity (U-diversity) and user-item non-interaction diversity (N-diversity) based on user historical behaviors. Then, we propose a succinct and effective method, named as Controllable Category Diversity Framework (CCDF) to achieve both high U-diversity and N-diversity simultaneously.Specifically, CCDF consists of two stages, User-Category Matching and Constrained Item Matching. The User-Categor
Whereas formal category theory is classically considered within a $2$-category, in this paper a double-dimensional approach is taken. More precisely we develop such theory within the setting of augmented virtual double categories, a notion extending that of virtual double category by adding cells with nullary target. [...] After this the notion of `weak' Kan extension within an augmented virtual double category is considered, together with three strengthenings. [...] The notion of yoneda embedding is then considered in an augmented virtual double category, and compared to that of a good yoneda structure on a $2$-category; the latter in the sense of Street-Walters and Weber. Conditions are given ensuring that a yoneda embedding $y \colon A \to \hat A$ defines $\hat A$ as the free small cocompletion of $A$, in a suitable sense. In the second half we consider formal category theory in the presence of algebraic structures. In detail: to a monad $T$ on an augmented virtual double category $\mathcal K$ several augmented virtual double categories $T\text-\mathsf{Alg}_{(v, w)}$ of $T$-algebras are associated, [...]. This is followed by the study of the creation of, amongst others, left Kan
For a Hausdorff space $X$, a free involution $τ:X\to X$ and a Hausdorff space $Y$, we discover a connection between the sectional category of the double covers $q:X\to X/τ$ and $q^Y:F(Y,2)\to D(Y,2)$ from the ordered configuration space $F(Y,2)$ to its unordered quotient $D(Y,2)=F(Y,2)/Σ_2$, and the Borsuk-Ulam property (BUP) for the triple $\left((X,τ);Y\right)$. Explicitly, we demonstrate that the triple $\left((X,τ);Y\right)$ satisfies the BUP if the sectional category of $q$ is bigger than the sectional category of $q^Y$. This property connects a standard problem in Borsuk-Ulam theory to current research trends in sectional category. As an application of our results, we show that the index of $(X,τ)$ coincides with the sectional category of the quotient map $q:X\to X/τ$ minus 1 for any paracompact space $X$. In addition, we present some new results relating Borsuk-Ulam theory and sectional category.
We construct a compact closed category out of any symmetric monoidal category by freely adding adjoints to its objects. The morphisms of the completion are defined as string diagrams annotated by objects and morphisms from the original category. The symmetric monoidal category embeds via a faithful monoidal functor into its completion, but in contrast to the non-symmetric case, this embedding is not full. Our construction factors through the Int construction, which yields another free construction: the free traced monoidal category on a symmetric monoidal category.
We study a stabilization of the symplectic category introduced by A. Weinstein as a domain for the geometric quantization functor. The symplectic category is a topological category with objects given by symplectic manifolds, and morphisms being suitable lagrangian correspondences. The main drawback of Weinstein's symplectic category is that composition of morphisms cannot always be defined. Our stabilization procedure rectifies this problem while remaining faithful to the original notion of composition. The stable symplectic category is enriched over the category of spectra (in particular, its morphisms can be described as infinite loop spaces representing the space of immersed lagrangians), and it possesses several appealing properties that are relevant to deformation, and geometric quantization.
We construct the t-motivic mixed Carlitz zeta category over $\F_q(t)$ and show that it contains all the (mixed) t-motives with Carlitz-Thakur multi-zeta values as periods constructed by Anderson and Thakur. Our construction is canonical and our category is Tannakian and neutral and every object is equipped with a weight filtration whose graded pieces are Carlitz motives over $\F_q(t)$. For any finite separable extension L/\F_q(t) we show that existence of a similar category over $L$ is a consequence of a version of a conjecture of L. Taelman. Along the way we also prove the existence of the category of mixed $t$-motives and the category of mixed Carlitz motives over any $L$ (these two existence results are independent of any conjectures).
Using the combinatorics of the underlying simplicial complex $K$, we give various upper and lower bounds for the Lusternik-Schnirelmann (LS) category of moment-angle complexes $\zk$. We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS category. In particular, we characterise the LS category of moment-angle complexes $\zk$ over triangulated $d$-manifolds $K$ for $d\leq 2$, as well as higher dimension spheres built up via connected sum, join, and vertex doubling operations. %This characterisation is given in terms of the combinatorics of $K$, the cup product length of $H^*(\zk)$, as well as a certain Massey products. We show that the LS category closely relates to vanishing of Massey products in $H^*(\zk)$ and through this connection we describe first structural properties of Massey products in moment-angel manifolds. Some of further applications include calculations of the LS category and the description of conditions for vanishing of Massey products for moment-angle manifolds over fullerenes, Pogorelov polytopes and $k$-neighbourly complexes, which double as important examples of hyperbolic manifolds.
In [17], we introduced ``picture groups'' and computed the cohomology of the picture group of type $A_n$. This is the same group what was introduced by Loday [20] where he called it the ``Stasheff group''. In this paper, we give an elementary combinatorial interpretation of the {\color{blue}``cluster morphism category'' constructed in [13] in the special case of the linearly oriented quiver of type $A_n$.} We prove that the classifying space of this category is locally $CAT(0)$ and thus a $K(π,1)$. We prove a more general statement that classifying spaces of certain ``cubical categories'' are locally $CAT(0)$. The objects of our category are the classical noncrossing partitions introduced by Kreweras [19]. The morphisms are binary forests. This paper is independent of [13] and [17] except in the last section where we use [13] to compare our category with the category with the same name given by Hubery and Krause [9].
Recently Feferman (Rev. Symb. Logic 6: 6-15, 2013) has outlined a program for the development of a foundation for naive category theory. While Ernst (ibid. 8: 306-327, 2015) has shown that the resulting axiomatic system is still inconsistent, the purpose of this note is to show that nevertheless some foundation has to be developed before naive category theory can replace axiomatic set theory as a foundational theory for mathematics. It is argued that in naive category theory currently a 'cookbook recipe' is used for constructing categories, and it is explicitly shown with a formalized argument that this 'foundationless' naive category theory therefore contains a paradox similar to the Russell paradox of naive set theory.
These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific mind. There is no need to know advanced mathematics, nor any of the disciplines where category theory is traditionally applied, such as algebraic geometry or theoretical computer science. The only knowledge that is assumed from the reader is linear algebra. All concepts are explained by giving concrete examples from different, non-specialized areas of mathematics (such as basic group theory, graph theory, and probability). Not every example is helpful for every reader, but hopefully every reader can find at least one helpful example per concept. The reader is encouraged to read all the examples, this way they may even learn something new about a different field. Particular emphasis is given to the Yoneda lemma and its significance, with both intuitive explanations, detailed proofs, and specific examples. Another common theme in these notes is the relationship between categories and directed multigraphs, which is treated in detail. From the applied point of view, this shows why categorical thin
Applied category theory often studies symmetric monoidal categories (SMCs) whose morphisms represent open systems. These structures naturally accommodate complex wiring patterns, leveraging (co)monoidal structures for splitting and merging wires, or compact closed structures for feedback. A key example is the compact closed SMC of design problems (DP), which enables a compositional approach to co-design in engineering. However, in practice, the systems of interest may not be fully known. Recently, Markov categories have emerged as a powerful framework for modeling uncertain processes. In this work, we demonstrate how to integrate this perspective into the study of open systems while preserving consistency with the underlying SMC structure. To this end, we employ the change-of-base construction for enriched categories, replacing the morphisms of a symmetric monoidal $\mathcal{V}$-category $\mathcal{C}$ with parametric maps $A \to \mathcal{C}(X,Y)$ in a Markov category induced by a symmetric monoidal monad. This results in a symmetric monoidal 2-category $N_*\mathcal{C}$ with the same objects as $\mathcal{C}$ and reparametrization 2-cells. By choosing different monads, we capture var
Category theory provides a unified language for organizing composable operations in many disciplines. In disciplines where unitarity is fundamental -- such as functional analysis, quantum field theory, and quantum logic -- this language must also capture adjoints, leading to the notion of dagger categories. Higher category theory, which extends this framework to encode operations between operations, has recently become indispensable in both theoretical physics and pure mathematics. Finding a higher categorical analogue of a dagger category is therefore key to the foundations of quantum field theory. In this work, we present a coherent definition of \emph{dagger $(\infty,n)$-category} in terms of equivariance data trivialized on parts of the category. Our main example is the bordism $(\infty,n)$-category $\mathbf{Bord}_{n}^X$. This allows us to define (fully-local) \emph{reflection-positive topological quantum field theories} to be higher dagger functors out of $\mathbf{Bord}_{n}^X$.
Drazin inverses are a fundamental algebraic structure which have been extensively deployed in semigroup theory, ring theory, and matrix theory. Drazin inverses can also be defined for endomorphisms in any category. However, beyond a paper by Puystjens and Robinson from 1987, there has been almost no further development of Drazin inverses in category theory. Here we provide a survey of the theory of Drazin inverses from a categorical perspective. We introduce Drazin categories, in which every endomorphism has a Drazin inverse, and provide various examples including the category of matrices over a field, the category of finite length modules over a ring, and finite set enriched categories. We also introduce the notion of expressive rank and prove that a category with expressive rank is Drazin. Moreover, we not only study Drazin inverses in mere categories, but also in additive categories and dagger categories. In an arbitrary category, we show how a Drazin inverse corresponds to an isomorphism in the idempotent splitting, as well as explain how Drazin inverses relate to Leinster's notion of eventual image duality. In additive categories, we consider core-nilpotent decompositions, ima
Building on the notion of normed category as suggested by Lawvere, we introduce notions of Cauchy convergence and cocompleteness which differ from proposals in previous works. Key to our approach is to treat them consequentially as categories enriched in the monoidal-closed category of normed sets. Our notions largely lead to the anticipated outcomes when considering individual metric spaces as small normed categories, but they can be challenging when considering some large categories, like those of semi-normed or normed vector spaces and all linear maps, or of generalized metric spaces and all mappings. These are the key example categories discussed in detail in this paper. Working with a general commutative quantale V as a value recipient for norms, rather than only with Lawvere's quantale of the extended real half-line, we observe that the categorically atypical structure gap between objects and morphisms in the example categories is already present in the underlying normed category of the enriching category of V-normed sets. To show that this normed category and, in fact, all presheaf categories over it, are Cauchy cocomplete, we assume the quantale V to satisfy a couple of lig
Generalizing the theory of parity sheaves on complex algebraic stacks due to Juteau-Mautner-Williamson, we develop a theory of twisted equivariant parity sheaves. We use this formalism to construct a modular incarnation of Lusztig and Yun's monodromic Hecke category. We then give two applications: (1) a modular categorification of the monodromic Hecke algebra, and (2) a monoidal equivalence between the monodromic Hecke category of parity sheaves and the ordinary Hecke category of parity sheaves on the endoscopic group.