Related papers: Archimedean Type Conditions in Categories

Let $(F,\le)$ be an ordered field and let $A,B$ be square matrices over $F$ of the same size. We say that $A$ and $B$ belong to the same archimedean class if there exists an integer $r$ such that the matrices $r A^T A-B^T B$ and $r B^T…

The usual notion of set-convexity, valid in the classical Euclidean context, metamorphoses into several distinct convexity types in the more general Riemannian setting. By studying this phenomenon in reverse, we characterize complete…

We give a necessary and sufficient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category ${\mathcal C}$ to be equivalent. This concludes the classification of such module…

The recent trend in mathematics is towards a framework of abstract mathematical objects, rather than the more concrete approach of explicitly defining elements which objects were thought to consist of. A natural question to raise is whether…

In this paper, we have studied the axiomatics of {\it Ann-categories} and {\it categorical rings.} These are the categories with distributivity constraints whose axiomatics are similar with those of ring structures. The main result we have…

We revisit the definition of Cartesian differential categories, showing that a slightly more general version is useful for a number of reasons. As one application, we show that these general differential categories are comonadic over…

For any length category, we establish a set of rules (necessary and sufficient) that ensure a partial order on the isomorphism classes of simple objects such that the category is equivalent to the category of finite dimensional…

We use double categories to obtain a single theorem characterizing certain exponentiable morphisms of small categories, topological spaces, locales, and posets.

We define the notion of 1-affineness for a prestack, and prove an array of results that establish 1-affineness of certain types of prestacks.

We define a notion of "theory of (1,infty)-categories", and we prove that such a theory is unique up to equivalence.

Working over an arbitrary field, we define compact semisimple 2-categories, and show that every compact semisimple 2-category is equivalent to the 2-category of separable module 1-categories over a finite semisimple tensor 1-category. Then,…

We define the notion of an enriched Reedy category, and show that if A is a C-Reedy category for some symmetric monoidal model category C and M is a C-model category, the category of C-functors and C-natural transformations from A to M is…

We prove a categorical duality between a class of abstract algebras of partial functions and a class of (small) topological categories. The algebras are the isomorphs of collections of partial functions closed under the operations of…

A categoricity theorem is established for patterns of resemblance of order 2 showing that the order in which patterns arise in a wide range of hierarchies is the same.

This is mostly an overview. Given finitely presentable abelian categories $A$ and $B$, we sketch the construction of an abelian category of continuous functors from $A$ to $B$ that has nice $2$-categorical behaviour and gives an explicit…

The categorified theories known as "doctrines" specify a category equipped with extra structure, analogous to how ordinary theories specify a set with extra structure. We introduce a new framework for doctrines based on double category…

There are different categorizations of the definition of a {\it ring} such as {\it Ann-category} (see N. T. Quang [6]), {\it ring category} (see M. Kapranov and V.Voevodsky [2]),... The main result of this paper is to prove that every axiom…

We classify the "quotients" of a tannakian category in which the objects of a tannakian subcategory become trivial, and we examine the properties of such quotient categories.

We consider limits over categories of extensions and show how certain well-known functors on the category of groups turn out as such limits. We also discuss higher (or derived) limits over categories of extensions.

We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of $\R^n$ in a way that is completely algebraic.…