Related papers: Categories internal to crossed modules
We explain how the computation of induced crossed modules allows the computation of certain homotopy 2-types and, in particular, second homotopy groups. We discuss various issues involved in computing induced crossed modules and give some…
We give a criterion for cohomological symmetry in a triangulated category. As an application, we show that such cohomological symmetry holds for all pairs of modules over any exterior algebra.
Since categories are graphs with additional "structure", one should start from fuzzy graphs in order to define a theory of fuzzy categories. Thus is makes sense to introduce categories whose morphisms are associated with a plausibility…
Some aspects of basic category theory are developed in a finitely complete category $\C$, endowed with two factorization systems which determine the same discrete objects and are linked by a simple reciprocal stability law. Resting on this…
We give a detailed proof of the following fundamental result: the singularity category of a ring is triangle equivalent to the stabilization of its stable module category. The result yields singular equivalences between rings of different…
We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but instead of having a single collection of "objects of D" with a map to the…
We obtain a classification of metaplectic modular categories: every metaplectic modular category is a gauging of the particle-hole symmetry of a cyclic modular category. Our classification suggests a conjecture that every weakly-integral…
We study derived categories of coherent sheaves on abelian varieties. We give a criterion for the equivalence of the derived categories on two abelian varieties. We describe the autoequivalence group for the derived category of coherent…
For a commutative ring $S$ and self-orthogonal subcategory $\mathsf{C}$ of $\mathsf{Mod}(S)$, we consider matrix factorizations whose modules belong to $\mathsf{C}$. Let $f\in S$ be a regular element. If $f$ is $M$-regular for every $M\in…
We propose a framework for producing interesting subcategories of the category ${}_A\mathsf{Mod}$ of left $A$-modules, where $A$ is an associative algebra over a field $k$. The construction is based on the composition, $Y$, of the Yoneda…
We prove a coherence theorem for actions of groups on monoidal categories. As an application we prove coherence for arbitrary braided $G$-crossed categories.
This article deals with a relationship between derived categories of modules over some partially ordered sets and triangulated categories arising from quasi-homogeneous isolated singularities. It produces heuristics for the existence of…
We classify module categories over the category of representations of quantum $SL(2)$ in a case when $q$ is not a root of unity. In a case when $q$ is a root of unity we classify module categories over the semisimple subquotient of the same…
We survey the extensions of a group by a group using crossed products instead of exact sequences of groups. The approach has various advantages, one of them being that the crossed product is an universal object. Several new applications are…
Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previously known results. We describe the close…
A crossed module is (A,H,d,\la) where d:A\to H is a homomorphism of groups and H acts on A, with conditions leading to a groupoid A\lcross H{\to\atop \to}H as an example of a strict 2-group. We give the corresponding notion of a quantum…
We establish a Dwyer-Kan equivalence of relative categories of combinatorial model categories, presentable quasicategories, and other models for locally presentable (infinity,1)-categories. This implies that the underlying quasicategories…
Normal monomorphisms in the sense of Bourn describe the equivalence classes of an internal equivalence relation. Although the definition is given in the fairly general setting of a category with finite limits, later investigations on this…
Recollements of triangulated categories may be seen as exact sequences of such categories. Iterated recollements of triangulated categories are analogues of geometric or topological stratifications and of composition series of algebraic…
A non-self-contained gathering of notes on category theory, including the definition of locally cartesian closed category, of the cartesian structure in slice categories, or of the pseudo-cartesian structure on Eilenberg-Moore categories.…