Related papers: Categories of partial equivalence relations as loc…
We formulate a connection between a topological and a geometric category. The former is the idempotent completion of the (horizontal) trace of the affine Hecke category, while the latter is the equivariant derived category of the…
Mal'tsev categories turned out to be a central concept in categorical algebra. On one hand, the simplicity and the beauty of the notion is revealed through a lot of characterizations of different flavour. Depending on the context, one can…
We study a number of categorical quasi-uniform structures induced by functors. We depart from a category $\mathcal{C}$ with a proper $(\mathcal{E}, \mathcal{M})$-factorization system, then define the continuity of a $\mathcal{C}$-morphism…
In this paper we generalise the notion of linearity (in the sense of Lawvere) to a category C equipped with a compatible sum structure and product structure. In this context, any morphism f from an n-fold sum to an n-fold product has a…
It is a deep fact that the homotopy classification of topological manifolds is convariantly functorial. In other words, a map from a topological manifold M to another N naturally induces a map from the structure set S(M) to S(N). We extend…
Given a homological epimorphism $\pi:\mathcal{C}\longrightarrow \mathcal{C}/\mathcal{I}$ between $K$-categories, we show that if the ideal $\mathcal{I}$ satisfies certain conditions, then there exists an equivalence between the singularity…
Let l be a commutative ring with unit. Garkusha constructed a functor from the category of l-algebras into a triangulated category D, that is a universal excisive and homotopy invariant homology theory. Later on, he provided different…
A category of FI type is one which is sufficiently similar to finite sets and injections so as to admit nice representation stability results. Several common examples admit a Grothendieck fibration to finite sets and injections. We begin by…
In a previous work, by extending the classical Quillen construction to the non-simply connected case, we have built a pair of adjoint functors, 'model' and 'realization', between the categories of simplicial sets and complete differential…
Let $F\colon \mathcal{C} \to \mathcal{E}$ be a functor from a category $\mathcal{C}$ to a homological (Borceux-Bourn) or semi-abelian (Janelidze-M\'arki-Tholen) category $\mathcal{E}$. We investigate conditions under which the homology of…
We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial…
The question of when the derived category of a ring satisfies Brown--Adams representability is revisited via studying the transfer of pure homological dimension along definable functors: it is shown that, for any ring, the pure global…
Let $X$ be a $G$-space. In this paper, we introduce the notion of sectional category with respect to $G$. As a result, we obtain $G$-homotopy invariants: the LS category with respect to $G$, the sequential topological complexity with…
In this note we define fibrations of topological stacks and establish their main properties. We prove various standard results about fibrations (fiber homotopy exact sequence, Leray-Serre and Eilenberg-Moore spectral sequences, etc.). We…
In this paper, we investigate the property (P) that finite products commute with arbitrary coequalizers in pointed categories. Examples of such categories include any regular unital or (pointed) majority category with coequalizers, as well…
In "Frobenius Categories versus Brauer Blocks" we have proved some universality of the so-called localizing functor associated with a Frobenius $P$-category $F$, where $P$ is a finite $p$-group, with respect to the coherent $F$-localities…
Given any pointed CW complex (X,x), it is well known that the fondamental group of X pointed at x is naturally isomorphic to the automorphism group of the functor which associates to a locally constant sheaf on X its fibre at x. The purpose…
Let G be a reductive groups over an algebraically closed field k. Let P^{(i)} be associated parabolic subgroups, and X^{(i)}:=T^*G/P^i. The bounded derived categories of coherent sheaves on X^{(i)} are equivalent, but there is no canonical…
We develop a representation theory of categories as a means to explore characteristic structures in algebra. Characteristic structures play a critical role in isomorphism testing of groups and algebras, and their construction and…
We study Verdier quotients of diverse homotopy categories of a full additive subcategory $\mathcal E$ of an abelian category. In particular, we consider the categories $K^{x,y}({\mathcal E})$ for $x\in\{\infty, +,-,b\}$, and…