Flat objects of a finitely accessible additive category C are described in terms of some objects of the associated functor category of C, called strongly flat functors. We study closure properties of the class of strongly flat functors, and we use them to deduce the known result that every object of a finitely accessible abelian category has a flat cover.

For any finite group G, we define a bifunctor from the Dress category of finite G-sets to the conjugation biset category, whose objects are subgroups of G, and whose morphisms are generated by certain bifree bisets. Any additive functor from the conjugation biset category to abelian groups yields a Mackey functor by composition. We characterize the Mackey functors which arise in this way.

For any finite groupG, we define a bivariant functor from the Dress category of finite G-sets to the conjugation biset category, whose objects are subgroups of G, and whose morphisms are generated by certain bifree bisets. Any additive functor from the conjugation biset category to abelian groups yields a Mackey functor by composition. We characterize the Mackey functors which arise in this way.

This paper consists of three results on Frobenius categories: (1)we give sufficient conditions on when a factor category of a Frobenius category is still a Frobenius category; (2) we show that any Frobenius category is equivalent to an extension-closed exact subcategory of the Frobenius category formed by Cohen–Macaulaymodules over some additive category; this is an analogue of Gabriel–Quillen’...

We show that, in any Mal’tsev (and a fortiori protomodular) category E, not only the fibre GrdXE of internal groupoids above the object X is a naturally Mal’tsev category, but moreover it shares with the category Ab of abelian groups the property following which the domain of any split epimorphism is isomorphic with the direct sum of its codomain with its kernel. This allows us to point at a ne...

In this note we study several ways of turning a small category into a ringoid. We refer to such processes as linearisation, for lack of a better name. Although this material is very abstract, our main application is concrete: the category Mod(X) of sheaves of modules over a scheme X can be viewed as a localisation of a category of modules over a ringoid, very naturally constructed from the stru...

Quasi-abelian categories are abundant in functional analysis and representation theory. It is known that a quasi-abelian category $\mathcal{E}$ cotilting torsionfree class of an abelian category. In fact, this property characterizes categories. This ambient derived equivalent to the $\mathcal{E}$, can be constructed as heart $\mathcal{LH}(\mathcal{E})$ $t$-structure on bounded $\mathbf{D}^b(\ma...

We describe a new family of topic-ranking algorithms for multi-labeled documents. The motivation for the algorithms stem from recent advances in online learning algorithms. The algorithms are simple to implement and are also time and memory efficient. We provide a unified analysis of the family of algorithms in the mistake bound model. We then discuss experiments with the proposed family of top...

According to a currently popular principle of classification, mathematics is the study of various "categories. " A category consists of certain "objects" (e.g., groups, topological spaces) and certain "mappings" (e.g., homomorphisms, continuous functions). One possible category has measure spaces for its objects and, correspondingly, measure-preserving transformations for its mappings. The usua...

We prove that any stable, additive, combinatorial model category M has a canonical model enrichment over Sp(sAb) (symmetric spectra based on simplicial abelian groups). So to any object X ∈ M one can attach an endomorphism ring object in this category, denoted hEndad(X). Since the homotopy theory of ring objects in Sp(sAb) is equivalent to the homotopy theory of differential graded algebras, on...