We prove that any rigid additive symmetric monoidal category can be mapped to a abelian in universal way. This yields novel approach Grothendieck's standard conjecture D and Voevodsky's smash nilpotence conjecture.

Any semi-abelian category A appears, via the discrete functor, as a full replete reflective subcategory of the semi-abelian category of internal groupoids in A. This allows one to study the homology of n-fold internal groupoids with coefficients in a semi-abelian category A, and to compute explicit higher Hopf formulae. The crucial concept making such computations possible is the notion of prot...

We investigate the abelian category which is the target of intersection homology. Recall that, given a stratified space X, we get intersection homology groups IHnX depending on the choice of an n-perversity p. The n-perversities form a lattice, and we can think of IHnX as a functor from this lattice to abelian groups, or more generally R-modules. Such perverse R-modules form a closed symmetric ...

Given a cohomological functor from a triangulated category to an abelian category, we construct under appropriate assumptions for any localization functor of the abelian category a lift to a localization functor of the triangulated category. This discussion of Bousfield localization is combined with a basic introduction to the concept of localization for arbitrary categories.

Abelian categories are the most general category in which one can develop homological algebra. The idea and the name “abelian category” were first introduced by MacLane [Mac50], but the modern axiomitisation and first substantial applications were given by Grothendieck in his famous Tohoku paper [Gro57]. This paper was motivated by the needs of algebraic geometry, where the category of sheaves ...