We call a finitely complete category diexact if every difunctional relation admits a pushout which is stable under pullback and itself a pullback. We prove three results relating to diexact categories: firstly, that a category is a pretopos if and only if it is diexact with a strict initial object; secondly, that a category is diexact if and only if it is Barr-exact, and every pair of monomorph...

We show that K1(E) of an exact category E agrees with K1(DE) of the associated triangulated derivator DE. More generally we show that K1(W) of a Waldhausen category W with cylinders and a saturated class of weak equivalences agrees with K1(DW) of the associated right pointed derivator DW. Introduction For a long time there was an interest in defining a nice K-theory for triangulated categories ...

The notion of normal subobject having an intrinsic meaning in any proto-modular category, we introduce the notion of normal functor, namely left exact conservative functor which reeects normal subobjects. The point is that for the category Gp of groups the change of base functors, with respect to the bration of pointed objects, are not only conservative (this is the deenition of a protomodular ...

An algebraically exact category is one that admits all of the limits and colimits which every variety of algebras possesses and every forgetful functor between varieties preserves, and which verifies the same interactions between these limits and colimits as hold in any variety. Such categories were studied by Adámek, Lawvere and Rosický: they characterised them as the categories with small lim...

The general notion of what is now called Hall algebra or Ringel-Hall algebra is an algebra defined out of a certain type of category, a "finitary abelian category". Finitary abelian categories are, roughly speaking, categories where the notions of exact sequences make sense and such that for any pair of objects A,B in the category in question, we have # Hom(A,B) <∞ and # Ext1(A,B) <∞. The Hall ...

Exact indecomposable module categories over the tensor category of representations of Hopf algebras that are liftings of quantum linear spaces are classified.

Any category A can be embedded in its right completion A. When A is small and abelian, this completion A is AB5 and the embedding is exact.

The Hochschild and (cotriple) cyclic homologies of crossed modules of (notnecessarily-unital) associative algebras are investigated. Wodzicki’s excision theorem is extended for inclusion crossed modules in the category of crossed modules of algebras. The cyclic and cotriple cyclic homologies of crossed modules are compared in terms of long exact homology sequence, generalising the relative cycl...

In the context of Mal’cev categories, a left exact root for the congruence distributive property is given and investigated, namely the property that there is no non trivial internal group inside the fibres of the fibration of pointed objects. Indeed, when moreover the basic category C is Barr exact, the two previous properties are shown to be equivalent.

We develop some techniques to the study of exact module categories over some families of pointed finite-dimensional Hopf algebras. As an application we classify exact module categories over the tensor category of representations of the small quantum groups uq(sl2).