Auslander’s formula shows that any abelian category C is equivalent to the category of coherent functors on C modulo the Serre subcategory of all effaceable functors. We establish a derived version of this equivalence. This amounts to showing that the homotopy category of injective objects of some appropriate Grothendieck abelian category (the category of ind-objects of C) is compactly generate...

This is an account of three 1-hour lectures given at the Instructional Conference on Representation Theory of Algebraic Groups and Related Finite Groups, Isaac Newton Institute, Cambridge, 6–11 January 1997. In section 1, we define abelian categories following Grothendieck [12]. We then characterize module categories among abelian categories. Finally we sketch a proof of Mitchell’s full embeddi...

The category of symmetric quandles is a Mal’tsev variety whose subvariety of abelian symmetric quandles is the category of abelian algebras. We give an algebraic description of the quandle extensions that are central for the adjunction between the variety of quandles and its subvariety of abelian symmetric quandles.

For a self-small abelian group A of torsion-free rank 1, we characterize A-reflexive abelian groups which are induced by the contravariant functor Hom(−, A) in two cases: the range of Hom(−, A) is the category of all abelian groups, respectively the range of Hom(−, A) is the category of all left E-modules, where E is the endomorphism ring of A.

We study the question of the validity of the Snake Lemma in a P-semi-abelian category. We also obtain a generalization of the Snake Lemma in a quasi-abelian category. Mathematics Subject Classification: 18A20, 18E05.

The theory of abelian categories proved very useful, providing an ax-iomatic framework for homology and cohomology of modules over a ring (in particular, abelian groups) [5]. A similar framework has been lacking for non-abelian (co)homology, the subject of which includes the categories of groups and Lie algebras etc. The point of my thesis is that semi-abelian categories (in the sense of Janeli...

We show that every semi-abelian category, as defined by Palamodov, possesses a maximal exact structure in the sense of Quillen and that the exact structure of a quasi-abelian category is a special case thereof.

Two different definitions of a chain theory K∗ lead to the same class of derived homology theories h∗( ). On the category of CW pairs, these are those homology theories E∗ admitting a classifying simplicial abelian group spectrum E. So one has a functor from the category of chain theories into the category of simplicial abelian group spectra. 2000 Mathematics Subject Classification: 55U15, 55U1...

MV-algebras were introduced by Chang as an algebraic counterpart of the Lukasiewicz infinite-valued logic. D. Mundici proved that the category of MV-algebras is equivalent to the category of abelian l-groups with strong unit. A. Di Nola and A. Lettieri established a categorical equivalence between the category of perfect MV-algebras and the category of abelian l-groups. In this paper we investi...

Deligne has shown that there is an equivalence from the category of ordinary abelian varieties over a finite field A: to a category of Z-modules with additional structure. We translate several geometric notions, including that of a polarization, into Deligne's category of Z-modules. We use Deligne's equivalence to characterize the finite group schemes over k that occur as kernels of polarizatio...