Cohomology theory in 2-categories

Hiroyuki Nakaoka

Recently, symmetric categorical groups are used for the study of the Brauer groups of symmetric monoidal categories. As a part of these efforts, some algebraic structures of the 2-category of symmetric categorical groups SCG are being investigated. In this paper, we consider a 2-categorical analogue of an abelian category, in such a way that it contains SCG as an example. As a main theorem, we construct a long cohomology 2-exact sequence from any extension of complexes in such a 2-category. Our axiomatic and self-dual definition will enable us to simplify the proofs, by analogy with abelian categories.

Keywords: symmetric categorical group, 2-category, cohomology, exact sequence

2000 MSC: 18D05

Theory and Applications of Categories, Vol. 20, 2008, No. 16, pp 543-604.

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