Cohomology of Groups in Arbitrary Categories

Cohomology of Groups in Arbitrary Categories

Tate Cohomology for Arbitrary Groups via Satellites

We define cohomology groups Ĥn(G;M), n ∈ Z, for an arbitrary group G and G-module M , using the concept of satellites. These cohomology groups generalize the Farrell-Tate groups for groups of finite virtual cohomological dimension and form a connected sequence of functors, characterized by a natural universal property. The classical Tate cohomology groups of finite groups have been generalized ...

Cohomology in Tensored Categories

Both [Mac Lane 1965, Beck 1967] have recently defined cohomology theories for algebras in abstract categorial setting. Mac Lane’s theory is an abstract formaization of the (normalised) bar construction (see [Mac Lane 1963, p. 144] for example). Since, however, his proof of the normalization theorem (p. 236) remains perfectly valid, the normalized bar construction can be replaced by the un-norma...

Digital cohomology groups of certain minimal surfaces

In this study, we compute simplicial cohomology groups with different coefficients of a connected sum of certain minimal simple surfaces by using the universal coefficient theorem for cohomology groups. The method used in this paper is a different way to compute digital cohomology groups of minimal simple surfaces. We also prove some theorems related to degree properties of a map on digital sph...

Cohomology Theory in 2-categories

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 ...