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In this note we show that the main results of the paper [PR] can be obtained as consequences of more general results concerning categories whose morphisms can be uniquely presented as compositions of morphisms of their two subcategories with the same objects. First we will prove these general results and then we will apply it to the case of finite noncommutative sets.

We construct a Hecke structure on equivariant Bredon cohomology with local coefficients and then describe some of its properties. We compare this structure with the Mackey structure defined by T. tom Dieck and with the Illman transfer.

Let p be a prime number. We prove that if G is a compact Lie group with a non-trivial p-subgroup, then the orbit space $\left(B_p\left(G\right)\right)/G$ of the classifying space of the category associated to the G-poset ${}_p\left(G\right)$ of all non-trivial elementary abelian p-subgroups of G is contractible. This gives, for every G-CW-complex X each of whose isotropy groups contains a non-trivial p-subgroup, a decomposition of X/G as a homotopy colimit of the functor $X^{Eₙ}/(NE₀\cap ...\cap NEₙ)$ defined over the poset $\left(s{d}_p\left(G\right)\right)/G$, where sd is the barycentric subdivision. We also...

Conditions which imply Morita equivalences of functor categories are described. As an application a Dold-Kan type theorem for functors defined on a category associated to associative algebras with one-side units is proved.

The homology theory of colored posets, defined by B. Everitt and P. Turner, is generalized. Two graph categories are defined and Khovanov type graph cohomology are interpreted as Ext* groups in functor categories associated to these categories. The connection, described by J. H. Przytycki, between the Hochschild homology of an algebra and the graph cohomology, defined for the same algebra and a cyclic graph, is explained from the point of view of homological algebra in functor categories.

We describe a unifying approach to a variety of homotopy decompositions of classifying spaces, mainly of finite groups. For a group G acting on a poset W and an isotropy presheaf d:W → (G) we construct a natural G-map $hocoli{m}_{{}_d}G/d(-)\to \left|W\right|$ which is a (non-equivariant) homotopy equivalence, hence $hocoli{m}_{{}_d}EG{\times }_G{F}_d\to EG{\times }_G\left|W\right|$ is a homotopy equivalence. Different choices of G-posets and isotropy presheaves on them lead to homotopy decompositions of classifying spaces. We analyze higher limits over the categories associated to isotropy presheaves...