G-functors, G-posets and homotopy decompositions of G-spaces

Abstract

Abstract. 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| 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 W d ; in some important cases they vanish in dimensions greater than the length of W and can be explicitly calculated in low dimensions. We prove a cofinality theorem for functors F : C → O(G) into the category of G-orbits which guarantees that the associated map α F : hocolim C EG × G F (−) → BG is a mod-p-homology decomposition.

Representations of categories and their applications

“…In this case, Theorem 1.7 becomes a result of Jackowski and Słomińska [22]. The representation theory of an EI-category C enables us to describe projective resolutions of RC-modules, and hence leads us to some other interesting results.…”

“…We note that Jackowski and Słomińska introduced the EI-categories with quotients in their paper [22], which is a concept dual to the EI-categories with subobjects in the sense that if (C, I) is a category with subobjects then (C op , I op ) is a category with quotients, and vice versa. All results in this section have their counterparts for EI-categories with quotients.…”

“…Proof. Jackowski-Słomińska [22] proved the morphisms in any EI-category with quotients are epimorphism. Thus all the morphisms in (C, I) are monomorphisms.…”

“…Let [y] be the isomorphism class of an object y ∈ Ob C. This preorder induces a partial order on the set Is C of isomorphism classes of Ob C (specified by [y] [x] if and only if Hom C (y, x) = ∅), which plays an important role in studying representations and cohomology of EI-categories. Because of the existence of an order for the isomorphism classes of objects in any EI-category, EI-categories are sometimes referred to as "ordered categories" by some authors, see Oliver [28] and Jackowski-Słomińska [22]. For any EI-category C, any subcategory D and any object x ∈ Ob C, we can define a full subcategory D x ⊂ D consisting of all y ∈ Ob D such that y x, or equivalently Hom D (y, x) = ∅.…”

Higher Limits via Subgroup Complexes

“…I am grateful to S. Jackowski for informing me of her unpublished work and for providing me with a copy. After the submission of the present paper Jackowski and S lomińska have finished the paper [43] which includes S lomińska's unpublished results.…”