Simplicial and Categorical Diagrams, and Their Equivariant Applications

We show that the homotopy category of simplicial diagrams I-SS indexed by a small category I is equivalent to a homotopy category of SS ↓ NI simplicial sets over the nerve NI. Then their equivalences, by means of the nerve functor N : Cat → SS from the category Cat of small categories, with respective homotopy categories associated to Cat are established. Consequently, an equivariant simplicial version of the Whitehead Theorem is derived. In his remarkable paper [14], Thomason shows the equivalence of the homotopy categories of Cat, the category of small categories, and SS, the category of simplicial sets, by means of the nerve functor N : Cat → SS and one of its homotopy inverses (see [7, 9] for details). By [11], the homotopy structure on Cat induces, for every small category I, homotopy structures on the category Cat ↓ I of small categories over I and the category I-Cat of contravariant functors from I to Cat. From [10], it follows that there is a pair of adjoint functors D : Cat ↓ I → I-Cat and I ∫ : I-Cat → Cat ↓ I which establishes an equivalence of respective homotopy categories. Similarly, by [5] the homotopy category of simplicial sets on which a fixed simplicial group G acts is equivalent to the homotopy category of simplicial sets over the classifying complex WG. From this it follows the wellknown fact that the homotopy category of topological spaces on which a fixed discrete group G acts is equivalent to the homotopy category of spaces over the classifying space K(G, 1). We were influenced by these papers to search for a link between the comma category SS ↓ N I and the category I-SS of contravariant functors from I to SS. In Section 1 we define, by means of [3, p.327], a pair of adjoint functors A : SS ↓ N I → I-SS and B : I-SS → SS ↓ N I, and examine in Proposition 1.4 their homotopy properties. Let N : Cat ↓ I → SS ↓ N I and Ñ : I-Cat → I-SS be the associated functors to the nerve one N : Cat → SS. Then by Theorem 1.5, from the diagram of functors