The object of this paper is to prove that the standard categories in which homotopy theory is done, such as topological spaces, simplicial sets, chain complexes of abelian groups, and any of the various good models for spectra, are all homotopically self-contained. The left half of this statement essentially means that any functor that looks like it could be a tensor product (or product, or sma...

Past studies of the Brauer group of a scheme tells us the importance of the interrelationship among Brauer groups of its finite étale coverings. In this paper, we consider these groups simultaneously, and construct an integrated object “BrauerMackey functor”. We realize this as a cohomological Mackey functor on the Galois category of finite étale coverings. For any finite étale covering of sche...

Namely, the assumption implies that for every prime p, one has induced isomorphisms H∗>n(BH;Z/p) → H∗>n(BG;Z/p) and thus, by applying the “reconstruction functor”, isomorphisms H∗(BH;Z/p)→ H∗(BG;Z/p). Since BG and BH are spaces of finite type, it follows that the induced map H∗(BH;Z)→ H∗(BG;Z) is an isomorphism too. Thus, by Jackowski [6] and Minami [10], ρ is an isomorphism of Lie groups. In s...

Given a cohomological functor from a triangulated category to an abelian category, we construct under appropriate assumptions for any localization functor of the abelian category a lift to a localization functor of the triangulated category. This discussion of Bousfield localization is combined with a basic introduction to the concept of localization for arbitrary categories.