We study topological properties of the correspondence of prime spectra associated to a noncommutative ring homomorphism R → S. Our main result provides criteria for the adjointness of certain functors between the categories of Zariski closed subsets of SpecR and SpecS; these functors arise naturally from restriction and extension of scalars. When R and S are left noetherian, adjointness occurs ...

We show that if U∗ is a hypercover of a topological space X then the natural map hocolimU∗ → X is a weak equivalence. This fact is used to construct topological realization functors for the A-homotopy theory of schemes over real and complex fields.

Weak factorization systems, important in homotopy theory, are related to injective objects in comma{categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not coobrantly generated. We also present a weak factorization system on the category of posets which is not coobrantly generated. No s...

14. Diagrams of spaces 68 14.1. The geometric realization and total singular complex functors 68 14.2. Filtered diagrams of simplicial sets 70 14.3. Commuting hocolim and holim: simplicial sets 72 14.4. Commuting hocolim and holim: topological spaces 74 References 75

This note establishes internal criteria on a category C and a separator 2 in C which characterize the condition that the 2-induced covariant hom-functor /i2: C -> Set is (epi, mono-source)-topological. Introduction. Hoffmann [4] showed how topological functors (of Herrlich [2]) may be recovered from factorizations of sources and sinks in the domain category. This process was extended to (F, M)-...

Mackey functors are a framework having the common properties of many natural constructions for finite groups, such as group cohomology, representation rings, the Bumside ring, the topological K-theory of classifying spaces, the algebraic K-theory of group rings, the Witt rings of Galois extensions, etc. In this work we first show that the Mackey functors for a group may be identified with the m...