Brown Representability and Spaces over a Category

Abstract

We prove a Brown Representability Theorem in the context of spaces over a category. We discuss two applications to the representability of equivariant cohomology theories, with emphasis on Bredon cohomology with local coefficients.

Computations in $$C_{pq}$$ C pq -Bredon cohomology

“…While the differentials of the equivariant Atiyah -Hirzebruch spectral sequence are a natural transformation, and even in the parametrized setting they are identified by Theorem 5.7 in [5] in homotopy theoretical terms as maps between classifying spectra for Bredon cohomology, and there exist constructions of the cohomology operations in [24], it has not been possible to give a complete list of candidates for the relevant cohomological operations between Bredon cohomology groups.…”

External Spanier-Whitehead duality and homology representation theorems for diagram spaces

“…If Y is a C-CW-complex and E is fibrant, the R can be omitted. The fact that every C-cohomology theory has this form, i. e. the generalisation of the classical Brown Representability Theorem, may be obtained by mimicking its original proof [Bár14,Lac16], or by citing a theorem of Neeman again [Nee01,Thm. 8.3.3].…”

“…Neeman [Nee01] has vastly generalised this argument to a triangulated category setup that is sufficient to treat the case of C-cohomology theories. Specific references for the case of C-spaces are [Bár14,Lac16]. The classical strategy for deducing the homological Theorem A from the cohomological one is the following: Use Spanier-Whitehead duality to switch between cohomology and homology, and then use Adams' version of Brown's representability theorem to deal with the arising difficulty that the duality functor is only defined on finite spectra.…”

External Spanier–Whitehead duality and homology representation theorems for diagram spaces

“…Neeman [42] has vastly generalized this argument to a triangulated category setup that is sufficient to treat the case of C-cohomology theories. Specific references for the case of C-spaces are Bárcenas [1] and Lackmann [27].…”

“…Theorem D Suppose that C D Or.G; F/ where G is finite and all members of the family F are cyclic of prime power order. 1 Then a Chern character exists for every 2 The closed bicategory DerMod.Sp O /…”