Brown Representability and Spaces over a Category

  title={Brown Representability and Spaces over a Category},
  author={No{\'e} B{\'a}rcenas},
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. 
The Completion Theorem in twisted equivariant $K$-Theory for proper and discrete actions
We compare different algebraic structures in twisted equivariant K-Theory for proper actions of discrete groups. After the construction of a module structure over untwisted equivariant K-Theory, we
External Spanier-Whitehead duality and homology representation theorems for diagram spaces
We construct a Spanier-Whitehead type duality functor relating finite $\mathcal{C}$-spectra to finite $\mathcal{C}^{\mathrm{op}}$-spectra and prove that every $\mathcal{C}$-homology theory is given


Equivariant cohomology with local coefficients
We show that for a discrete group G, the equivariant cohomology of a G-space X with G-local coefficients M is isomorphic to the Bredon-Illman cohomology of X with equivariant local coefficients M.
Representing Bredon cohomology with local coefficients
Equivariant Cohomological Chern Characters
  • W. Lück
  • Mathematics
    Int. J. Algebra Comput.
  • 2005
It is shown that the coefficients of the equivariant cohomology theory possess a Mackey structure, which is present in many interesting examples, and this structure can be assigned to CW-complexes.
Segal's Spectral Sequence in Twisted Equivariant K-theory for Proper and Discrete Actions
Abstract We use a spectral sequence developed by Graeme Segal in order to understand the twisted G-equivariant K-theory for proper and discrete actions. We show that the second page of this spectral
Steenrod's operations in simplicial Bredon-Illman cohomology with local coefficients
In this paper we use Peter May’s algebraic approach to Steenrod operations to construct Steenrod’s reduced power operations in simplicial Bredon-Illman cohomology with local coefficients of a one
The equivariant Serre spectral sequence
For spaces with a group action, we introduce Bredon cohomology with local (or twisted) coefficients and show that it is invariant under weak equivariant homotopy equivalence. We use this new
The Grothendieck duality theorem via Bousfield’s techniques and Brown representability
Grothendieck proved that if f: X ) Y is a proper morphism of nice schemes, then Rf* has a right adjoint, which is given as tensor product with the relative canonical bundle. The original proof was by
Categories for the Working Mathematician
I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large
Equivariant Homotopy and Cohomology Theory