Bredon theory of modules over a category

A module over a small category $\Gamma$ is a contravariant functor from $\Gamma$ to the category of $R$-modules, for some commutative ring $R$ (cf. also Module). Taking morphisms to be natural transformations, the category of modules over $\Gamma$ is an Abelian category, so one can do homological algebra with these objects.

G.E. Bredon introduced modules over a category in [a1] in order to study obstruction theory for $G$-spaces. If $G$ is a finite group, let $\mathcal{O}_G$ be the category of orbits $G/H$ and $G$-mappings between them. $\mathbf{Z}$-modules (Abelian groups) over $\mathcal{O}_G$ play the role of the coefficients in Bredon's ($\mathbf{Z}$-graded) equivariant ordinary cohomology theories. The equivariant ordinary cohomology of a $G$-CW-complex can be computed using the equivariant chain complex of the space, which is a chain complex in the category of $\mathbf{Z}$-modules over $\mathcal{O}_G$. Similarly, equivariant local cohomology can be described using modules over a category depending on the space in question.

For $RO(G)$-graded equivariant ordinary cohomology one has to replace the orbit category $\mathcal{O}_G$ with the stable orbit category, in which the morphisms are the stable mappings between orbits, stabilization being by all representations of $G$. A module over the stable orbit category (i.e., an additive contravariant functor) is equivalent to a Mackey functor as studied by A.W.M. Dress in [a3].

For later examples of the use of modules over a category in equivariant algebraic topology, see [a4] and [a6]. In particular, [a4] includes a discussion of the elementary algebra of modules over a category. For examples involving Mackey functors, see [a2] and [a5].

References

[a1]	G.E. Bredon, "Equivariant cohomology theories" , Lecture Notes Math. , 34 , Springer (1967)
[a2]	T. tom Dieck, "Transformation groups and representation theory" , Lecture Notes Math. , 766 , Springer (1979)
[a3]	A.W.M. Dress, "Contributions to the theory of induced representations" , Algebraic $K$-theory, II (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) , Lecture Notes Math. , 342 , Springer (1973) pp. 183–240
[a4]	W. Lück, "Tranformation groups and algebraic $K$-theory" , Lecture Notes Math. , 1408 , Springer (1989)
[a5]	J.P. May, et al., "Equivariant homotopy and cohomology theory" , Regional Conf. Ser. Math. , 91 , Amer. Math. Soc. (1996)
[a6]	I. Moerdijk, J.A. Svensson, "The equivariant Serre spectral sequence" Proc. Amer. Math. Soc. , 118 (1993) pp. 263–278