View source for 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 [[#References|[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 [[#References|[a3]]].

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

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G.E. Bredon,   "Equivariant cohomology theories" , ''Lecture Notes Math.'' , '''34''' , Springer  (1967)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  T. tom Dieck,   "Transformation groups and representation theory" , ''Lecture Notes Math.'' , '''766''' , Springer  (1979)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Lück,   "Tranformation groups and algebraic $K$-theory" , ''Lecture Notes Math.'' , '''1408''' , Springer  (1989)</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  J.P. May,   et al.,   "Equivariant homotopy and cohomology theory" , ''Regional Conf. Ser. Math.'' , '''91''' , Amer. Math. Soc.  (1996)</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  I. Moerdijk,   J.A. Svensson,   "The equivariant Serre spectral sequence"  ''Proc. Amer. Math. Soc.'' , '''118'''  (1993)  pp. 263–278</TD></TR>
</table>

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