Difference between revisions of "Bredon theory of modules over a category"
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− | A module over a small 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.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 | + | 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]]]. | 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==== | ====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 | + | <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> | ||
+ | |||
+ | {{TEX|done}} |
Latest revision as of 18:52, 28 October 2016
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 |
Bredon theory of modules over a category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bredon_theory_of_modules_over_a_category&oldid=39510