# cofibrations in $O_G$-spaces

For a finite group $G$, let $O_G$ denote the orbit category of $G$. Is there a explicit/nice description of cofibrations in the functor category $Top^{O_G^{op}}$ where the weak equivalences and fibrations are defined objectwise?

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I guess you mean, other than by lifting? – David White Jan 18 '12 at 23:39
You can give generating cofibrations, starting with the usual generating cofibrations $S^{n−1}\to D^n$ for $Top$ and applying to these the functors $Top\to Top^{O^{op}_G}$ that are left adjoint to the evaluation functors at the various objects. Calling these the cells, then a cofibration is any retract of a cellular inclusion. – Tom Goodwillie Jan 19 '12 at 1:11
Thank you Prof. Goodwillie for the response. – sen-deba Jan 20 '12 at 23:08
@David: yes; some other useful criterion to check for a map to be cofibration. – sen-deba Jan 20 '12 at 23:09

Tom's comment gives a generic answer that applies to $Top^\mathcal{J}$ for any small topological category $\mathcal{J}$. For the orbit category $\mathcal{O}_G$, where $G$ is any discrete group (not necessarily finite), something special and much nicer happens. We have the fixed point functor $\Phi$ from $G$-spaces to $\mathcal{O}_G$-spaces. It sends $X$ to the functor on the orbit category that sends $G/H$ to $X^H$. The functor $\Phi$ actually specifies an isomorphism from the category of $G$-cell complexes (or the category of $G$-CW complexes) to the category of cell diagrams or (CW diagrams) in the diagram category of $\mathcal{O}_G$-spaces. Modulo a few obvious typos, the easy proof is given on page 56 of Equivariant Homotopy and Cohomology Theory (by some of my friends and me), CBMS Regional Conference Series in Mathematics Number 91, AMS 1996. That book is posted on my web page (http://math.uchicago.edu/~may/). The cofibrations are the retracts of the relative cell diagrams.