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Let $M$ be a (discrete) monoid acting on a space $X.$ We may take the quotient of $X,$ by this action, $X/M$ that is the coequalizer of the action map $M \times X \to X$ against the projection map. $X$ with its action can also be encoded as a functor $$F_X:M \to Top$$ with $F_X(*)=X,$ and $X/M$ is simply the colimit of this functor. One may also take the homotopy colimit of $F_X$ and call this the homotopy quotient of $X$ by its action, $X//_hM$. (Of course, an internal version of this can be used for non-discrete topological monoids, however the discrete case is what I care about right now) By a theorem of Thomason, we have that the homotopy quotient of the $1$-point space by its trivial $M$-action, $pt//_hM$ i.e. the homotopy colimit of the constant functor $$M \to Top$$ with value $pt,$ is $BM$- the classifying space of $M.$ When $M=G$ is in fact a group, this fact is well known and can be proven by replacing $pt$ with $EG$ with its free principle $G$-action and taking the ordinary quotient to obtain $BG$. Moreover, when $M=G$ is a group, $X//_hG$ can be modelled by a Borel construction.

Question: Is there a model for $X//_hM$ by a Borel construction for monoids? If so, how close to the story for groups is this?

Question: Has the case of homotopy quotients by monoid actions been well studied? (Can someone point to some references if so?) Has, e.g., equivariant cohomology been developed to calculate cohomology of $X//_hM$ in terms of the action $F_X$?

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This probably depends on your definition of homotopy colimit, but it you mean ``the geometric realisation of the simplicial replacement" then it seems to me that $X /\ \!/_h M$ is homeomorphic to $X \times_M EM$, where $EM$ is the simplicial model (i.e. the nerve of $M \wr M$). The main difference from the case of groups is that $\pi : EM \to BM := *\times_M EM$ is not anything like a bundle.

Homotopy quotients by monoid actions play a central role in "group-completion", for which I would recommend D. McDuff, G. Segal: Homology Fibrations and the "Group-Completion" Theorem, as well as G. Segal: Classifying spaces and spectral sequences.

For example, in the situation you describe there is a spectral sequence $$Tor_{k[M]}^s(k, H_t(X;k)) \Rightarrow H_{s+t}(X /\ \!/_hM;k)$$ given by filtering $EM$ by skeleta.

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  • $\begingroup$ I didn't realize the phrase "homotopy colimit" had any ambiguity. I am thinking about the model category structure on $Top$ (i.e. the colimit in the associated infinity category). At any rate, is $M \wr M$ notation for the "action category of $M$ on itself", i.e. the Grothendieck construction of the functor $M \to Set$ sending $*$ to $M$ and each $m \in M$ to the morphism $m:M \to M$ induced by composition? If so, this is what I hoped would work! Anyhow, thanks for your references, and for the spectral sequence! (Btw, where can I find a reference for this SS?) $\endgroup$
    – David Carchedi
    Commented May 20, 2013 at 18:39
  • $\begingroup$ (ofc, the notion of homotopy colimit I mean, is only well defined up to weak equivalence). $\endgroup$
    – David Carchedi
    Commented May 20, 2013 at 18:42
  • $\begingroup$ Yes, that's what I meant by $M \wr M$. The spectral sequence, in a much more general formulation, is in "Classifying spaces and spectral sequences". $\endgroup$
    – Oscar Randal-Williams
    Commented May 20, 2013 at 19:03

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