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Added a space between \ and \!/_h.
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Michael Albanese
Michael Albanese
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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$$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 $$Tor_{k[M]}^s(k, H_t(X;k)) \Rightarrow H_{s+t}(X /\\!/_hM;k)$$$$Tor_{k[M]}^s(k, H_t(X;k)) \Rightarrow H_{s+t}(X /\ \!/_hM;k)$$ given by filtering $EM$ by skeleta.

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.

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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Oscar Randal-Williams
Oscar Randal-Williams
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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.