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Let $S$ be a (small) symmetric monoidal category and $X$ a (small) category on which $S$ acts. $\pi_0(S) = \pi_0(BS)$ is naturally an abelian monoid, with $[A] + [B] := [A+B]$, where $[A]$ denotes the path component containing the 0-cell, i.e., object $A$.

$\pi_0(S)$ acts on $H_p(BX, \mathbb Z)$. How is this?

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  • $\begingroup$ $BS$ acts on $BX$, therefore you have an action map $H_0(BS)\otimes H_p(BX)\to H_p(BX)$. $H_0(BS)=\mathbb{Z}[\pi_0(BS)]$, therefore you have an action of the group algebra $\mathbb{Z}[\pi_0(BS)]$ on $H_p(BX)$ or equivalently an action of the group $\pi_0(S)$ on $H_p(BX)$. $\endgroup$
    – Geoffroy Horel
    Commented Mar 31, 2013 at 0:26
  • $\begingroup$ $S \times X \to X$ gives rise to $B(S \times X ) \to BX$. Unless we know that $BS$ is locally compact, in which case the natural map $B(S \times X) \to BS \times BX$ is a homeomorphism, how do we get a map $BS \times BX \to BX$?/ $\endgroup$
    – Joshua Seaton
    Commented Mar 31, 2013 at 14:14

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The action of $S$ on $X$ induces an action of $BS$ on $BX$. One way to see it is to do it at the level of $p$ simplices and construct a map $B_pS\times B_pX\to B_pX$ for all $p$. For any category $C$, the $p$ simplices $B_pC$ are the functors $[p]\to C$. Since $S$ acts on $X$, we have a functor $S\times X\to X$. Mapping $[p]$ to this map, we construct a map: $B_pS\times B_pX\to B_p X$ (note that $Fun([p],-)$ preserves product). It is formal to check that this map induce a map of simplicial sets $B_\bullet(S)\times B_\bullet(X)\to B_\bullet(X)$. If you would rather work with topological space, you can use the fact that geometric realization is product preserving and you can turn this map into a map of topological spaces.

This action induces an action map $H_0(BS)\otimes H_p(BX)\to H_p(BX)$ making $H_p(BX)$ into an $H_0(BS)$-module. But $H_0(BS)=\mathbb{Z}[\pi_0(BS)]$, therefore you have an action of the group algebra $\mathbb{Z}[\pi_0(BS)]$ on $H_p(BX)$ or equivalently an action of the group $\pi_0(S)$ on $H_p(BX)$.

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  • $\begingroup$ @Geoffroy: May you please elaborate on the first point. And by 'at the level of p-simplices' do you mean within the nerve or within the classifying space? $\endgroup$
    – Joshua Seaton
    Commented Apr 1, 2013 at 1:16
  • $\begingroup$ Done. Let me know if you would like to see more details. $\endgroup$
    – Geoffroy Horel
    Commented Apr 1, 2013 at 18:32
  • $\begingroup$ Sorry for the delay. -- You write that geometric realisation is product-preserving. The resources I've checked though say that this is not true in general, but is true when the spaces involved are compactly-generated. Am I missing something? $\endgroup$
    – Joshua Seaton
    Commented Apr 26, 2013 at 22:01
  • $\begingroup$ The geometric realization of a simplicial set is a CW-complex so it is certainly compactly generated. But you don't even need that. It's perfectly fine to define homology of a simplicial set by applying the free abelian group functor degreewise and then take the alternating sum of the faces as differential and take homology of the resulting chain complex. $\endgroup$
    – Geoffroy Horel
    Commented Apr 27, 2013 at 1:24
  • $\begingroup$ Got it - thanks for the clarification. Accepted. $\endgroup$
    – Joshua Seaton
    Commented Apr 27, 2013 at 1:42

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