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Consider a simplicial set $X^\bullet$ with face maps $d_i$ (assume the set is finite in each degree so there are no measure issues). Then given two functions $f,g:X^1\to \mathbb{C}$ one can form their "convolution" by doing a push-pull:

$\displaystyle (f\star g)(x)=d_{1*}\big((d_0^*f)(d_2^*g)\big)(x)=\sum_{\Delta} f(a)g(b),$

where the sum is over all $\Delta$ such that $d_0=a, d_1=x, d_2=b.$ If we consider the basis of functions given by $\delta_a(x)=1$ if $x=a$ and equals $0$ otherwise, we have that $\displaystyle (\delta_a\star\delta_b)(x)=w(x),$ where $w(x)$ is the number of ways of "composing" $a$ and $b$ to give $x.$

This isn't necessarily an algebra because this product might not be associative, however using $\star$ we can associate to each $f$ the linear map $g\mapsto f\star g$ and therefore we get a natural subalgebra of the endomorphisms of a finite dimensional vector space. If $X^{\bullet}$ is a groupoid then this gives the groupoid convolution algebra.

Has this been studied in the general case? I haven't been able to find anything written about it (though I have found it implicitly used in at least one paper).

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  • $\begingroup$ Any simplicial set is weakly equivalent to the nerve of a simplicial groupoid, see Dwyer–Kan, “Homotopy theory and simplicial groupoids”. Since the construction respects connected components, we can restrict to the connected case for simplicity, in which case we recover the usual presentation of a connected simplicial set as the delooping of a simplicial group, where the simplicial group is given by (say) the Kan loop group functor. Once we pass to simplicial groups, we can simply take the convolution algebra degreewise. $\endgroup$
    – Dmitri Pavlov
    Commented Apr 4, 2023 at 14:33
  • $\begingroup$ The advantage of doing these weakly equivalent replacements is that now the degreewise convolution algebra is associative, i.e., an honest ordinary algebra. One possible area that might be relevant here (or perhaps somebody has already studied the above construction there) is (higher) Dijkgraaf–Witten theories. $\endgroup$
    – Dmitri Pavlov
    Commented Apr 4, 2023 at 14:35
  • $\begingroup$ @DmitriPavlov Thanks for your response. I think one issue is, if you apply this construction to groupoids, do you really get the groupoid convolution back, or just something that is in some sense equivalent to it? $\endgroup$
    – Josh Lackman
    Commented Apr 4, 2023 at 17:59
  • $\begingroup$ If you apply this construction to the nerve of an ordinary groupoid G, then there is no need to invoke Dwyer–Kan equivalences. Instead, you can see directly that the construction recovers precisely the convolution algebra of the groupoid: X_1 is precisely the set of morphisms of G, so finitely-supported functions on X_1 are isomorphic to the underlying vector space of the groupoid algebra of G, and the product defined by the above formula is precisely the convolution product (you already proved it for basis elements). $\endgroup$
    – Dmitri Pavlov
    Commented Apr 4, 2023 at 18:17
  • $\begingroup$ @DmitriPavlov Right, you don't need to apply it, but what I mean is you can still apply it and I'm not sure if there is a relation between the algebras you get. If this were to generalize the groupoid convolution algebra then there should be some way of recovering the convolution algebra. This is what I'm wondering. $\endgroup$
    – Josh Lackman
    Commented Apr 4, 2023 at 22:47

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