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People here may be familiar with my preprint the universal simplicial bundle is a simplicial group (aka ’W is a monad’). With a bit of tweaking I’ve found the following result (contractible means having the point as a deformation retract using simplicial homotopies):
Let $T$ be a Lawvere theory containing that of monoids (better, consider a Lawvere theory equipped with an inclusion $Th(Mon) \hookrightarrow T$, giving a specified monoid operation $m$) and $C$ a finite product category. Then there is a monad $W_T$ on $s T alg(C)$ that takes a simplicial $T$-algebra $A$ in $C$ and returns a contractible simplicial $T$-algebra $W_T A$ with the unit a monomorphism (i.e. containing $A$). If the specified $m$ is left cancellative (e.g. it underlies a group operation), then the canonical action of $A$ on $W_T A$ given by the inclusion map is free. (In any case, the action is as free as the action of $A$ on itself is.)
If we pass to the underlying simplicial object $u(W_T A)$ in $C$ then there is an $A$-equivariant weak homotopy equivalence $WA \to u(W_T A)$ where $WA$ is the usual bar construction for $m$ (covered in Moore’s treatment of the functor $W$, but not in May’s later book). If $m$ underlies a group operation then this weak homotopy equivalence is an isomorphism, and the quotient $u(W_T A)/A$ exists in any finite-product category $C$ and is given by $W A/A = \overline{W}A$. More generally, one could consider the simplicial action category, or perhaps pass to the category of simplicial presheaves on $C$, if $C$ itself doesn’t have enough colimits. (Or even take the nerve of the action category to get a bisimplicial object in $C$ and take the diagonal.)
Thus we have a simplicial $T$-algebra structure on the total space of the ’universal $A$-bundle’, for whenever we can find an interpretation of ’universal’ and ’bundle’ that makes sense. I wonder if this is good for anything. Given the recent result by Thomas, Urs and Danny about $\infty$-groups in $\infty$-toposes being presented by groups in simplicial sheaves, and $\infty$-bundles being represented by simplicial bundles, it seems natural to consider whether this result can be extended to $\infty$-algebras for other Lawvere theories. Then my result above is really a statement about $\infty$-algebras. What that conjectured result is good for I do not know either, but it has a sort of cohomological feel to it.
What can you say in general about (the existence of) the quotient map $W A \to W A / A$?
For $A$ a simplicial group, the single fact that makes the whole theory tick is that this map is a Kan fibration resolution of the point inclusion $* \to \bar W G$. It would be interesting for non-groupal $A$ if you get an inner Kan fibration.
Perhaps we can think about this as finding a fibrant replacement for the canonical point $\ast \to \mathbf{B}A$, where we take $\mathbf{B}A$ to be some internal connected $(\infty,1)$-category… Or is all this just hiding basic results behind fancy words?
Perhaps we can think about this as finding a fibrant replacement for the canonical point *→BA, where we take BA to be some internal connected (∞,1)-category…
Yes, that’s what I mean.
Our comments crossed, Urs; I didn’t see what you had written when I posted my second one :-) I’ll reply in more detail later.
Oh, I see. :-)
On second thoughts, taking the quotient by the monoid action is the wrong ging to do. I’m not sure what the right thing to do is, yet.
The lax quotient.
@Urs, Are you thinking of the diagonal of the nerve of the action simplicial category, or some other sort of quotient?
I may have over-reacted about the monoid action. The whole point of $W_T$ is to get something on which one can take an ordinary quotient, rather than some sort of homotopy quotient. Let me think some more before having a brain-vomit.
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