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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 29th 2012
   • (edited Jun 29th 2012)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexPeople here may be familiar with my preprint [the universal simplicial bundle is a simplicial group](http://arxiv.org/abs/1204.4886) (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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 29th 2012
   • (edited Jun 29th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWhat 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 29th 2012
   • (edited Jun 29th 2012)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexPerhaps 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 $\ast \to \mathbf{B}A$, 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?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 29th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 29th 2012
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexOur comments crossed, Urs; I didn't see what you had written when I posted my second one :-) I'll reply in more detail later.

   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.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 29th 2012
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   Author: Urs
   Format: MarkdownItexOh, I see. :-)

   Oh, I see. :-)

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 29th 2012
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexOn 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.

   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.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJun 29th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe lax quotient.

   The lax quotient.

9. • CommentRowNumber9.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 30th 2012
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   Author: DavidRoberts
   Format: MarkdownItex@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.

   @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.