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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 14th 2013
• (edited Apr 14th 2013)

Added to A-infinity category the references pointed to by Bruno Valette here.

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeApr 15th 2013
• (edited Apr 15th 2013)

Is there a model structure known on the category of $A_\infty$-categories, where we use topological $A_\infty$ algebras instead of chain complexes or what-have-you? I’m imagining that we might have such a model structure presenting $(\infty,1)Cat$, but perhaps there are obstructions to this idea.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 15th 2013
• (edited Apr 15th 2013)

For a fixed set of objects this is given by the pertinent model structure on algebras over an operad.

From there one can try to take a kind of colimit as the set of objects varies. For low $n$ this is discussed a bit in section 5 of

• Andor Lucacs, Cyclic Operads, Dendroidal Structures, Higher Categories (pdf)
• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeApr 15th 2013
• (edited Apr 15th 2013)

I’m not sure how $n$ relates to anything I asked, but to elicit a clarification, let me say I’m interested in when I have a topological or Lie group $G$ or for the present, a Lie group, and when I can properly claim that $\mathbb{B}G$ is equivalent to $\mathbb{B}\Omega BG$, where $BG$ is a classifying space for $G$, as simplicial sheaves on $Top$ or $Diff$. I’m not just interested in one-object things, though, but the many-object things I think will all be ordinary internal groupoids.

EDIT: Am I correct in guessing that I need $n=\infty$, in that loop spaces are generally not algebras for $A_n$?

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeApr 15th 2013
• (edited Apr 15th 2013)

The “$n$” was for “$n$-category” as usual. When you look at that thesis you’ll see that it discusses (“weak”) $n$-categories for low $n$ as algebras over homotopy associative operads, hence as $A_\infty$-categories.

For $A_\infty$-spaces such as loop spaces you can use (any of the) model structures on algebras over an operad, for instance. As well as of course many other models which do not use operads explicitly.

For the classifying space of a topological group to me a model of its delooping in the category-theoretic sense it is sufficient for the group to be well-pointed, as I think you know. See at geometric realization of simplicial topological spaces – Classifying spaces.

• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeApr 15th 2013
• (edited Apr 15th 2013)

Here I’m thinking of $\mathbb{B}\Omega BG$ as an $A_\infty$-category, so really I’m thinking of a lift of the standard equivalence $G\sim \Omega BG$ in spaces (which is what you linked to) to one of $A_\infty$-categories, for some appropriate model structure. I’ll have to have a bit of a read …

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeApr 15th 2013

Yeah but for the one-object case this is all equivalent. See around “May recognition theorem” in Luries Higher Algebra, for instance.

• CommentRowNumber8.
• CommentAuthorDavidRoberts
• CommentTimeApr 16th 2013

ok, I’ll check it out. Thanks, but I may have questions :-)