I am looking for a description of the $A_\infty$ operad in the category of simplicial sets. More specifically, I am looking for a formulation of the loop space recognition principle for simplicial sets. Is such a thing written down anywhere?
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2$\begingroup$ Why the version described in "Geometry of iterated loop spaces" isn't enough? Ok it's written for topological spaces but it should work for simplicial sets too. Alternatively you could look in "Higher Algebra" where everything is done in terms of simplicial sets (but with a much more sophisticated approach) $\endgroup$– Denis NardinCommented Nov 22, 2014 at 15:01
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$\begingroup$ Hmm, yes. I guess my underlying hope was that there was a small concrete model with finitely many nondegenerate simplices described in a combinatorial manner. A subset of simplices of the Barratt-Eccles operad or something similar. Did you have a particular section/statement in Higher Algebra in mind by the way? $\endgroup$– KajCommented Nov 22, 2014 at 15:33
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3$\begingroup$ Theorem 5.2.6.10 for $k=1$ should be what you're looking for, with example 5.1.0.7 providing the small combinatorial model for $E_1$. $\endgroup$– Denis NardinCommented Nov 22, 2014 at 15:57
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2$\begingroup$ The question has already been answered in the comments. $\endgroup$– David WhiteCommented Nov 23, 2014 at 15:44
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$\begingroup$ @DenisNardin I think you could leave this as an answer, so that the question does not remain in the queue of "unanswered" questions $\endgroup$– Yemon ChoiCommented Nov 23, 2014 at 16:36
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2 Answers
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In my opinion the construction in "the geometry of iterated loop spaces" by P. May should carry through the simplicial world. If you want a completely simplicial treatment you can find it in theorem 5.2.6.10 of "Higher Algebra" by J.Lurie. Example 5.1.0.7 in the same book provides you with a small simplicial model for the $E_1$-operad.
However you need to be careful because the simplicity of this model for $E_1$ is offset by the higher sofistication required to work with $\infty$-operads.
answered Nov 23, 2014 at 18:12
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To answer your first question: you can describe the $A_\infty$-operad as the Boardman-Vogt W -construction of the associative operad $\mathsf{Ass}$. One can in turn describe this in terms of trees with valence $n$ vertices labelled by elements of $\Sigma_n$, and edges decorated by $\Delta^1$, together with the natural segment structure.
Maybe this isn't concrete enough for you, but it certainly gives a description of the operad $\mathsf{A}_\infty$ as an operad in simplicial sets.
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1$\begingroup$ Could you be a little more explicit about the simplicial W-construction? Is an element of $W(O)(k)_n$ equal to a tree with $k$ leaves and internal edges labeled by $\{1,\dots,n\}$ with face and degeneracy maps given by contracting edges or splitting edges in half? $\endgroup$ Commented Apr 2, 2023 at 17:48