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?



Simplicial version of the A-infinity operad
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.


