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Given a non symmetric operad $\mathcal{O}$, is there an explicit description of its (André-Quillen or other) cohomology in low degrees in terms of infinitesimal composition?

I ask because I am interested in a collection parametrised by objects that are 2-cocycles in the Hochschild cohomology of the following algebra with coefficients in an appropriate bimodule.

Define $A$ to be the algebra with underlying vector space generated by symbols $$\{(f,i)| f\in\mathcal{O}(n), 1\leq i\leq n\}$$ and define the multiplication via infinitesimal composition to be $$(f,i)(g,j):=(f\circ_i g, i+j-1)$$

This is obviously a very artificial construction, and relies only on the operad structure and the existence of this "infinitesimal" bimodule, so should be somehow intrinsic to the operad, but I cannot see how to relate it to any of the cohomology theories I know.

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    $\begingroup$ You can compute deformation cohomology using the complex $\hom(B\mathcal O,\mathcal O)$ (interpreted as derivations $\mathrm{Der}(\Omega B\mathcal O,\mathcal O)$ and the condition for an element to be a $2$-cocycle can be given in terms of the bar differential which indeed is given in terms of infinitesimal compositions. Is this what you want? $\endgroup$
    – Pedro
    Commented Apr 9, 2021 at 16:27
  • $\begingroup$ (Why did you include the tag "commutative algebra" here?) $\endgroup$
    – Pedro
    Commented Apr 9, 2021 at 16:29
  • $\begingroup$ @PedroTamaroff I think commutative algebra was a misclick. I was posting from my phone. This deformation cohomology seems like it could be exactly what I am looking for. I will have to write out the computation and see $\endgroup$
    – Aidan
    Commented Apr 10, 2021 at 11:03

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In his thesis, Truong Hoang gives a description of Andre-Quillen cohomology in terms of infinitesimal bimodules. He actually identifies the tangent categories of a dg-operad $O$ for the following three categories:

  • operads.
  • $O$-bimodules.
  • infinitesimal $O$-bimodules.

He has a paper https://arxiv.org/abs/2005.01198 about this (I'll try to find a link to the manuscript).

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