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On a higher structure on operadic deformation complexes

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Boris Shoikhet

In this paper, we prove that there is a canonical homotopy (n+1)-algebra
structure on the shifted operadic deformation complex
$\Def(e_n\to\mathcal{P})[-n]$ for any operad $\mathcal{P}$ and a map of
operads $f\colon e_n\to\mathcal{P}$. This result generalizes a result of
Tamarkin, who considered the case $\mathcal{P}=\End_\Op(X)$. Another more
computational proof of the same result was recently sketched by Calaque
and Willwacher.

Our method combines the one of Tamarkin, with the categorical algebra on
the category of symmetric sequences, introduced by Rezk and further
developed by Kapranov-Manin and Fresse. We define suitable deformation
functors on n-coalgebras, which are considered as the
"non-commutative" base of deformation, prove their representability, and
translate properties of the functors to the corresponding properties of
the representing objects. A new point, which makes the method more
powerful, is to consider the argument of our deformation theory as an
object of the category of symmetric sequences of dg vector spaces, not as
just a single dg vector space .

Keywords:
Deformation theory, operads, higher structures

2010 MSC:
Primary: 13D10, Secondary: 18D50

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 32, pp 988-1030.

Published 2018-10-14.

http://www.tac.mta.ca/tac/volumes/33/32/33-32.pdf

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