On a higher structure on operadic deformation complexes

Boris Shoikhet

Theory and applications of categories, Tome 33 (2018), pp. 988-1030.

Résumé

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 .

Détail

Publié le : 2018-10-14

Classification : Primary: 13D10, Secondary: 18D50
Keywords: Deformation theory, operads, higher structures