• A
  • A
  • A
  • ABC
  • ABC
  • ABC
  • А
  • А
  • А
  • А
  • А
Regular version of the site

Article

On a higher structure on operadic deformation complexes

Theory and Applications of Categories. 2018. Vol. 33. No. 32. P. 988-1030.
Shoikhet B.

In this paper, we prove that there is a canonical homotopy (n+1)-algebra structure on the shifted operadic deformation complex Def(e_n → P)[−n] for any operadP and a map of operads f : e_n → P. This result generalizes a result of Tamarkin, who considered the case P = End_{O_p}(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 consid- ered 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 argu- ment 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 .