Strictifying Homotopy Coherent Actions on Hochschild Complexes

TL;DR

This paper demonstrates that dg-props acting on dg-algebras induce actions on Hochschild complexes up to homotopy, and provides a functorial strictification method, with applications to commutative Hopf algebras.

Contribution

It introduces a functorial dg-replacement that strictifies homotopy coherent actions of dg-props on Hochschild complexes.

Findings

Homotopy coherent actions can be strictified via a functorial dg-replacement.

Explicit strictification of the homotopy coherent commutative Hopf algebra structure.

Extension of dg-operad actions to dg-prop actions on Hochschild complexes.

Abstract

If P is a dg-operad acting on a dg-algebra A via algebra homomorphisms, then P acts on the Hochschild complex of A. In the more general case when P is a dg-prop, we show that P still acts on the Hochschild complex, but only up to coherent homotopy. We moreover give a functorial dg-replacement of P that strictifies the action. As an application, we obtain an explicit strictification of the homotopy coherent commutative Hopf algebra structure on the Hochschild complex of a commutative Hopf algebra.