topological Hochschild cohomology

Hochschild cohomology of \(A\) is a derive mapping object \begin{align*} \mathop{\mathrm{\mathbb{R}Hom}}_{({A}, {A}){\hbox{-}}\mathsf{biMod}} (A, A) .\end{align*}

The higher Hochschild cohomology of an E_n algebra \(A\) is the derived mapping object of \(E_n{\hbox{-}}A{\hbox{-}}\)modules \begin{align*} \mathop{\mathrm{\mathbb{R}Hom}}_{\mathsf{E_n-A}{\hbox{-}}\mathsf{Mod}}(A, A) .\end{align*}