Let \(\mathcal O\) be an \(\infty\)-operad, \(C\) an \(\mathbb A_2\)-algebra (with \(\mathbb A_2\)-structure denoted by \(\alpha \in \mathbb A_2^{\mathcal O}(C)\)) and let \(A \xrightarrow{f} B \xrightarrow{g}C\) be morphisms of \(\mathbb E_0\)-algebras. We define the space \(\mathbb T_2^{\mathcal O}(f)_{/C}\) of \(\mathbb T_2\)-structures on \(f\) relative to \(C\) as the pullback of the span \[\{ \alpha\} \rightarrow\mathbb A_2^{\mathcal O}(C) = \mathbb T^{\mathcal O}_2(\mathrm{id}_C) \xrightarrow{- \circ (g\circ f, g\circ g)} \mathbb T_2(g\circ f) \xleftarrow{g \circ -}\mathbb T_2(f).\]
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2