Let \(\mathcal C\) be a symmetric monoidal \(\infty\)-category and \(A \in \mathrm{Alg}_{\mathbb E_0}(\mathcal C)\). If the center \(Z_0(A) \in \mathrm{Alg}_{\mathbb E_1}(\mathcal C)\) exists, then the space \[\mathbb A_2(A) \coloneqq \mathrm{Hom}_{\mathrm{Op}_{ \mathbb E_0/}}(\mathbb A_2, \mathcal C) \simeq \mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A\otimes A, A) \times_{\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A,B)^2} \{(\mathrm{id}_A,\mathrm{id}_A)\}\] of \(\mathbb A_2\)-structures on \(A\) is equivalent to the following space of (dashed) lifts in \(\mathrm{Alg}_{\mathbb E_0}(\mathcal C)\): 
Proof.
Apply proposition 7.4.11 for \(f=\mathrm{id}_A\) using that \(Z_0(A) = Z_0(\mathrm{id}_A)\). ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2