ScalingStacks

Observation 7.4.13.

Let \(\mathcal C\) be a symmetric monoidal \(\infty\)-category, and \(A\in \mathrm{Alg}_{\mathbb E_1}(\mathcal C)\). Assume the center \(Z_0(A)\in \mathrm{Alg}_{\mathbb E_1}(\mathcal C)\) of the underlying \(\mathbb E_0\)-algebra \(A \in \mathrm{Alg}_{\mathbb E_0}(\mathcal C)\) exists. Since \(A\) has a left action on itself, the universal property of \(Z_0(A) \in \mathrm{Alg}_{\mathbb E_1}(\mathcal C)\) induces an \(\mathbb E_1\)-homomorphism \(A\rightarrow Z_0(A)\). Moreover, since the multiplication of \(A\) is right unital, it follows that the composite \(A\rightarrow Z_0(A)\rightarrow A\) in \(\mathcal C\) is isomorphic to \(\mathrm{id}_A\). Hence, by corollary 7.4.12, this induces an \(\mathbb A_2\)-structure on \(A\). Unpacked, this \(\mathbb A_2\)-structure coincides with the one induced from the operad map \(\mathbb A_2 \rightarrow\mathbb E_1\) from example 7.2.7.