ScalingStacks

Let \(\mathcal C\) be a symmetric monoidal \(\infty\)-category and \(f\colon A\rightarrow B\) be a morphism in \(\mathrm{Alg}_{\mathbb E_0}(\mathcal C)\). If the centralizer \(Z_0(f)\) in \(\mathrm{Alg}_{\mathbb E_0}(\mathcal C)\) exists, then the space \[\mathbb{T}_2(f)\coloneqq \mathrm{Hom}_{\mathrm{Op}_{[1] \otimes \mathbb E_0/}}(\mathbb{T}_2, \mathcal C) \simeq \mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A\otimes A, A) \times_{\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A,B)^2} \{(f,f)\}\] of \(\mathbb{T}_2\)-structures on \(f\) is equivalent to the following space of (dashed) lifts in \(\mathrm{Alg}_{\mathbb E_0}(\mathcal C)\):