ScalingStacks

Proof.

The space of lifts ([00EZ]) is by definition the fiber of \[\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, Z_0(f)) \xrightarrow{\mathrm{ev}_{1_A} \circ -} \mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, B)\] at \(f\). By the universal property of the centralizer and since \(I\) is initial in \(\mathrm{Alg}_{\mathbb E_0}(\mathcal C)\), the space \(\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, Z_0(f))\) is equivalent to the space \(\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A\otimes A, B)\times_{\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, B)}\{f\}\) of lifts in \(\mathrm{Alg}_{\mathbb E_0}(\mathcal C)\), where \(1_A \colon I \rightarrow A\) denotes the unique morphism in \(\mathrm{Alg}_{\mathbb E_0}(\mathcal C)\). Under this identification, the map \(\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, Z_0(f)) \xrightarrow{\mathrm{ev}_{1_A} \circ -}\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, B)\) unpacks to the composite \[\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A\otimes A, B)\times_{\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, B)}\{f\} \rightarrow\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A\otimes A, B) \rightarrow\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, B)\] of the projection and the precomposition with \(A\simeq I \otimes A \xrightarrow{1_A \otimes \mathrm{id}_A}A\otimes A\). Hence, the fiber of \(\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, Z_0(f)) \xrightarrow{\mathrm{ev}_{1_A} \circ -}\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, B)\) at \(f\) is equivalent to the space \[\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A\otimes A, B)\times_{\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, B)^{\times 2}} \{(f,f)\} =: \mathbb T_2(f). \qedhere\] ◻