ScalingStacks

Notation 7.4.10.

Let \(\mathcal C\) be a symmetric monoidal \(\infty\)-category whose tensor unit \(I\) is initial and let \(f \colon A \rightarrow B\) be a morphism in \(\mathcal C\) whose centralizer \(\mathfrak{Z}(f) \in \mathcal C\) exists. Then, we define \(\mathrm{ev}_{1_A} \colon \mathfrak{Z}(f) \rightarrow B\) to be the composite \[\mathfrak{Z}(f) \simeq \mathfrak{Z}(f) \otimes I \xrightarrow{\mathrm{id}_{\mathfrak{Z}(f)} \otimes 1_A} \mathfrak{Z}(f) \otimes A \xrightarrow{\mathrm{ev}} B,\] where \(1_A\in \mathrm{Hom}_{\mathcal C}(I, A) \simeq *.\) Below, we will mostly use this morphism in the case that \(\mathcal C= \mathrm{Alg}_{\mathbb E_k}(\mathcal A)\) for a presentably symmetric monoidal \(\infty\)-category \(\mathcal A\), in which case this defines an \(\mathbb E_k\)-algebra map \(Z_k(f) \rightarrow B\) for any \(\mathbb E_k\)-algebra map \(f \colon A \rightarrow B\) in \(\mathrm{Alg}_{\mathbb E_k}(\mathcal A)\).