ScalingStacks

A.8.11 Adjunctions of \(\mathcal O\)-algebras[00J2]

For an \(\infty\)-operad \(\mathcal O\), an \(\mathcal O\)-monoidal left adjoint is an \(\mathcal O\)-monoidal functor \(F \colon \mathcal C\rightarrow\mathcal D\) between \(\mathcal O\)-monoidal \(\infty\)-categories such that for each color \(X \in \mathcal O\), the underlying functor \(F_X \colon \mathcal C\rightarrow\mathcal D\) is a left adjoint.

An important fact which we use repeatedly is that given an \(\mathcal O\)-monoidal left adjoint \(F\), its right adjoint \(G\) is canonically laxly \(\mathcal O\)-monoidal [Lur17, Cor. 7.3.2.7]. Conversely, given a laxly \(\mathcal O\)-monoidal right adjoint, it is merely a condition for its left adjoint to be \(\mathcal O\)-monoidal [Lur17, Cor. 7.3.2.12]. Moreover, such an adjunction determines an adjunction on \(\mathcal O\)-algebra objects [Lur17, Rem 7.3.2.13], whose adjoints both commute with the forgetful functors⁶¹, i.e., defines a morphism of adjunction: If \(\mathcal C, \mathcal D\) are symmetric monoidal \(\infty\)-categories, and \(F \colon \mathcal C\rightarrow\mathcal D\) is a symmetric monoidal left adjoint, then the symmetric monoidal functor \(\mathrm{Alg}_{\mathcal O}(F) \colon \mathrm{Alg}_{\mathcal O}(\mathcal C) \rightarrow\mathrm{Alg}_{\mathcal O}(\mathcal D)\) is a symmetric monoidal left adjoint.