ScalingStacks

A.8.8 Laxly \(\mathcal O\)-monoidal functors[00IY]

If \(\mathcal C\) and \(\mathcal D\) are \(\mathcal O\)-monoidal \(\infty\)-categories, then a morphism \(\mathcal C\rightarrow\mathcal D\) in \(\mathrm{Op}_{/\mathcal O}\) is called a laxly \(\mathcal O\)-monoidal functor.⁵⁸ Let us denote the coCartesian fibrations \(\mathcal C^{\otimes} \rightarrow\mathcal O^{\otimes}\), \(\mathcal D^{\otimes} \rightarrow\mathcal O^{\otimes}\) by \(p\) and \(q\), then an \(\mathcal O\)-monoidal functor is a laxly \(\mathcal O\)-monoidal functor that takes \(p\)-coCartesian morphisms in \(\mathcal C^{\otimes}\) to \(q\)-coCartesian morphisms in \(\mathcal D^{\otimes}\).

Whereas an \(\mathcal O\)-monoidal functor respects the \(\mathcal O\)-monoidal structure up to coherent natural equivalence, a laxly \(\mathcal O\)-monoidal functor \(\mathcal C\rightarrow\mathcal D\) respects it only up to certain (generally noninvertible) coherent natural transformations, which nevertheless suffices to obtain an induced functor \(\mathrm{Alg}_\mathcal O(\mathcal C) \rightarrow\mathrm{Alg}_\mathcal O(\mathcal D)\) on \(\infty\)-categories of \(\mathcal O\)-algebra objects (simply by composition in \(\mathrm{Op}_{/\mathcal O}\)). For instance, given a laxly monoidal functor \(\mathcal C\xrightarrow{F} \mathcal D\) and an algebra object \(A \in \mathrm{Alg}_{\mathbb E_1}(\mathcal C)\), we obtain structure maps \(F(A) \otimes^\mathcal DF(A) \rightarrow F(A \otimes^\mathcal CA) \xrightarrow{F(\mu_A)} F(A)\) and \(\mathbbm{1}_\mathcal D\rightarrow F(\mathbbm{1}_\mathcal C) \xrightarrow{F(\eta_A)} F(A)\) giving the multiplication and unit of \(F(A) \in \mathrm{Alg}(\mathcal D)\).

Furthermore, \(\mathrm{Alg}_{\mathcal O}(-)\) takes (laxly) symmetric monoidal functors between symmetric monoidal \(\infty\)-categories to (laxly) symmetric monoidal functors.