ScalingStacks

A.8.4 \(\mathcal O\)-algebra objects[00IT]

Given an \(\mathcal O\)-monoidal \(\infty\)-category \(\mathcal C\), an \(\mathcal O\)-algebra object in \(\mathcal C\) is a section of the structure map \(\mathcal C\rightarrow\mathcal O\) in \(\mathrm{Op}\). These assemble into an \(\infty\)-category \(\mathrm{Alg}_\mathcal O(\mathcal C) \coloneqq \underline{\mathrm{Hom}}_{\mathrm{Op}_{/\mathcal O}}(\mathcal O,\mathcal C)\). As special cases, we write \(\mathrm{Alg}(\mathcal C) \coloneqq \mathrm{Alg}_{\textup{Assoc}}(\mathcal C)\) for the \(\infty\)-category of (associative) algebra objects in \(\mathcal C\) and \(\mathrm{CAlg}(\mathcal C) \coloneqq \mathrm{Alg}_\text{Comm}(\mathcal C)\) for the \(\infty\)-category of commutative algebra objects in \(\mathcal C\).

More generally, given a morphism \(\mathcal P\xrightarrow{p} \mathcal O\) in \(\mathrm{Op}\), we analogously define the \(\infty\)-category \(\mathrm{Alg}_{\mathcal P/\mathcal O}(\mathcal C) \coloneqq \underline{\mathrm{Hom}}_{\mathrm{Op}_{/\mathcal O}}(\mathcal P,\mathcal C)\) of \(\mathcal P\)-algebras in \(\mathcal C\) (relative to \(p\)). Equivalently, the base change \(p^*\mathcal C\rightarrow\mathcal P\) defines the underlying \(\mathcal P\)-monoidal \(\infty\)-category of \(\mathcal C\in \mathrm{Alg}_\mathcal O(\mathrm{Cat}_\infty)\), and we have \(\mathrm{Alg}_{\mathcal P/\mathcal O}(\mathcal C) \simeq \mathrm{Alg}_\mathcal P(p^*\mathcal C)\). For example, there is a natural morphism \(\mathrm{LM} \rightarrow{\textup{Assoc}}\), and so we can contemplate \(\mathrm{LM}\)-algebras in any monoidal \(\infty\)-category \(\mathcal C\in \mathrm{Alg}(\mathrm{Cat}_\infty)\). Note that when \(\mathcal O= \mathrm{Fin}_*\) we also write this as \(\mathrm{Alg}_{\mathcal P}(\mathcal C)\). ⁵⁶ Altogether, for an \(\mathcal O\)-monoidal \(\infty\)-category \(\mathcal C\) we obtain a functor \[(\mathrm{Op}_{/\mathcal O})^\mathrm{op} \xrightarrow{\mathrm{Alg}_{(-)/\mathcal O}(\mathcal C)} \mathrm{Cat}_\infty ~,\] whose functoriality is given by precomposition.

As a matter of terminology, it is common to refer to \(\mathcal O\)-algebra objects in a Cartesian symmetric monoidal \(\infty\)-category as \(\mathcal O\)-monoids (e.g. in \(\mathcal S\) or \(\mathrm{Cat}_\infty\)). In particular, \(\mathcal O\)-monoidal \(\infty\)-categories are indeed \(\mathcal O\)-monoids in \(\mathrm{Cat}_\infty\). When referring to notions in spaces, one generally simply prepends “\(\infty\)-” to the classical terms, so e.g. the objects of \(\mathrm{Alg}(\mathcal S)\) may be referred to as “\(\infty\)-monoids”.

Of particular relevance to this paper is the case \(\mathcal O= \mathbb E_2\), and we generally use the term braided in place of the prefix “\(\mathbb E_2\)-”: in particular, a braided monoidal \((\infty,2)\)-category is an \(\mathbb E_2\)-algebra in \(\mathrm{Cat}_{(\infty,2)}\). Indeed, a braided monoidal \(\infty\)-category in the classical sense defines an \(\mathbb E_2\)-monoid in \(\mathrm{Cat}_\infty\).