Theorem B.3.1 (00L3) · ScalingStacks

Fix a small \(\infty\)-operad \(\mathcal O\) such that \(\underline{\mathcal O} \simeq {\sf pt}\) and a presentably \(\mathcal O\)-monoidal \(\infty\)-category \(\mathcal C\) equipped with a compatible factorization system \((\mathcal L,\mathcal R)\) of small generation.

For any \(\mathcal A\in \mathrm{Op}_{/\mathcal O}\), the presentable \(\infty\)-category \(\mathrm{Alg}_{\mathcal A/\mathcal O}(\mathcal C)\) admits a factorization system \((\mathcal L_\mathcal A,\mathcal R_\mathcal A)\) of small generation with \(\mathcal L_\mathcal A= U^{-1}(\mathcal L^{\underline{\mathcal A}})\) and \(\mathcal R_\mathcal A= U^{-1}(\mathcal R^{\underline{\mathcal A}})\), where we write \(\mathrm{Alg}_{\mathcal A/\mathcal O}(\mathcal C) \xrightarrow{U} \mathrm{Fun}(\underline{\mathcal A},\underline{\mathcal C})\) for the forgetful functor and (as the exponential notation suggests (and as in Lemma B.2.3)) a morphism in \(\mathrm{Fun}(\underline{\mathcal A},\underline{\mathcal C})\) lies in \(\mathcal L^{\underline{\mathcal A}}\) (resp. \(\mathcal R^{\underline{\mathcal A}}\)) if and only if its components all lie in \(\mathcal L\) (resp. \(\mathcal R\)).
Fix a morphism \(\mathcal A\rightarrow\mathcal B\) in \(\mathrm{Op}_{/\mathcal O}\). Then, in the adjunction we have \(F_\mathcal A^\mathcal B(\mathcal L_\mathcal A) \subseteq \mathcal L_\mathcal B\) and \(U_\mathcal A^\mathcal B(\mathcal R_\mathcal B) \subseteq \mathcal R_\mathcal A\) (using the notation of part ([00L4])). In particular, the factorization systems of part ([00L4]) determine a lift through the indicated forgetful functor.
The total \(\infty\)-category of the Cartesian unstraightening of the horizontal functor in diagram ([00L6]) admits a factorization system \((\mathcal L_\mathrm{Alg},\mathcal R_\mathrm{Alg})\) of small generation, described as follows: an arbitrary morphism \((A \in \mathrm{Alg}_{\mathcal A/\mathcal O}(\mathcal C)) \xrightarrow{\widetilde{\alpha}} (B \in \mathrm{Alg}_{\mathcal B/\mathcal O}(\mathcal C))\) therein is specified by its image \(\mathcal A\xrightarrow{\alpha} \mathcal B\) in \(\mathrm{Op}_{/\mathcal O}\) along with a morphism \(A \rightarrow\alpha^* B\) in \(\mathrm{Alg}_{\mathcal A/\mathcal O}(\mathcal C)\), and
1. it lies in \(\mathcal L_\mathrm{Alg}\) if and only if \(\alpha\) is an equivalence and moreover for every color \(X \in \underline{\mathcal A}\) the morphism \(A_X \rightarrow(\alpha^* B)_X\) in \(\underline{\mathcal C}\) lies in \(\mathcal L\), and
2. it lies in \(\mathcal R_\mathrm{Alg}\) if and only if for every color \(X \in \underline{\mathcal A}\) the morphism \(A_X \rightarrow(\alpha^* B)_X\) in \(\underline{\mathcal C}\) lies in \(\mathcal R\).
In the case that \(\mathcal O= \mathbb E_\infty\) and \(\mathcal C\) is a presentably symmetric monoidal \(\infty\)-category with a compatible factorization system, \(\mathrm{Alg}_{\mathcal A}(\mathcal C)\) has a canonical symmetric monoidal structure ⁷² (see subsection A.8.5). The factorization system \((\mathcal L_\mathcal A, \mathcal R_\mathcal A)\) defined above is compatible with the symmetric monoidal structure.

[00L9]

Proof.

We begin with part ([00L4]).

First of all, observe the equivalences and the adjunction Original paper diagram Lemma B.2.3 furnishes the factorization system \((\mathcal L^{\underline{\mathcal A}} , \mathcal R^{\underline{\mathcal A}})\) of small generation on \(\mathrm{Fun}(\underline{\mathcal A}, \underline{\mathcal C})\). Hence, we prove part ([00L4]) by applying Lemma B.2.7, whose hypotheses it remains to show are satisfied.

We first show that the adjunction ([00LA]) is monadic and that its underlying monad preserves geometric realizations. For monadicity, by [Lur17, Thm. 4.7.0.3] it suffices to show that \(U\) is conservative and preserves sifted colimits. The former follows from [Lur17, Lem. 3.2.2.6], while the latter follows from [Lur17, Prop. 3.2.3.1]. Of course, \(F\) preserves geometric realizations (being a left adjoint), and so the monad \(T \coloneqq UF\) preserves geometric realizations as well.

We now claim that this monad \(T\) preserves \(\mathcal L^{\underline{\mathcal A}}\). This follows from the explicit description of the free algebra functor as an operadic left Kan extension (see particularly [Lur17, Props. 3.1.1.15, 3.1.1.16, and 3.1.1.20]). So indeed, the hypotheses of Lemma B.2.7 are satisfied, and we obtain a factorization system \((\mathcal L_\mathcal A, \mathcal R_\mathcal A)\) of small generation on \(\mathrm{Alg}_{\mathcal A/\mathcal O}(\mathcal C)\) as asserted.

For part ([00L5]), it suffices to note that the containment \(U_\mathcal A^\mathcal B(\mathcal R_\mathcal B) \subseteq \mathcal R_\mathcal A\) follows directly from the commutative square Original paper diagram in \(\mathrm{Op}_{/\mathcal O}\).

With parts ([00L4]) and ([00L5]) in hand, part ([00L7]) follows from Lemma B.2.4.

Lastly, part ([00L8]) follows from part ([00L4]) and the fact that the forgetful functor \(U \colon \mathrm{Alg}_{\mathcal A}(\mathcal C) \rightarrow\mathrm{Alg}_{\mathcal A_{\mathrm{Triv}}}(\mathcal C) = \mathrm{Fun}(\underline{\mathcal A}, \underline{\mathcal C})\) is symmetric monoidal (see subsection A.8.5). ◻