Section B (00J9) · ScalingStacks

B Factorization systems for enriched $\infty$-categories[00J9]

As the terminology suggests, a factorization system on an $\infty$-category gives a functorial way of factoring its morphisms. As a basic example, every morphism $X \xrightarrow{f} Y$ in the $\infty$-category of spaces (and in particular in the category of sets) admits a factorization Original paper diagram as a surjection followed by a monomorphism. In fact, every morphism of spaces admits a unique such factorization. This uniqueness persists in the case of a general factorization system, arising from a certain orthogonality relation that is required of the two factors.⁶⁵

In this appendix, given a presentably monoidal $\infty$-category $\mathbb V$, we prove as Theorem B.4.1 that a factorization system on $\mathbb V$ that is compatible with its monoidal structure determines a factorization system on the $\infty$-categories $\mathrm{Cat}[\mathbb V]$ of $\mathbb V$-enriched $\infty$-categories. In fact, we prove a more general result as Theorem B.3.1: if $\mathbb V$ is presentably $\mathcal O$-monoidal, under mild hypotheses we obtain a factorization system on the $\infty$-category of algebras over any $\infty$-operad $\mathcal A$ equipped with a morphism $\mathcal A\rightarrow\mathcal O$. We use Theorem B.4.1 to obtain factorization systems on $(\infty,k)$-categories (Theorem 5.3.7), on enriched $(\infty, 2)$-categories (Corollary 6.2.4), and on $\infty$-operads (Proposition 7.5.3). Along the way, we establish a number of useful results concerning factorization systems, some of which are also used in the main body of the paper.

We begin in Subsection B.1 by recalling some basic definitions and properties of factorization systems, including some convenient features that result from specializing to presentable $\infty$-categories. We then proceed in Subsection B.2 to establish a number of ways of obtaining new factorization sytems from old ones. We then prove our two main results Theorem B.3.1 (concerning algebras over $\infty$-operads) in Subsection B.3 and Theorem B.4.1 (concerning enriched $\infty$-categories) in Subsection B.4.

B.1 Recollections on factorization systems[00JA]

B.1.1 Basics of factorization systems[00JB]

[00JC]

Definition B.1.1. ([Lur09, Def. 5.2.8.1]).

Given morphisms $a \xrightarrow{l} b$ and $c \xrightarrow{r} d$ in an $\infty$-category, we say that $l$ is left orthogonal to $r$ or that $r$ is right orthogonal to $l$ if for any solid commutative square Original paper diagram the space of dashed lifts $b \rightarrow c$ is contractible. In this situation, we may write $l \bot r$. More broadly, given classes $\mathcal L$ and $\mathcal R$ of morphisms in an $\infty$-category, we write $\mathcal L\bot \mathcal R$ to indicate that $l \bot r$ for every $l \in \mathcal L$ and every $r \in \mathcal R$.

[00JE]

Example B.1.2.

A morphism $f$ in an $\infty$-category satisfies the relation $f \bot f$ if and only if it is an equivalence.

[00JF]

Observation B.1.3.

Given an adjunction Original paper diagram and morphisms $f$ and $g$ in $\mathcal C$ and $\mathcal D$ respectively, the orthogonality relations $f \bot G(g)$ and $F(f) \bot g$ are equivalent. We use this fact without further comment.

[00JG]

Notation B.1.4.

Given a class $S$ of morphisms in an $\infty$-category, we write $S^\perp$ (resp. $^\perp S$) for the class of morphisms that are right (resp. left) orthogonal to those in $S$.

[00JH]

Definition B.1.5. ([Lur09, Def. 5.2.8.8]).

A factorization system on an $\infty$-category $\mathcal C$ is a pair $(\mathcal L, \mathcal R)$ of classes of morphisms in $\mathcal C$ satisfying the following conditions.

The classes $\mathcal L$ and $\mathcal R$ are stable under the formation of retracts (in $\mathrm{Fun}([1],\mathcal C)$).
We have the orthogonality relation $\mathcal L\perp \mathcal R$.
Every morphism $c \xrightarrow{f} d$ in $\mathcal C$ admits a factorization with $l \in \mathcal L$ and $r \in \mathcal R$.

We respectively write $\mathrm{Cat}_\infty^{\text{f.s.},\mathcal L}$, $\mathrm{Cat}_\infty^{\text{f.s.},\mathcal R}$, and $\mathrm{Cat}_\infty^{\text{f.s.},\mathcal L,\mathcal R}$ for the $\infty$-categories of $\infty$-categories equipped with factorization systems, in which a morphism is a functor that respectively preserves the left class, the right class, or both classes.

[00JM]

Notation B.1.6.

To simplify our notation, we take the following conventions when studying a class $S$ of morphisms in an $\infty$-category $\mathcal C$.

Assuming that $S$ consists of precisely the morphisms in a subcategory of $\mathcal C$ (e.g. both classes in a factorization system on $\mathcal C$), we simply write $S$ to denote this subcategory.
Assuming that $S$ is stable under homotopy (e.g. both classes in a factorization system on $\mathcal C$), we also simply write $S$ to denote the full subcategory of $\mathrm{Fun}([1],\mathcal C)$ on the morphisms in $S$.
We simply write $\mathcal C^\simeq$ for the class of equivalences in $\mathcal C$, and we simply write $\mathcal C$ for the class of all morphisms in $\mathcal C$.
For any object $c \in \mathcal C$, we write ${\mathcal C}_{\small{/^{S}}{c}} \subseteq \mathcal C_{/c}$ for the full subcategory on those objects $(d \rightarrow c) \in \mathcal C_{/c}$ that lie in $S$ (when considered as morphisms in $\mathcal C$). In the special case that $c \simeq {\sf pt}_\mathcal C$ is terminal, we simply write $\mathcal C^S \coloneqq {\mathcal C}_{\small{/^{S}}{{\sf pt}_{\mathcal C}}}$.

[00JN]

Example B.1.7.

For any $\infty$-category $\mathcal C$, the pairs $(\mathcal C^\simeq,\mathcal C)$ and $(\mathcal C,\mathcal C^\simeq)$ define factorization systems on $\mathcal C$.

[00JP]

Example B.1.8.

Let $\mathbb N^\times \coloneqq \{1, 2, 3, \ldots \}^\times$ denote the (commutative) monoid of natural numbers under multiplication. Given two elements $s,t \in \mathbb N^\times$, their corresponding morphisms in $B \mathbb N^\times$ satisfy $s \bot t$ (and thereafter $t \bot s$) if and only if $s$ and $t$ are coprime. From here, it is easy to check that e.g. the pairs (powers of 2, odds) and (odds, powers of 2) define factorization systems on $B \mathbb N^\times$. More generally, if $\{2, 3, 5, \ldots \} = P_1 \sqcup P_2$ denotes a two-element partition of the set of prime numbers, then \[\text{(powers of elements of $P_1$, powers of elements of $P_2$)}\] determines a factorization system on $B \mathbb N^\times$, and moreover every factorization system on $B \mathbb N^\times$ arises in this way.

[00JQ]

Observation B.1.9.

A factorization system $(\mathcal L,\mathcal R)$ on an $\infty$-category is completely determined by either $\mathcal L$ or $\mathcal R$ (since $\mathcal R= \mathcal L^\bot$ and $\mathcal L= {}^\bot \mathcal R$). We use this fact without further comment.

[00JR]

Observation B.1.10.

By [Lur09, Prop. 5.2.8.17], the factorization in part ([00JL]) of Definition B.1.5 is unique. We often use this fact without further comment.

[00JS]

Notation B.1.11.

Justified by Observation B.1.10, given a morphism $c \xrightarrow{f} d$ in an $\infty$-category $\mathcal C$ equipped with a factorization system $(\mathcal L,\mathcal R)$, we write $\mathrm{Fact}(f) \coloneqq \mathrm{Fact}_{(\mathcal L,\mathcal R)}(f) \in \mathcal C$ for the unique object through which $f$ factors via the factorization system.

We introduce the following notion for future use.

[00JT]

Definition B.1.12.

Let $\mathcal O$ be an $\infty$-operad and let $\mathcal C$ be an $\mathcal O$-monoidal $\infty$-category. Suppose that for every color $X \in \underline{\mathcal O}$, the $\infty$-category $\mathcal C_X$ of $X$-colored objects in $\mathcal C$ is equipped with a factorization system $(\mathcal L_X,\mathcal R_X)$. We say that the $\mathcal O$-monoidal structure of $\mathcal C$ is compatible with these factorization systems if for every $n \geq 0$ and every $n$-ary operation $(X_1,\ldots,X_n) \rightarrow X$ in $\mathcal O$, the corresponding functor $\mathcal C_{X_1} \times \cdots \times \mathcal C_{X_n} \rightarrow\mathcal C_X$ carries morphisms in $\mathcal L_{X_1} \times \cdots \times \mathcal L_{X_n}$ to morphisms in $\mathcal L_X$.⁶⁶

B.1.2 Factorization systems on presentable $\infty$-categories[00JU]

We now discuss factorization systems of small generation on presentable $\infty$-categories. We then proceed to make some further observations about factorization systems that admit specializations when applied to those of small generation.

For motivation, observe that both classes of a factorization system necessarily contain all equivalences. As a result, both classes of a factorization system on a large $\infty$-category must be large. However, on a presentable $\infty$-category one can define a factorization system in terms of a small set of morphisms (which then generate the left class), as we now recall.

[00JV]

Definition B.1.13. ([Lur09, Def. 5.5.5.1]).

We say that a class of morphisms $S$ in an $\infty$-category $\mathcal C$ is saturated if it satisfies the following conditions.

The class $S$ contains all equivalences and is closed under composition.⁶⁷
The full subcategory $S \subseteq \mathrm{Fun}([1],\mathcal C)$ is closed under (small) colimits.
The class $S$ is stable under cobase change.

[00JZ]

Proposition B.1.14. ([Lur09, Prop. 5.5.5.7]).

Fix a presentable $\infty$-category $\mathcal C$ and a small set of morphisms $S$ in $\mathcal C$. Then, there exists a factorization system $(\mathcal L,\mathcal R)$ on $\mathcal C$ with $\mathcal R= S^\perp$. Moreover, $\mathcal L$ is the smallest saturated class of morphisms in $\mathcal C$ that contains $S$. 0◻

[00K0]

Definition B.1.15.

In the context of Proposition B.1.14, we say that the factorization system $(\mathcal L,\mathcal R)$ (or simply the left class $\mathcal L$) is of small generation, or more specifically that it is generated by $S$. Moreover, we may write $\overline{S}$ for $\mathcal L$. We define the subcategories \[{\Pr}^{L,\text{f.s.},\mathcal L} \subset \widehat{\mathrm{Cat}}_\infty^{\text{f.s.},\mathcal L} \qquad \text{and} \qquad {\Pr}^{R,\text{f.s.},\mathcal R} \subset \widehat{\mathrm{Cat}}_\infty^{\text{f.s.},\mathcal R}\] to be those on the presentable $\infty$-categories whose factorization systems are of small generation, whose morphisms are respectively required to be left or right adjoints (in addition to preserving the indicated class of the factorization system).

[00K1]

Example B.1.16.

Fix any integer $n \geq -2$. By [Lur09, Ex. 5.2.8.16], the $\infty$-category $\mathcal S$ of spaces admits a factorization system ($n$-connected, $n$-truncated),⁶⁸ which is generated by the singleton $\{S^{n+1} \rightarrow{\sf pt}\}$.⁶⁹

[00K2]

Observation B.1.17.

Given an adjunction between $\infty$-categories equipped with factorization systems, the left adjoint preserves the left class if and only if the right adjoint preserves the right class. It follows that passing to adjoints determines an equivalence $\Pr^{L,\text{f.s.},\mathcal L} \simeq (\Pr^{R,\text{f.s.},\mathcal R})^\mathrm{op}$. We use these facts without further comment.

[00K3]

Lemma B.1.18.

If $\mathcal C$ is a presentably $\mathcal O$-monoidal category for a small operad $\mathcal O$ and suppose that for every color $X\in \underline{\mathcal O}$, the presentable $\infty$-category $\mathcal C_{X}$ is equipped with a factorization system $(\mathcal L_X, \mathcal R_X)$ generated by a set $S_X$. Then, the factorization systems are compatible with the $\mathcal O$-monoidal structure if and only if for every operation $(X_1, \ldots, X_n) \rightarrow X$ in $\mathcal O$, the corresponding functor $\mathcal C_{X_1} \times \cdots \mathcal C_{X_n} \rightarrow\mathcal C_{X}$ carries morphisms in $S_{X_1} \cdots \times \cdots S_{X_n}$ to morphisms in $\mathcal L_X$.

[00K4]

Proof.

Assume that $\mathcal C_{X_1} \times \cdots \mathcal C_{X_n} \rightarrow\mathcal C_{X}$ carries morphisms in $S_{X_1} \cdots \times \cdots S_{X_n}$ to morphisms in $\mathcal L_X$. By assumption, the functor $\mathcal C_{X_1} \times \cdots \times \mathcal C_{X_n} \rightarrow\mathcal C_{X}$ preserves small colimits separately in all variables. Since for every $Y \in \underline{\mathcal O}$, the class of morphisms $\mathcal L_Y$ is by proposition B.1.14 the smallest saturated class of morphisms in $\mathcal C_Y$ that contains $S_Y$, the functor $\mathcal C_{X_1} \times \cdots \times \mathcal C_{X_n} \rightarrow\mathcal C_{X}$ therefore also carries morphisms in $\mathcal L_{X_1} \times \cdots \times \mathcal L_{X_n}$ to morphisms in $\mathcal L_X$. ◻

[00K5]

Observation B.1.19.

Fix an $\infty$-category $\mathcal C$ with a factorization system $(\mathcal L,\mathcal R)$.

For any object $c \in \mathcal C$, we obtain factorization systems on both $\mathcal C_{c/}$ and $\mathcal C_{/c}$ in which both classes are pulled back from $\mathcal C$ via the respective forgetful functors.
Suppose that $\mathcal C$ is presentable and that $(\mathcal L,\mathcal R)$ is of small generation. Then, the factorization systems of part ([00K6]) are both of small generation as well. Specifically, if $S$ denotes a set of morphisms in $\mathcal C$ that generates $(\mathcal L,\mathcal R)$, then they are respectively generated by the evident (small) spaces of morphisms indexed by \[\bigsqcup_{(a \rightarrow b) \in S} \mathrm{Hom}_\mathcal C(c,a) \qquad \text{and} \qquad \bigsqcup_{(a \rightarrow b) \in S} \mathrm{Hom}_\mathcal C(b,c) ~.\]

[00K7]

Observation B.1.20.

Fix an $\infty$-category $\mathcal C$ with a factorization system $(\mathcal L,\mathcal R)$.

Assume that $\mathcal C$ contains a terminal object. Then, there exists a left adjoint to the fully faithful inclusion, which is given by the formula $c \mapsto \mathrm{Fact}(c \rightarrow{\sf pt}_\mathcal C)$. Moreover, the right adjoint is the inclusion of the $\mathcal L$-local objects, and hence the left adjoint exhibits $\mathcal C^\mathcal R$ as the localization $\mathcal C[\mathcal L^{-1}]$.⁷⁰
Assume that $\mathcal C$ is presentable and that $(\mathcal L,\mathcal R)$ is generated by a set $S$ of morphisms in $\mathcal C$. Then, the reflective localization ([00K9]) also identifies $\mathcal C^\mathcal R$ with the (accessible) localization $\mathcal C[S^{-1}]$.
Furthermore, if $\mathcal C$ has a symmetric monoidal structure compatible with the factorization system and which has the terminal object as monoidal unit, then it induces a symmetric monoidal structure on $\mathcal C^\mathcal R$, for which the left adjoint $\mathcal C\rightarrow\mathcal C^R$ is symmetric monoidal.
Given a morphism $(\mathcal C_0,(\mathcal L_0,\mathcal R_0)) \xrightarrow{F} (\mathcal C_1,(\mathcal L_1,\mathcal R_1))$ in $\mathrm{Cat}_\infty^{\text{f.s.},\mathcal L,\mathcal R}$ in which both $\mathcal C_0$ and $\mathcal C_1$ admit terminal objects and $F({\sf pt}_{\mathcal C_0}) \simeq {\sf pt}_{\mathcal C_1}$, the reflective localizations of part ([00K8]) assemble into a morphism of adjunctions.

B.2 Induced factorization systems[00KC]

In this subsection, we record an assortment of useful results that allow us to obtain new factorization systems from old ones, in roughly increasing order of complexity. Specifically, we give sufficient conditions for inducing factorization systems on subcategories (Observation B.2.1), on $\infty$-categories of functors (Lemma B.2.3), on the total $\infty$-categories of Cartesian fibrations (Lemma B.2.4), through reflective localizations (Lemma B.2.5), and through monadic adjunctions (Lemma B.2.7).

We begin by recording necessary and sufficient conditions for a factorization system to restrict to a subcategory.

[00KD]

Observation B.2.1.

Let $\mathcal C$ be an $\infty$-category equipped with a factorization system $(\mathcal L,\mathcal R)$, let $\mathcal C_0 \subseteq \mathcal C$ be a subcategory, and let us write $\mathcal L_0 \coloneqq \mathcal L\cap \mathcal C_0$ and $\mathcal R_0 \coloneqq \mathcal R\cap \mathcal C_0$. Then, the pair $(\mathcal L_0,\mathcal R_0)$ forms a factorization system on $\mathcal C_0$ if and only if the following conditions are satisfied.

For any solid commutative square ([00JD]) in $\mathcal C_0$ with $l \in \mathcal L$ and $r \in \mathcal R$, the unique lift in $\mathcal C$ (guaranteed by the orthogonality relation $\mathcal L\bot \mathcal R$) also lies in $\mathcal C_0$.
For any morphism $c \xrightarrow{f} d$ in $\mathcal C_0$, both morphisms in the factorization $c \rightarrow\mathrm{Fact}_{(\mathcal L,\mathcal R)}(f) \rightarrow d$ also lie in $\mathcal C_0$.

In particular, if $\mathcal C_0$ is a full subcategory of $\mathcal C$, then $(\mathcal L_0,\mathcal R_0)$ forms a factorization system if and only if $\mathrm{Fact}_{(\mathcal L,\mathcal R)}(f)$ lies in $\mathcal C_0$ for any morphism $f$ in $\mathcal C_0$.

[00KG]

Lemma B.2.2.

Let $\mathcal C$ be an $\infty$-category equipped with a factorization system $(\mathcal L,\mathcal R)$, let $\mathcal C_0 \subseteq \mathcal C$ be a subcategory, and suppose that $\mathcal L_0 \coloneqq \mathcal L\cap \mathcal C_0$ and $\mathcal R_0 \coloneqq \mathcal R\cap \mathcal C_0$ fulfill conditions (1) and (2) of observation B.2.1, so that they induce a factorization system $(\mathcal L_0, \mathcal R_0)$ on $\mathcal C_0$. Then we have:

If $\mathcal C_0$ and $\mathcal C$ are presentable, $(\mathcal L, \mathcal R)$ is of small generation, and the inclusion $\mathcal C_0 \rightarrow\mathcal C$ admits a left adjoint $L\colon \mathcal C\rightarrow\mathcal C_0$, then $(\mathcal L_0, \mathcal R_0)$ is of small generation and $L(\mathcal L) \subseteq \mathcal L_0$.
If $\mathcal C_0$ and $\mathcal C$ are furthermore presentably $\mathcal O$-monoidal for a small operad $\mathcal O$, the left adjoint $L\colon \mathcal C\rightarrow\mathcal C_0$ is $\mathcal O$-monoidal, and $(\mathcal L, \mathcal R)$ is of small generation and compatible with the $\mathcal O$-monoidal structure, then so is $(\mathcal L_0, \mathcal R_0)$.

[00KH]

Proof.

For (1), let $L\colon \mathcal C\rightarrow\mathcal C_0$ denote the left adjoint, let $S$ be a generating set for $\mathcal L$ and define $S_0$ to be the set of $\mathcal C_0$-morphisms $S_0 \coloneqq L(S)$. By adjunction, a morphism $f$ in $\mathcal C_0$ is in $S_0^{\bot}$ iff it is in $S^{\bot} = \mathcal R$, and hence that $S_0^{\bot} = \mathcal R\cap \mathcal C_0,$ proving that $(\mathcal L_0, \mathcal R_0)$ is generated by $S_0$. $L(\mathcal L) \subseteq \mathcal L_0$ follows from the adjunction.

For (2), it follows from the proof of (1) that for every $X\in \underline{\mathcal{O}}$, the class $(\mathcal L_0)_X$ is generated by $(S_0)_X \coloneqq L_X(S_X)$ where $S_X$ is a generating set for $\mathcal L_X$. Hence, to check that $(\mathcal L_0, \mathcal R_0)$ is compatible with the $\mathcal O$-monoidal structure, it suffices by lemma B.1.18 to show that for every operation $(X_1, \ldots, X_n) \rightarrow X$ in $\mathcal O$, the induced functor $(\mathcal C_0)_{X_1}\times \cdots \times (\mathcal C_0)_{X_n} \rightarrow(\mathcal C_0)_{X}$ carries morphisms in $L_{X_1}(S_{X_1}) \times \cdots \times L_{X_n}(S_{X_n})$ to a morphism in $(\mathcal L_0)_X$. But since $L$ is $\mathcal O$-monoidal, such a family of morphism is carried to the image under $L_X$ of their product in $\mathcal C_X$. Since $\mathcal L$ is compatible with the monoidal structure, that product is in $\mathcal L_X$ and hence the morphisms are carried to a morphism in $L_X(\mathcal L_X) \subseteq (\mathcal L_0)_X$. ◻

We now show that an $\infty$-category of functors automatically inherits a factorization system from one on the target.

[00KI]

Lemma B.2.3.

Let $\mathcal C$ be an $\infty$-category equipped with a factorization system $(\mathcal L, \mathcal R)$, and let $\mathcal I$ be a small $\infty$-category.

The $\infty$-category $\mathrm{Fun}(\mathcal I, \mathcal C)$ admits a factorization system $(\mathcal L^\mathcal I,\mathcal R^\mathcal I)$, in which (as the exponential notation suggests) a natural transformation between functors lies in $\mathcal L^\mathcal I$ (resp. $\mathcal R^\mathcal I$) if and only if its components all lie in $\mathcal L$ (resp. $\mathcal R$).
If $\mathcal C$ is presentable and $(\mathcal L,\mathcal R)$ is of small generation, then $\mathrm{Fun}(\mathcal I,\mathcal C)$ is presentable and $(\mathcal L^\mathcal I,\mathcal R^\mathcal I)$ is of small generation.
If $\mathcal O$ is a small operad, $I$ is $\mathcal O$-monoidal, $\mathcal C$ is presentably $\mathcal O$-monoidal and $(\mathcal L, \mathcal R)$ is compatible with the $\mathcal O$-monoidal structure on $\mathcal C$, then $(\mathcal L^I, \mathcal R^I)$ is compatible with the Day convolution $\mathcal O$-monoidal structure on $\mathrm{Fun}(I, \mathcal C)$.

[00KM]

Proof.

Part ([00KJ]) is a restatement of [Lur09, Cor. 5.2.8.18]. To prove part ([00KK]), assume that $\mathcal C$ is presentable and let $S$ be a set of morphisms in $\mathcal C$ that generates $\mathcal L$. We note first that $\mathrm{Fun}(\mathcal I,\mathcal C)$ is presentable by [Lur09, Prop. 5.5.3.6]. Now, for each functor ${\sf pt}\xrightarrow{i} \mathcal I$ (selecting an object of $\mathcal I$) we obtain an adjunction Original paper diagram It follows that $\mathcal R^\mathcal I$ is precisely the right orthogonal to the (small) space of morphisms \[S' \coloneqq \bigsqcup_{i \in \iota_0 \mathcal I} \bigsqcup_{f \in S} i_!(f)\] in $\mathrm{Fun}(\mathcal I,\mathcal C)$. From here, Proposition B.1.14 implies that $(\mathcal L^\mathcal I,\mathcal R^\mathcal I)$ is generated by $S'$ (and in particular that $\mathcal L^\mathcal I$ is the smallest saturated class of morphisms containing $S'$).

For part ([00KL]), given an n-ary operation $(X_1, \ldots, X_n) \rightarrow X$ in $\mathcal O$, the induced functor $\mathrm{Fun}(I_{X_1}, \mathcal C_{X_1}) \times \cdots \times \mathrm{Fun}(I_{X_n}, \mathcal C_{X_n}) \rightarrow\mathrm{Fun}(I_X, \mathcal C_X)$ is computed as the left Kan extension of $I_{X_1} \times \cdots \times I_{X_n} \rightarrow\mathcal C_{X_1} \times \cdots \times \mathcal C_{X_n} \rightarrow\mathcal C_X$ against $I_{X_1}\times \cdots \times I_{X_n} \rightarrow I_X$. As the left class of a factorization system, $\mathcal L$ is closed under colimits in $\mathcal C$. Therefore, the image of natural transformations $f_i \in \mathrm{Fun}(I_{X_i}, \mathcal C_{X_i})$ which are componentwise in $\mathcal L$ will again be componentwise in $\mathcal L$. ◻

A simple example of a factorization system arises from a Cartesian fibration $\mathcal E\xrightarrow{p} \mathcal B$: the total $\infty$-category $\mathcal E$ admits a factorization system $(\mathcal L,\mathcal R)$ in which $\mathcal L= p^{-1}(\mathcal B^\simeq)$ and $\mathcal R$ consists of the $p$-Cartesian morphisms. This can be generalized as follows.

[00KN]

Lemma B.2.4.

Fix an $\infty$-category $\mathcal B$ and a functor $\mathcal B^\mathrm{op}\xrightarrow{F} \widehat{\mathrm{Cat}}_\infty^{\text{f.s.}, \mathcal R}$ (recall Definition B.1.5). For each $b \in \mathcal B$, let us write $(\mathcal L_b,\mathcal R_b)$ for the given factorization system on $F(b)$. Moreover, let us write $\mathcal E\xrightarrow{p} \mathcal B$ for the Cartesian fibration associated to $F$.

The $\infty$-category $\mathcal E$ admits a factorization system $(\mathcal L,\mathcal R)$, described as follows.
1. A morphism $e \xrightarrow{\alpha} f$ lies in $\mathcal L$ if and only if the morphism $p(e) \xrightarrow{p(\alpha)} p(f)$ in $\mathcal B$ is an equivalence and moreover the morphism $e \rightarrow p(\alpha)^*(f)$ in $\mathcal E_{p(e)} \simeq F(p(e))$ lies in $\mathcal L_{p(e)}$.
2. A morphism $e \xrightarrow{\alpha} f$ lies in $\mathcal R$ if and only if the morphism $e \rightarrow p(\alpha)^*(f)$ lies in $\mathcal R_{p(e)}$.
Suppose that $\mathcal B$ is small and that $F$ factors through the subcategory $\Pr^{R,\text{f.s.},\mathcal R} \subset \widehat{\mathrm{Cat}}_\infty^{\text{f.s.},\mathcal R}$ (recall Definition B.1.15). Then, the factorization system of part ([00KP]) is also of small generation. More specifically, if for each $b \in \mathcal B$ the set $S_b$ generates the class $\mathcal L_b$, then the set $S \coloneqq \bigsqcup_{b \in \mathcal B} S_b$ generates the class $\mathcal L$ (considering each $S_b$ as defining a set of morphisms in the fiber $\mathcal E_b \simeq F(b)$).

[00KR]

Proof.

Part ([00KP]) is straightforward. Thereafter, for part ([00KQ]) it suffices to show that $\mathcal R= S^\bot$ (so that $(\mathcal L,\mathcal R)$ is indeed the factorization system generated by $S$ via proposition B.1.14). The containment $\mathcal R\subseteq S^\bot$ follows from the fact that the Cartesian monodromy functors preserve the right classes, while the containment $\mathcal R\supseteq S^\bot$ follows from the explicit description of $\mathcal R$. ◻

We now provide sufficient conditions for descending a factorization system through a reflective localization.

[00KS]

Lemma B.2.5.

Suppose that is a reflective localization and that $(\mathcal L,\mathcal R)$ is a factorization system on $\mathcal C$. Suppose further that $L(\mathcal L)$ is stable under retracts and that $RL(\mathcal R) \subseteq \mathcal R$.

The pair $(L(\mathcal L),L(\mathcal R))$ forms a factorization system on $\mathcal D$, in which the factorization of a morphism $d \xrightarrow{f} d'$ is given by the lower composite in the commutative diagram Moreover, considering $\mathcal D$ as a full subcategory of $\mathcal C$ (via $R$), the right class $L(\mathcal R)$ is intersected from $\mathcal R$ (i.e. we have $RL(\mathcal R) = \mathcal R\cap R(\mathcal D)$).
Suppose further that $\mathcal C$ and $\mathcal D$ are presentable and that $(\mathcal L,\mathcal R)$ is generated by a set $S$ of morphisms in $\mathcal C$. Then, $(L(\mathcal L),L(\mathcal R))$ is generated by the set $L(S)$ of morphisms in $\mathcal D$.
Suppose that $\mathcal C$ is (symmetric) monoidal compatible with both the reflective localization (recall Subsection A.8.9) and with the factorization system (recall Definition B.1.12). Then, the induced (symmetric) monoidal structure on $\mathcal D$ is compatible with the factorization system $(L(\mathcal L),L(\mathcal R))$.

[00KX]

Warning B.2.6.

Although the right adjoint to a reflective localization is (by definition) the inclusion of a full subcategory, the factorization system given by Lemma B.2.5.([00KT]) is generally not the same as that of Observation B.2.1. More specifically, although its right class is simply the restriction of the larger right class, its left class is not generally the restriction of the larger left class: using the notation of Lemma B.2.5, although we do have $RL(\mathcal R) = \mathcal R\cap R(\mathcal D)$, in general $RL(\mathcal L)$ and $\mathcal L\cap R(\mathcal D)$ are distinct. Indeed, by Observation B.2.1, these factorization systems coincide if and only if for every morphism $f$ in $\mathcal D$ the object $\mathrm{Fact}_{(\mathcal L,\mathcal R)}(R(f)) \in \mathcal C$ lies in the image of $R$.⁷¹

[00KY]

Proof of Lemma B.2.5.

We begin with part ([00KT]).

Note first that in diagram ([00KU]), the lower diagonal morphisms respectively lie in $L(\mathcal L)$ and $L(\mathcal R)$, so this is indeed a factorization of the desired type.

Next, $L(\mathcal L)$ is stable under retracts by assumption. To see that $L(\mathcal R)$ is also stable under retracts, consider a retract $g$ in $\mathrm{Fun}([1],\mathcal D)$ of some $L(f) \in L(\mathcal R)$. Applying $R$, we find that $R(g)$ is a retract in $\mathrm{Fun}([1],\mathcal C)$ of $RL(f) \in RL(\mathcal R)$. Since by assumption $RL(\mathcal R) \subseteq \mathcal R$, it follows that $RL(f) \in \mathcal R$, and hence $R(g) \in \mathcal R$ since $\mathcal R$ is stable under retracts. It follows that $g \simeq LR(g) \in L(\mathcal R)$, as desired.

We now verify the orthogonality relation $L(\mathcal L) \bot L(\mathcal R)$. For this, given any $f \in \mathcal L$ and any $g \in \mathcal R$, we must show that $L(f) \bot L(g)$. By adjunction, this is equivalent to showing that $f \bot RL(g)$. But by assumption we have $RL(g) \in RL(\mathcal R) \subseteq \mathcal R$, and so the claim follows from the fact that $\mathcal L\bot \mathcal R$.

We now verify both containments that together constitute the claim that $RL(\mathcal R) = \mathcal R\cap R(\mathcal D)$. First of all, by assumption we have $RL(\mathcal R) \subseteq \mathcal R$, and moreover clearly $L(\mathcal R) \subseteq \mathcal D$ and hence $RL(\mathcal R) \subseteq R(\mathcal D)$. So indeed, we have $RL(\mathcal R) \subseteq \mathcal R\cap R(\mathcal D)$. In the other direction, consider an arbitrary element $R(f) \in \mathcal R\cap R(\mathcal D)$. In particular we have $R(f) \in \mathcal R$, so $LR(f) \in L(\mathcal R)$, so $R(f) \simeq RLR(f) \in RL(\mathcal R)$. So indeed, we have $RL(\mathcal R) \supseteq \mathcal R\cap R(\mathcal D)$.

We now prove part ([00KV]). By Proposition B.1.14, it suffices to show that $L(\mathcal R) = L(S)^\bot$. To verify the containment $L(\mathcal R) \subseteq L(S)^\bot$, it is equivalent by adjunction to check that $RL(\mathcal R) \subseteq S^\bot$, and this follows from the fact that $RL(\mathcal R) \subseteq \mathcal R= S^\bot$. To verify the containment $L(\mathcal R) \supseteq L(S)^\bot$, we observe that for any $f \in L(S)^\bot$, by adjunction we have $R(f) \in S^\bot = \mathcal R$, so indeed $f \simeq LR(f) \in L(\mathcal R)$.

We conclude by proving part ([00KW]). Note that it suffices to prove the claim in the monoidal case. And here, the claim follows from the observation that \[L(\mathcal L) \otimes^\mathcal DL(\mathcal L) \coloneqq L(RL(\mathcal L) \otimes^\mathcal CRL(\mathcal L)) \simeq L(\mathcal L\otimes^\mathcal C\mathcal L) \subseteq L(\mathcal L) ~,\] in which the equivalence and the containment respectively follow from the compatibilities of the reflective localization with the monoidal structure $\otimes^\mathcal C$ and with the factorization system $(\mathcal L,\mathcal R)$. ◻

We now turn to our final auxiliary result, which gives sufficient conditions for inducing a factorization system through a monadic adjunction.

[00KZ]

Lemma B.2.7.

Fix a monadic adjunction between presentable $\infty$-categories (where $T \coloneqq UF$ denotes the underlying monad). Suppose that $(\mathcal L,\mathcal R)$ is a factorization system on $\mathcal C$ of small generation, and suppose further that $T$ commutes with geometric realizations and preserves $\mathcal L$. Then, $\mathcal D$ admits a factorization system $(\mathcal L',\mathcal R')$ of small generation, where $\mathcal L' = U^{-1}(L)$ and $\mathcal R' = U^{-1}(\mathcal R)$.

Furthermore, assume that $\mathcal C$, $\mathcal D$ are presentably $\mathcal O$-monoidal for a small $\infty$-operad $\mathcal O$, and the left adjoint $\mathcal C\rightarrow\mathcal D$ is $\mathcal O$-monoidal. If the $\mathcal O$-monoidal structure on $\mathcal C$ is compatible with the factorization system, then the $\mathcal O$-monoidal structure on $\mathcal D$ is compatible with the induced factorization system on $\mathcal D$.

[00L0]

Proof.

We begin by fixing a small set $S$ of morphisms in $\mathcal C$ that generates $\mathcal L$. By Proposition B.1.14, we obtain a factorization system $(\mathcal L',\mathcal R')$ on $\mathcal D$ generated by its image $F(S)$. So, it remains to show that $\mathcal L' = U^{-1}(L)$ and that $\mathcal R' = U^{-1}(R)$.

We first show that $\mathcal R' = U^{-1}(R)$. For this, note that by definition $\mathcal R' = F(S)^\perp$. Hence, it suffices to show that a morphism lies in $F(S)^\perp$ precisely if its image under $U$ lies in $\mathcal R$, which follows from the adjunction $F \dashv U$ (and the fact that $\mathcal R= S^\perp$).

We now show that $\mathcal L' = U^{-1}(\mathcal L)$. For this, let us write $\mathcal L'' \coloneqq U^{-1}(\mathcal L)$, so that our goal is to show that $\mathcal L' = \mathcal L''$. Since $\mathcal L' = \overline{F(S)}$, it is equivalent to show that $\overline{F(S)} = \mathcal L''$. In other words, it suffices to verify that $\mathcal L''$ is the smallest saturated class of morphisms in $\mathcal D$ that contains $F(S)$.

We deduce this in steps. First of all, the fact that $\mathcal L''$ contains $F(S)$ follows from the assumption that the monad $T$ preserves $\mathcal L$ and the fact that $\mathcal L$ contains $S$.

We now show that $\mathcal L''$ is saturated by verifying the conditions of Definition B.1.13.

Condition ([00JW]) is clear: $\mathcal L''$ defines a wide subcategory of $\mathcal D$.

We now verify condition ([00JX]), i.e. we show that the full subcategory $\mathcal L'' \subseteq \mathrm{Fun}([1],\mathcal D)$ is closed under small colimits. For this, fix a small $\infty$-category $\mathcal I$ as well as a functor $\mathcal I\xrightarrow{X} \mathrm{Fun}([1],\mathcal D)$ that factors through $\mathcal L''$. We wish to show that the colimit $\mathrm{colim}_\mathcal I(X)$ (computed in $\mathrm{Fun}([1],\mathcal D)$) also lies in $\mathcal L''$, i.e. that $U(\mathrm{colim}_\mathcal I(X)) \in \mathrm{Fun}([1],\mathcal C)$ lies in $\mathcal L$. Now, since the adjunction $F \dashv U$ is monadic, every object $D \in \mathcal D$ admits a functorial bar resolution: it is the geometric realization of the levelwise free simplicial object $FT^\bullet U(D) \in \mathrm{Fun}(\Delta^{\mathrm{op}}, \mathcal D)$ [Lur17, Prop. 4.7.3.14]. Using this, we may compute $\mathrm{colim}_\mathcal I(X)$ as the colimit of the functor Original paper diagram Namely, we obtain the string of equivalences \[\begin{aligned} U(\mathrm{colim}_\mathcal I(X)) & \simeq U(\mathrm{colim}_{\mathcal I\times \Delta^{\mathrm{op}}}(X')) \\ & \simeq U( \mathrm{colim}_{[n] \in \Delta^{\mathrm{op}}} ( \mathrm{colim}_{i \in \mathcal I} ( X'(i,[n])))) \\ & \simeq \mathrm{colim}_{[n] \in \Delta^{\mathrm{op}}} U(\mathrm{colim}_{i \in \mathcal I} ( X'(i,[n]))) \\ & \eqqcolon \mathrm{colim}_{[n] \in \Delta^{\mathrm{op}}} U(\mathrm{colim}_{i \in \mathcal I}(FT^nU(X(i)))) \\ & \simeq \mathrm{colim}_{[n] \in \Delta^{\mathrm{op}}} UF(\mathrm{colim}_{i \in \mathcal I}(T^nU(X(i)))) \\ & \eqqcolon \mathrm{colim}_{[n] \in \Delta^{\mathrm{op}}} T(\mathrm{colim}_{i \in \mathcal I}(T^n U(X(i)))) ~, \end{aligned}\] in which the third equivalence follows from the fact that $U$ commutes with geometric realizations since $T$ does by [Lur17, Cor. 4.2.3.5]. Now, by definition of $\mathcal L'' \coloneqq U^{-1}(\mathcal L)$, for each object $i \in \mathcal I$ the object $U(X(i)) \in \mathrm{Fun}([1],\mathcal C)$ lies in $\mathcal L$. Using repeatedly both the fact that $T$ preserves $\mathcal L$ and that $\mathcal L\subseteq \mathrm{Fun}([1],\mathcal C)$ is closed under colimits, we find that $U(\mathrm{colim}_\mathcal I(X)) \in \mathrm{Fun}([1],\mathcal C)$ lies in $\mathcal L$, as desired. So indeed, $\mathcal L'' \subseteq \mathrm{Fun}([1],\mathcal D)$ is closed under colimits.

The verification of condition ([00JY]) (that $\mathcal L'' \subseteq \mathrm{Fun}([1],\mathcal D)$ is stable under cobase change) follows from an essentially identical argument (inasmuch as it involves the computation of a colimit (specifically a pushout) in $\mathrm{Fun}([1],\mathcal D)$). So indeed, the class $\mathcal L''$ of morphisms in $\mathcal D$ is saturated.

In order to conclude that $\mathcal L'' = \overline{F(S)}$, it therefore remains to show that any saturated class $\mathcal L'''$ of morphisms in $\mathcal D$ that contains $F(S)$ also contains $\mathcal L''$. Since $F$ preserves colimits, certainly $F(\mathcal L) \subseteq \mathcal L'''$. From here, to show the containment $\mathcal L'' \subseteq \mathcal L'''$, choose any $f \in \mathcal L'' \coloneqq U^{-1}(\mathcal L)$. Recall that the aforementioned bar resolution yields an equivalence $|FT^\bullet U(f)| \simeq f$. Note that $U(f) \in \mathcal L$, and since $T$ preserves $\mathcal L$ then $T^nU(f) \in \mathcal L$, and so all values of the simplicial object $FT^\bullet U(f)$ lie in ${\sf{F}}(\mathcal L) \subseteq \mathrm{Fun}([1],\mathcal D)$. Hence, its geometric realization – namely, $f$ – must lie in $\mathcal L'''$. So indeed, $\mathcal L''$ is the smallest saturated class of morphisms in $\mathcal D$ containing $F(S)$.

It remains to prove the compatibility with $\mathcal O$-monoidal structure. Given an operation $(a_1, ..., a_m) \rightarrow b$ in $\mathcal O$, we want to show that the induced functor $\mu \colon \mathcal D_{a_1} \times \cdots \times \mathcal D_{a_m} \rightarrow\mathcal D_b$ carries $\mathcal L'_{a_1} \times \cdots \times \mathcal L'_{a_m}$ to $\mathcal L'_b$. Explicitly, given morphisms $X_i \colon [1] \rightarrow\mathcal D_{a_i}$ in $\mathcal L'_{a_i}$, we would like to show that $U(\mu(X_1, \cdots, X_m)) \in \mathcal L$. As above, the bar resolution gives us a simplicial object $X'_i \colon \Delta^{\mathrm{op}}\rightarrow\mathrm{Fun}([1], \mathcal D_{a_i})$ for each $1 \leq i \leq n$, with $X'_i([n]) = FT^nU(X_i)$.

We have a string of equivalences: \[ \begin{aligned} U(\mu(X_1, \cdots, X_n)) & \simeq U(\mu( \mathrm{colim}_{[n_1] \in \Delta^{\mathrm{op}}} X'_1([n_1]), \cdots, \mathrm{colim}_{[n_m] \in \Delta^{\mathrm{op}}} X'_m([n_m]))) \\ & \simeq U(\mathrm{colim}_{([n_1], \cdots, [n_m]) \in {(\Delta^{\mathrm{op}})}^m} \mu(X'_1([n_1]), \cdots, X'_m([n_m])))\\ & \simeq U(\mathrm{colim}_{[n]\in \Delta^{\mathrm{op}}} \mu(X'_1([n]), \cdots, X'_m([n])))\\ & \simeq \mathrm{colim}_{[n]\in \Delta^{\mathrm{op}}} U(\mu(X'_1([n]), \cdots, X'_m([n])))\\ & \simeq \mathrm{colim}_{[n]\in \Delta^{\mathrm{op}}} U(\mu(FT^nU(X_1), \cdots, FT^nU(X_m)))\\ & \simeq \mathrm{colim}_{[n]\in \Delta^{\mathrm{op}}} UF(\mu(T^nU(X_1), \cdots, T^nU(X_m)))\\ & \simeq \mathrm{colim}_{[n]\in \Delta^{\mathrm{op}}} T(\mu(T^nU(X_1), \cdots, T^nU(X_m))) ~, \end{aligned}\] in which the third line uses the fact that the diagonal $\Delta^{\mathrm{op}}$ in ${(\Delta^{\mathrm{op}})}^n$ is cofinal, which is equivalent to the statement that $\Delta^{\mathrm{op}}$ is sifted [Lur17, Def. 5.5.8.1, Lem. 5.5.8.4]. For $1 \leq i \leq n$, $U(X_i) \in \mathcal L$ by asssumption. It follows that $T^nU(X_i)$ is also in $\mathcal L$. Since the $\mathcal O$-monoidal structure on $\mathcal C$ is compatible with the factorization system and $T$ preserves $\mathcal L$, we see that $T(\mu(T^nU(X_1), \cdots, T^nU(X_m)))$ is in $\mathcal L$. The result now follows from ([00L1]) and the fact that $\mathcal L$ is closed under colimits. ◻

B.3 Factorization systems for algebras over $\infty$-operads[00L2]

We now prove the first main theorem of this appendix, which gives factorization systems for algebras over $\infty$-operads.

[00L3]

Theorem B.3.1.

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). ◻

B.4 Factorization systems for enriched $\infty$-categories[00LB]

We now prove the second main theorem of the appendix, which gives factorization systems for enriched $\infty$-categories (using those for algebras over $\infty$-operads).

[00LC]

Theorem B.4.1.

Let $\mathbb V$ be a presentably monoidal $\infty$-category equipped with a compatible factorization system $(\mathcal L, \mathcal R)$.

The $\infty$-category $\mathrm{Cat}[\mathbb V]$ of $\mathbb V$-enriched $\infty$-categories admits a factorization system $(\mathcal L_\mathrm{Cat},\mathcal R_\mathrm{Cat})$, described as follows.
1. A morphism lies in $\mathcal L_\mathrm{Cat}$ if and only if it is surjective on objects (i.e. $\iota_0$-surjective) and lies in $\mathcal L$ homwise.
2. A morphism lies in $\mathcal R_\mathrm{Cat}$ if and only if it lies in $\mathcal R$ homwise.
If $S$ is a set of generators for $\mathcal L$, then the localization of $\Sigma[S]$ is a set of generators for $\mathcal L_\mathrm{Cat}$.
If $\mathbb V$ is symmetric monoidal, then this factorization system is compatible with the resulting symmetric monoidal structure on $\mathrm{Cat}[\mathbb V]$.

[00LG]

Remark B.4.2.

Theorem B.4.1 generalizes the (fully faithful, essentially surjective) factorization system on enriched $\infty$-categories established by Haugseng in the recent work [Hau23] (without any presentability assumptions).

Before we prove Theorem B.4.1, let us first consider factorization systems on categorical algebras:

[00LH]

Lemma B.4.3.

Fix a presentably monoidal $\infty$-category $\mathbb V$ equipped with a compatible factorization system $(\mathcal L,\mathcal R)$.

The $\infty$-category $\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]$ of categorical $\mathbb V$-algebras admits a factorization system $(\mathcal L_{\mathrm{Alg}_{\mathrm{Cat}}},\mathcal R_{\mathrm{Alg}_{\mathrm{Cat}}})$, described as follows.
1. A morphism lies in $\mathcal L_{\mathrm{Alg}_{\mathrm{Cat}}}$ if and only if it is an $\iota_0$-equivalence and it lies in $\mathcal L$ homwise.
2. A morphism lies in $\mathcal R_{\mathrm{Alg}_{\mathrm{Cat}}}$ if and only if it lies in $\mathcal R$ homwise.
If $S$ is a set of generators for $\mathcal L$, then $\Sigma[S] \coloneqq \{ \Sigma(s)\}_{s \in S}$ is a set of generators for $\mathcal L_{\mathrm{Alg}_{\mathrm{Cat}}}$.
If $\mathbb V$ is symmetric monoidal, then the factorization system $(\mathcal L_{\mathrm{Alg}_{\mathrm{Cat}}},\mathcal R_{\mathrm{Alg}_{\mathrm{Cat}}})$ is compatible with the resulting symmetric monoidal structure on $\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]$.

[00LL]

Proof.

By definition, the Cartesian fibration $\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V] \xrightarrow{\iota_0} \mathcal S$ is the unstraightening of a composite functor \[\mathcal S^{\mathrm{op}} \xrightarrow{{\textup{codisc}}} (\mathrm{Op}_{/\mathbb E_1})^{\mathrm{op}} \xrightarrow{\mathrm{Alg}_{(-)/\mathbb E_1}(\mathbb V)} {\Pr}^R ~.\]^* For a space $X \in \mathcal S$, its corresponding $\infty$-operad ${\textup{codisc}}(X) \in \mathrm{Op}_{/\mathbb E_1}$ has space of colors given by pairs of points $x,y \in X$ (up to a symmetrization (i.e. the quotient by the $\mathfrak S_2$-action) coming from [Lur17, Thm. 4.1.3.14]), and a categorical $\mathbb V$-algebra $\mathcal C$ with space of objects $X$ assigns to these the hom-object $\mathrm{Hom}_\mathcal C(x,y) \in \mathbb V$. Hence, checking conditions on morphisms in $\underline{\mathbb V}$ colorwise over ${\textup{codisc}}(X)$ indeed corresponds to checking conditions on morphisms homwise, and thereafter part ([00LI]) follows by combining Theorem B.3.1.([00L7]) and Lemma B.2.4.

Thereafter, part ([00LJ]) follows from the observation that $(\Sigma[S])^\bot = \mathcal R_{\mathrm{Alg}_{\mathrm{Cat}}}$, which is immediate from the universal property of $\Sigma[-]$.

Lastly, part ([00LK]) follows from the assumption that $(\mathcal L,\mathcal R)$ is compatible with the monoidal structure of $\mathbb V$. ◻

[00LM]

Proof of Theorem B.4.1.

Given the factorization system on $\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]$ of Lemma B.4.3, we wish to apply Lemma B.2.5 to the reflective localization Original paper diagram We note preliminarily that the morphisms in $\mathrm{Cat}[\mathbb V]$ that are localizations of $\iota_0$-equivalences in $\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]$ are precisely the $\iota_0$-surjections.

We first show that the hypotheses of Lemma B.2.5 are satisfied. To show that $RL(\mathcal R_{\mathrm{Alg}_{\mathrm{Cat}}}) \subseteq \mathcal R_{\mathrm{Alg}_{\mathrm{Cat}}}$, we simply observe that if a morphism $F$ in $\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]$ is homwise in $\mathcal R$ then so is its localization $L(F)$ and hence so is $RL(F)$. To show that $L(\mathcal L_{\mathrm{Alg}_{\mathrm{Cat}}})$ is stable under retracts, it suffices to observe that $\mathcal L$ is stable under retracts (by definition of a factorization system) and that surjections in $\mathcal S$ are stable under retracts (since surjections in $\mathrm{Set}$ are).

From here, the three parts of Lemma B.2.5 respectively imply the three parts of the present result. ◻

We refer the reader to Section A for a brief introduction to $\infty$-categories.↩︎
Indeed, as mentioned in [MR20], such a construction “requires working in a suitable homotopical setting ($\mathbb{A}_{\infty}$- or $\infty$-categories), and this creates technical complications for the full construction of a braided monoidal $2$-categorical structure”, also see [Rou19].↩︎
Such $\infty$-categories can be modelled by small pretriangulated idempotent-complete $k$-linear dg-categories [Coh16].↩︎
We stress the distinction from the pointwise symmetric monoidal structure; Day convolution uses the symmetric monoidal structure of $B\mathbb{Z}$.↩︎
This is directly analogous to the classical situaton in which $\mathrm{Rep}(\mathbb{Z}) \simeq \mathrm{Fun}(U(1), \mathrm{Vec})$ has two canonical symmetric monoidal structures: a pointwise one, and one arising from Day convolution.↩︎
Although $\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]$ can be seen as an $(\infty,3)$-category (in fact it is self-enriched), the notion of an $\mathbb E_2$-algebra therein only makes references to its underlying $(\infty,1)$-category.↩︎
Beware that these are not generally inclusions of subobjects, i.e. monomorphisms, in $\mathrm{Cat}_{(\infty,2)},$ see warning 5.3.9.↩︎
The reader should be warned that the inclusion $\mathrm{Sbim}\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})$ is merely a faithful functor but not a monomorphism in $\mathrm{Cat}_{(\infty,2)}$, see warning 5.3.9.↩︎
A pointed object $\mathcal A$ is an object together with a morphism $\eta_{\mathcal A} \colon \mathbbm{1}_{\mathcal V} \rightarrow\mathcal A$, equivalently, an $\mathbb E_0$-algebra. A pointed morphism is a morphism that respect that pointing, equivalently, an $\mathbb E_0$-algebra map.↩︎
On the level of ordinary $1$-categories, this statement is usually implicitly assumed in the literature on Bott-Samelson and Soergel bimodules and crucial when passing between algebraic or Lie theoretic definitions and diagrammatical versions of these important categories.↩︎
Note that this is not the same concept as a $k$-linear category that is also graded by a group.↩︎
By a finitely generated graded-projective $A$-module for a graded algebra $A$ we mean an $A$-module that is a retract of a finite direct sum of grading shifts of the rank 1 free $A$-module along grading preserving maps.↩︎
The chain homotopy category $\mathrm{K}^b(\mathcal C)$ is often just called ‘homotopy category’; we here reserve the latter term for the more general construction of a $1$-category from a higher category, see Definition 5.4.11.↩︎
[Lur17, Prop. 1.3.3.14] does not apply directly since the full subcategory $\mathcal A^{\mathrm{c}}$ of compact objects in a presentable abelian category $\mathcal A$ is not necessarily itself abelian.↩︎
Explicitly, under the equivalence $\mathrm{Hom}_{\mathbb V}(\underline{\mathrm{Hom}}_{\mathbb V}(v,v'), G\underline{\mathrm{Hom}}_{\mathbb W}(Fv, Fv')) \simeq \mathrm{Hom}_{\mathbb W}(F\underline{\mathrm{Hom}}_{\mathbb V}(v,v'), \underline{\mathrm{Hom}}_{\mathbb W}(Fv, Fv')) \simeq \mathrm{Hom}_{\mathbb W}(F\underline{\mathrm{Hom}}_{\mathbb V}(v,v') \otimes Fv, Fv') \simeq\mathrm{Hom}_{\mathbb W}(F(\underline{\mathrm{Hom}}_{\mathbb V}(v,v') \otimes v), Fv')$, the morphism becomes the $F$- image of the counit $\underline{\mathrm{Hom}}_{\mathbb V}(v, v') \otimes v \rightarrow v'$.↩︎
A $\mathbb V$-enriched $\infty$-category $\mathcal C$ admits tensors if the $\mathbb V$-enriched functor $\underline{\mathrm{Hom}}_{\mathbb V}(v, \underline{\mathrm{Hom}}_{\mathcal C}(c, -)) \colon \mathcal C\rightarrow\mathbb V$ is corepresentable for all $v\in \mathbb V$ and $c \in \mathcal C$. Denoting the corepresenting objects by $v\otimes c \in \mathcal C$, this induces an action of $\mathbb V$ on the underlying $\infty$-category of $\mathcal C$.↩︎
Equivalently, by [MS21, Lem. A.2.14], $\mathbb V$-enriched functors whose underlying functor is left adjoint and preserves the induced $\mathbb V$-action.↩︎
If $\mathcal Z$ is a grouplike commutative monoid, the delooping $B\mathcal Z$ is simply the classifying space of $\mathcal Z$ with its induced grouplike commutative monoid structure.↩︎
Recall that a module $M$ over a ring $k$ is called flat if $- \otimes_k M$ is an exact functor.↩︎
For $\mathbb V$ a Cartesian symmetric monoidal $\infty$-category, the induced symmetric monoidal structure on $\mathrm{Cat}[\mathbb V]$ is also Cartesian: by construction, the symmetric monoidal structure on $\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]$ is Cartesian; since the inclusion $\mathrm{Cat}[\mathbb V] \hookrightarrow \mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]$ is a right adjoint and preserves products, it follows that the symmetric monoidal structure on $\mathrm{Cat}[\mathbb V]$ is also Cartesian.↩︎
One can see that the functor $\mathrm{Cat}_\infty \xrightarrow{|-|} \mathcal S$ preserves finite products e.g. by observing that it can be computed as the geometric realization of complete Segal spaces and $\Delta^{\mathrm{op}}$ is sifted [Lur09, Cor. 4.2.3.5].↩︎
In the case that $k=0$, we may instead respectively refer to these as the $\infty$-groupoid completion and maximal sub-$\infty$-groupoid (or even simply maximal subgroupoid) functors.↩︎
Note that the functor $[1] = c_1 \rightarrow c_0 = {\sf pt}$ is a localization at the universal $1$-morphism: it is merely a condition that a functor $c_1 \xrightarrow{\alpha} \mathcal C$ admit an extension along it, namely that $\alpha$ selects an equivalence. By induction (using the universal property of $\Sigma[-]$), it follows that the functor $c_{n+1} \xrightarrow{\Sigma^n[c_1 \rightarrow c_0]} c_n$ is a localization at the universal $(n+1)$-morphism: it is merely a condition that a functor $c_{n+1} \xrightarrow{\alpha} \mathcal C$ admit an extension along it, namely that $\alpha$ selects an invertible $(n+1)$-morphism.↩︎
Here, $\mathbb V\xrightarrow{\Sigma[-]} \mathrm{Cat}[\mathbb V]$ denotes the “categorical suspension” functor (see Subsection A.10). As the name suggests, $c_n$ is the free $(\infty,n)$-category on an $n$-morphism.↩︎
Note that $\partial c_1 = S^0 = c_{0} \sqcup_{\partial c_{0}} c_{0}$. Since $\Sigma$ commutes with colimits, we see that $\partial c_n = c_{n-1} \sqcup_{c_{n-1}} c_{n-1}$ for $n \geq 1$, i.e., $\partial c_n$ indeed corepresents pairs of parallel $(n-1)$-morphisms.↩︎
Readers be aware that it is also common to refer to $n$-connected spaces and maps as “$(n+1)$-connective”, see for example [Lur09, Terminology].↩︎
We nevertheless continue to use the terms “$n$-connected” and “$n$-truncated” (referring to maps of spaces) since they play a fundamental role as the base case of our inductive definitions.↩︎
This is the unique $(\infty,k)$-category equipped with a factorization $\mathcal C\rightarrow\mathrm{Fact}_n(F) \rightarrow\mathcal D$ of $F$ via an $n$-surjective functor followed by an $n$-faithful functor (see Observation B.1.10 and Notation B.1.11).↩︎
For general $k$, a morphism is $n$-truncated if and only if it is so on spaces of $i$-morphisms for all $0 \leq i \leq k$.↩︎
Using the notation of Definition B.1.5, this is to say that the diagram $\mathrm{Cat}_{(\infty, {0})} \xrightarrow{i_1} \mathrm{Cat}_{(\infty, {1})} \xrightarrow{i_2} \cdots$ lies in $\widehat{\mathrm{Cat}}_\infty^{\text{f.s.},\mathcal L,\mathcal R}$ when we equip all of these $\infty$-categories with their ($n$-surjective, $n$-faithful) factorization systems.↩︎
At first glance, it may be confusing that e.g. for an $(\infty,2)$-category $\mathcal C$ to be a $(2,2)$-category we only need to impose a condition on its spaces of $2$-morphisms: one might wonder whether e.g. its space of $1$-morphisms $\mathrm{Hom}_{\mathrm{Cat}_{(\infty, {2})}}(c_1,\mathcal C)$ might have higher homotopy groups. But actually, path spaces of the latter contribute path components of the former. (Indeed, this same reasoning shows that the space of objects of a $(1,1)$-category must in fact be a $1$-groupoid.)↩︎
Beware that (despite our usage of the letter $\tau$) this is not the abstractly-defined “$n$-truncation” functor in $\mathrm{Cat}_{(\infty, {k})}$ (see Warning 5.3.9)↩︎
All of the results in this section which do not specifically refer to the categories of Bott-Samelson and Soergel bimodules apply generally to arbitrary connective ring spectra $k \in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})$, and to arbitrary commutative monoids $Z \in \mathrm{CAlg}(\mathcal S)$ in place of $\mathbb{Z}$.↩︎
In this section, and the rest of the paper, we abuse notation and let $k$ denote both the commutative ring, and the induced commutative ring spectrum $Hk$ and simply write $\mathrm{add}_k$ and $\mathrm{st}_k$ instead of $\mathrm{add}_{Hk}$ and $\mathrm{st}_{Hk}$ as in §4.↩︎
As a subcategory of $\mathrm{Sbim}$, $\mathrm{BSbim}$ then inherits its local $\mathbb{Z}$-action from $\mathrm{Sbim}$.↩︎
Given $f,g \in \mathrm{Ar}^{-1}(\mathcal C)$, the fiber of the induced map on hom-spaces $\mathrm{Hom}_{\mathrm{Ar}(\mathcal C)} (f, g) \rightarrow\mathrm{Hom}_{\mathcal C}(tf, tg)$ at an $h \in \mathrm{Hom}_{\mathcal C}(tf, tg)$ is the space of lifts $\mathrm{Hom}_{\mathcal C_{/tg}}(h \circ f, g)$. Since $g$ is $(-1)$-truncated, this space is $(-1)$-truncated.↩︎
As in section 6, all results in this section which do not specifically refer to the categories of Bott-Samelson and Soergel bimodules apply more generally to arbitrary connective ring spectra $k \in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})$, and to arbitrary commutative monoids $\mathcal Z\in \mathrm{CAlg}(\mathcal S)$ in place of $\mathbb{Z}$.↩︎
The forgetful functor $\mathrm{st}^{B\mathbb{Z}}_{k}\rightarrow\mathrm{Cat}_{\infty}$ is right adjoint to the composite $\mathrm{Cat}_{\infty}\xrightarrow{\mathrm{Lin}_k(- \times \mathbb{Z})} \mathrm{add}_{k}^{B\mathbb{Z}}\xrightarrow{{\mathbf K}^b}\mathrm{st}^{B\mathbb{Z}}_{k}$ of the symmetric monoidal functors from ([00CN]) and proposition 4.3.2 ([0084]) and is hence laxly symmetric monoidal.↩︎
The left diagram exists since $Q \rightarrow A$ and $Q \rightarrow C$ are in $\mathcal L$, while $B|_{Q} \rightarrow B$ and $D|_{Q} \rightarrow D$ are in $\mathcal R$.↩︎
For instance, the associativity isomorphisms must satisfy the pentagon axiom, which articulates a coherence condition among the five possible ways of composing a sequence of four composable morphisms.↩︎
Indeed, the homotopy hypothesis can be taken as one motivation (of many) for a theory of weak $n$-categories in the first place: it can be seen directly that strict 3-groupoids do not model all homotopy 3-types [Sim98].↩︎
Recall that for $0 \leq k \leq n$, the term “$(n,k)$-category” refers to an $n$-category whose $i$-morphisms are all invertible for all $i > k$. So an $(n,n)$-category is just an $n$-category (without further conditions), while an $(n,0)$-category is an $n$-groupoid.↩︎
This is equivalent to the condition that the commutative square is a pullback square. (In particular, if a functor commutes with pullbacks then it preserves monomorphisms.)↩︎
This claim is easy to check in the model for $\infty$-categories given by complete Segal spaces [Rez01].↩︎
The classical notion of geometric realization that carries a simplicial set to a topological space is a homotopy colimit (of the corresponding levelwise discrete simplicial topological space), which provides a concrete model for the colimit in $\mathcal S$ of the corresponding levelwise discrete simplicial space. So, these two uses of the term are spiritually compatible.↩︎
If $\mathcal C$ admits a terminal object ${\sf pt}_\mathcal C\in \mathcal C$, then this locality condition is equivalent to the orthogonality relation $(d \rightarrow e) \bot (c \rightarrow{\sf pt}_\mathcal C)$ (see Definition B.1.1).↩︎
As a non-example, binary coproducts in $\mathcal C_{c/}$ are computed by pushouts in $\mathcal C$.↩︎
For instance, the commutative square ([00IJ]) selects an associator for $M$ (i.e. a path in $\mathrm{Hom}_\mathcal S(M^{\times 3} , M)$), and thereafter we can locate the pentagon axiom as the image in $\mathrm{Hom}_\mathcal S(M^{\times 4},M)$ of the (tautological and unique) nullhomotopy of a certain pentagon in $\mathrm{Hom}_{\Delta^\mathrm{op}}([4],[1]) \in \mathrm{Set}\subset \mathcal S$.↩︎
See [Maz19] for a more leisurely description.↩︎
These are also frequently referred to as “Beck–Chevalley squares”.↩︎
This is frequently referred to as a “Beck–Chevalley condition” on the original commutative square.↩︎
In any symmetric monoidal $\infty$-category the unit object is canonically a commutative algebra object, and here this recovers the Cartesian symmetric monoidal structure on $\mathcal S$ (which is a presentably symmetric monoidal structure).↩︎
The usage of an abstract finite set $I$ here (as opposed to $\{1,\ldots,n\}$) is convenient since it naturally builds in the relevant symmetric group actions.↩︎
This topological space is homotopy equivalent to that of framed embeddings $(\mathbb R^k)^{\sqcup n} \hookrightarrow\mathbb R^k$, under which composition of multimorphisms in $\mathbb E_k$ corresponds to composition of framed embeddings.↩︎
This latter perspective is effectively a generalization of the bar construction indicated in Subsection A.3, and it further generalizes to $\mathcal O$-algebra objects in any Cartesian symmetric monoidal $\infty$-category [Lur17, § 2.4.2] (which notion is defined shortly).↩︎
Hence, this framework adheres closely to the “microcosm/macrocosm principle”: it is precisely an $\mathcal O$-monoidal structure on an $\infty$-category that allows us to contemplate $\mathcal O$-algebra objects therein. In particular, one can make sense of $\mathcal O$-algebra objects for any $\infty$-operad $\mathcal O$ inside of a symmetric monoidal $\infty$-category (since $\text{Comm}\in \mathrm{Op}$ is terminal).↩︎
This follows from the explicit construction of the $\infty$-operad $\mathrm{Alg}_{\mathcal O}(\mathcal P)$ in [Lur17, Const. 3.2.4.1]↩︎
This is also technically a $\mathcal C$-algebra object in $\mathcal D$ (relative to $\mathcal O$), although we find the present terminology to be more illuminating.↩︎
Even if $\otimes^\mathcal C$ is a symmetric monoidal structure, this compatibility only depends on its underlying monoidal structure.↩︎
An illustrative example is the completed tensor product of modules over a topological commutative ring (e.g. a local commutative ring $(R,\mathfrak{m})$ equipped with the $\mathfrak{m}$-adic topology).↩︎
Indeed, such data define an adjunction in the $(\infty,2)$-category $\mathrm{Alg}_\mathcal O(\mathrm{Cat}_\infty)^{\textup{lax}}$, to which we may apply $\underline{\mathrm{Hom}}_{\mathrm{Alg}_\mathcal O(\mathrm{Cat}_\infty)}(\mathcal O\leftarrow \mathcal O_\mathrm{Triv},-)$.↩︎
This theory is effectively a generalization of the theory of complete Segal spaces [Rez01], with the univalence being obtained by restricting to those from all Segal spaces. In particular, using the terminology introduced just below, an ordinary category in the classical sense (defined in terms of a set of objects) is equivalently a categorical $\mathrm{Set}$-algebra whose space of objects is discrete (or equivalently a Segal set). By contrast, an object of $\mathrm{Cat}[\mathrm{Set}] \subset \mathrm{Cat}[\mathcal S] \simeq \mathrm{Cat}_\infty$ – equivalently, an $\infty$-category whose hom-spaces are all discrete – merely has a 1-truncated $\infty$-groupoid of objects, a surjection to which from a set determines a presentation thereof as an ordinary category in the classical sense.↩︎
Beware that the functor $\mathrm{Cat}[\mathbb V] \xrightarrow{\iota_0} \mathcal S$ is not generally symmetric monoidal: for $\mathcal C,\mathcal D\in \mathrm{Cat}[\mathbb V]$, the morphism $\iota_0(\mathcal C\otimes^{\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]} \mathcal D) \rightarrow\iota_0(L(\mathcal C\otimes^{\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]} \mathcal D)) \eqqcolon \iota_0(\mathcal C\otimes^{\mathrm{Cat}[\mathbb V]} \mathcal D)$ is generally not an equivalence. A simple example is given by taking $\mathbb V= {\sf Ab}$ to be the category of abelian groups, and taking $\mathcal C= B R$ and $\mathcal D= B S$ to be the one-object categorical ${\sf Ab}$-algebras associated to associative rings $R$ and $S$. We have $\iota_0 ( L(B R)) \simeq B(R^\times)$, and $B R \otimes^{\mathrm{Alg}_{\mathrm{Cat}}[{\sf Ab}]} B S \coloneqq B (R \otimes S)$, but the canonical map $R^\times \times S^\times \rightarrow(R \otimes S)^\times$ is generally not an isomorphism of sets. (For instance, if $R$ and $S$ are fields of different characteristic, then $R \otimes S = 0$.)↩︎
In general, a categorical $\mathbb V$-algebra with space of objects $X \in \mathcal S$ has an underlying $\mathbb V$-graph, i.e. a functor $X^{\times 2} \rightarrow\mathbb V$ (which encodes the hom-objects but not composition), and this forgetful functor has a left adjoint free functor (using e.g. that $\mathbb V$ is presentably monoidal). Then, given an object $V \in \mathbb V$, the categorical algebra $\Sigma[V] \in \mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]$ is free on the $\mathbb V$-graph with space of objects $\{0,1\} \in \mathrm{Set}\subset \mathcal S$ given by \[(i,j) \longmapsto \left\{ \begin{array}{ll} V~, & (i,j) = (0,1) \\ \emptyset_\mathbb V~, & \text{otherwise} \end{array} \right. ~.\]↩︎
As illustrated in Example B.1.8, this orthogonality relation may be seen as a sort of generalized coprimality requirement on the factors.↩︎
The data of an $\mathcal O$-monoidal $\infty$-category equipped with compatible factorization systems is equivalent to that of a functor $\mathcal O\rightarrow\mathrm{Cat}_\infty^{\text{f.s.},\mathcal L}$ satisfying certain Segal conditions (as in Subsection A.8.3). (Observe that $\mathrm{Cat}_\infty^{\text{f.s.},\mathcal L}$ admits products, which are defined in the evident way.)↩︎
Said differently, the morphisms in $S$ are precisely those that lie in a wide subcategory of $\mathcal C$.↩︎
These notions are recalled in Subsection 5.2.↩︎
In the case that $n = -2$ this recovers $(\mathcal S,\mathcal S^\simeq)$, and in the case that $n = -1$ this recovers (surjections, monomorphisms).↩︎
However, beware that a morphism in $\mathcal C$ may be sent to an equivalence in $\mathcal C^\mathcal R$ even if it is not in $\mathcal L$.↩︎
Notably, we do not generally have this coincidence in a case of primary interest for us, namely the reflective localization from categorical $\mathbb V$-algebras to $\mathbb V$-enriched $\infty$-categories (see Theorem B.4.1 and Lemma B.4.3). Rather, given a morphism of $\mathbb V$-enriched $\infty$-categories, if we consider it as a morphism of categorical $\mathbb V$-algebras then its factorization will have the same underlying space as the source, but this will generally not be univalent. (In this case, $RL(\mathcal L)$ consists of the surjective functors between $\mathbb V$-enriched $\infty$-categories that are homwise in $\mathcal L$, whereas $\mathcal L\cap R(\mathcal D)$ consists of the $\iota_0$-equivalences between $\mathbb V$-enriched $\infty$-categories that are homwise in $\mathcal L$.)↩︎
Which may not be presentably symmetric monoidal, see warning A.8.1.↩︎

Additional source footnote

See [GH15, Def. 4.3.1], and note that that nonsymmetric (a.k.a. planar) $\infty$-operads are equivalent to $\infty$-operads over $\mathbb E_1$ by [Lur17, Thm. 4.1.3.14].

↩

Yu Leon Liu, Aaron Mazel-Gee, David Reutter, Catharina Stroppel, Paul Wedrich

Original source: arXiv:2401.02956v2

previous section

Original source · 2401.02956v2

ScalingStacks

B Factorization systems for enriched \(\infty\)-categories[00J9]

B.1 Recollections on factorization systems[00JA]

B.1.1 Basics of factorization systems[00JB]

B.1.2 Factorization systems on presentable \(\infty\)-categories[00JU]

B.2 Induced factorization systems[00KC]

B.3 Factorization systems for algebras over \(\infty\)-operads[00L2]

B.4 Factorization systems for enriched \(\infty\)-categories[00LB]

Additional source footnote