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 
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 
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 \(\mathcal L_X\).
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.
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), which is generated by the singleton \(\{S^{n+1} \rightarrow{\sf pt}\}\).
[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 \(\mathcal C[\mathcal L^{-1}]\).
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.