Subsection B.4 (00LB) · ScalingStacks

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

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Original source · 2401.02956v2