aa r X i v : . [ m a t h . A T ] J un FREE ALGEBRAS THROUGH DAY CONVOLUTION
HONGYI CHU AND RUNE HAUGSENG
Abstract.
Building on the foundations in our previous paper, we study Segal conditions thatare given by finite products, determined by structures we call cartesian patterns. We set up Dayconvolution on presheaves in this setting and use it to give conditions under which there is acolimit formula for free algebras and other left adjoints. This specializes to give a simple proof ofLurie’s results on operadic left Kan extensions and free algebras for symmetric ∞ -operads. Contents
1. Introduction 12. Cartesian Patterns and Monoids 33. Examples of Cartesian Patterns 54. O -Monoidal ∞ -Categories and Algebras 65. Monoids as Algebras 96. Day Convolution over Cartesian Patterns 127. O -Monoidal Localizations and Presentability of Algebras 218. Extendability and Free Algebras 269. Examples of Extendability 3110. Morita Equivalences 33References 361. Introduction
A key feature of symmetric ∞ -operads, as defined in Lurie’s book [Lur17], is that there is anexplicit formula for their free algebras. More generally, for any morphism f : O → P of ∞ -operadsthere is a formula for the corresponding left operadic Kan extension, i.e. the left adjoint to thefunctor f ∗ : Alg P ( C ) → Alg O ( C ) between ∞ -categories of algebras given by composition with f .However, the construction of these left adjoints in [Lur17] is by a cumbersome simplex-by-simplexinduction using a delicate analysis of the inert–active factorization system on finite pointed sets.Part of our goal in this paper is to give a new, simpler construction of these left adjoints in thefollowing three steps:(1) We first consider algebras in the ∞ -category of spaces (with the cartesian monoidal structure).Here it is easy to see that the left adjoint is just given by an ordinary left Kan extension.(2) Next we consider algebras in presheaves on small symmetric monoidal ∞ -categories. Here theuniversal property of the Day convolution structure allows us to reduce to the previous case.(3) Finally, we use that any presentably symmetric monoidal ∞ -category is a symmetric monoidallocalization of a presheaf ∞ -category with Day convolution. Since the localization functor issymmetric monoidal and colimit-preserving, we can use it to transport the colimit formula forthe left adjoint from the previous step.Symmetric ∞ -operads in Lurie’s sense are certain ∞ -categories over the category F ∗ of pointedfinite sets; these are, in a sense, the universal objects that have algebras in symmetric monoidal Date : June 16, 2020. ∞ -categories. In practice, however, it can be useful to describe algebraic structures by more general ∞ -categories over F ∗ . For example, we can sometimes find a combinatorially simpler description byusing an ∞ -category that is not an ∞ -operad; as a somewhat trivial example, associative algebrascan be described in terms of the simplex category „ op , which is less complicated than the symmetricassociative operad. In other cases, even though it is formally known that a certain structure isdescribed by a symmetric ∞ -operad, this object may be difficult to describe explicitly; for instance,the structure of n compatible associative algebra structures can trivially be described in terms of theproduct „ n, op , while describing the associated ∞ -operad E n amounts to proving the Dunn–Lurieadditivity theorem.It is therefore desirable to understand when free algebras and other left adjoints can be describedby an explicit formula without passing to the associated symmetric ∞ -operads. Our main goal inthis paper is to obtain a simple criterion for this by following the same three steps we outlinedabove.We start in § ∞ -categories over F ∗ we are interested in, whichwe call cartesian patterns . This builds on the foundations in our previous paper [CH19], where westudied Segal conditions in general; here we are interested in those Segal-type limit conditions thatare given by finite products. We then give some examples of cartesian patterns in § ∞ -categories over a cartesian pattern and algebras therein in § • In § O is a cartesian pattern and C is an ∞ -category with finite products,then O -monoids in C are equivalent to O -algebras in an O -monoidal structure on C given bycartesian products. • In § ∞ -categories over a cartesianpattern. • In § ∞ -category is a monoidal localization of a Day convolution structure.These foundations allows us to prove our main result in § Theorem 1.1. (i) Suppose f : O → P is a morphism of cartesian patterns that is extendable in the sense ofDefinition 8.3 and V is a presentably P -monoidal ∞ -category. Then the functor f ∗ : Alg P ( V ) → Alg O / P ( V ) between ∞ -categories of algebras given by restriction along f has a left adjoint f ! , which for P ∈ P el satisfies ( f ! A )( P ) ≃ colim ( O, φ : f ( O ) P ) ∈ O act /P φ ! A ( O ) . (ii) Suppose O is a cartesian pattern that is extendable in the sense of Definition 8.8 and V is apresentably O -monoidal ∞ -category. Then the functor U O : Alg O ( V ) → Fun / O el ( O el , V ) given by restriction to the subcategory O el of elementary objects has a left adjoint F O , whichfor E ∈ O el satisfies U O F O Φ( E ) ≃ colim φ : O E ∈ Act O ( E ) φ ! (Φ( O ) , . . . , Φ( O n )) . Moreover, the adjunction F O ⊣ U O is monadic. This theorem is a combination of Corollaries 8.13 and 8.14; we refer the reader to §§ §
9. The explicit formula for free algebras leads to a simple criterionfor a morphism of cartesian patterns to induce equivalences on ∞ -categories of algebras, which weconsider in §
10 together with some simple applications.
REE ALGEBRAS THROUGH DAY CONVOLUTION 3
Acknowledgments.
The first author thanks the Labex CEMPI (ANR-11-LABX-0007-01) andMax Planck Institute for Mathematics for their hospitality and financial support during the processof writing this article.Some work on this paper was carried out while the second author was employed by the IBSCenter for Geometry and Physics in Pohang, in a position funded by grant IBS-R003-D1 of theInstitute for Basic Science of the Republic of Korea; it was completed while he was in residenceat the Matematical Sciences Research Institute in Berkeley, California, during the Spring 2020semester, and is thereby based upon work supported by the National Science Foundation undergrant DMS-1440140.We also thank Hadrian Heine for sharing his work on Day convolution [Hei18, § Cartesian Patterns and Monoids
In this section we review some basic definitions from [CH19] and introduce the algebraic structureswe will study in this paper, namely cartesian patterns and algebras and monoids over them.
Definition 2.1. An algebraic pattern consists of an ∞ -category O equipped with a factorizationsystem, whereby each map factors essentially uniquely as an inert map followed by an active map,together with a collection of elementary objects. We write O act and O int for the subcategories of O containing only the active and inert morphisms, respectively, and O el ⊆ O int for the full subcategoryof elementary objects and inert morphisms among them. A morphism of algebraic patterns from O to P is a functor f : O → P that preserves inert and active morphisms and elementary objects. Notation 2.2. If O is an algebraic pattern we will indicate an inert map between objects O, O ′ of O as O O ′ and an active map as O O ′ . These symbols are not meant to suggest any intuitionabout the nature of inert and active maps. Example 2.3.
Let F ∗ denote a skeleton of the category of finite pointed sets with objects h n i :=( { , , . . . , n } , φ : h n i → h m i is • inert if it is an isomorphism away from the base point, i.e. | φ − ( i ) | = 1 if i = 0, • active if it doesn’t send anything except the base point to the base point, i.e. φ − (0) = { } .We write F ♭ ∗ for the algebraic pattern given by this factorization system with h i as the only ele-mentary object. Notation 2.4. If O is an algebraic pattern and O is an object of O , we write O el O/ := O el × O int O int O/ for the ∞ -category of inert maps from O to elementary objects, and inert maps between them. Notation 2.5.
We write ρ i : h n i → h i , i = 1 , . . . , n , for the inert map given by ρ i ( j ) = ( , j = i, , j = i. Then ( F ♭ ∗ ) el h n i / is equivalent to the discrete set { ρ , . . . , ρ n } . Definition 2.6. A cartesian pattern is an algebraic pattern O equipped with a morphism of algebraicpatterns | – | : O → F ♭ ∗ such that for every object O ∈ O the induced map O el O/ → F el ∗ , | O | / is an equivalence. A morphism of cartesian patterns is a morphism of algebraic patterns over F ♭ ∗ . Notation 2.7. If O is a cartesian pattern and O is an object of O such that | O | ∼ = h n i , then the ∞ -category O el O/ is equivalent to a discrete set consisting of n inert morphisms with source O , withan essentially unique such morphism lying over ρ i : h n i → h i for i = 1 , . . . , n . We denote this inertmorphism by ρ Oi : O → O i . HONGYI CHU AND RUNE HAUGSENG
Lemma 2.8. If O is a cartesian pattern then the ∞ -category O el is an ∞ -groupoid.Proof. If O ∈ O lies over h i in F ∗ then it follows from the definition that the ∞ -category O el O/ isa contractible ∞ -groupoid. This holds in particular for E ∈ O el (since E must map to the uniqueelementary object h i in F ♭ ∗ ), so that the fibres of the cartesian fibrationev : Fun(∆ , O el ) → O el are contractible ∞ -groupoids. This functor is therefore an equivalence; in particular, O el is localwith respect to ∆ → ∆ , which means that it is an ∞ -groupoid. (cid:3) Definition 2.9. If O is a cartesian pattern and C is an ∞ -category with finite products, then afunctor F : O → C is an O -monoid if for O ∈ O lying over h n i the natural map F ( O ) → n Y i =1 F ( O i ) , induced by the maps ρ Oi : O → O i , is an equivalence. We write Mon O ( C ) for the full subcategory ofFun( O , C ) consisting of O -monoids. Remark 2.10. If O is a general algebraic pattern, then in [CH19] we defined a Segal O -object to bea functor F : O → C such that F | O int is a right Kan extension of F | O el , or equivalently if for everyobject O ∈ O the canonical map F ( O ) → lim E ∈ O el O/ F ( E )is an equivalence. Thus an O -monoid is just a special case of a Segal O -object. We choose touse different terminology for this and a few other concepts to emphasize the special features ofcartesian patterns and the parallels between our definitions and the special cases that are studiedin [Lur17, Bar18]. Lemma 2.11.
Any morphism of cartesian patterns f : O → P gives by composition a functor f ∗ : Mon P ( C ) → Mon O ( C ) . Proof.
Immediate since for O ∈ O we have | O | ∼ = | f ( O ) | and f ( ρ Oi ) ≃ ρ f ( O ) i . (cid:3) Remark 2.12.
Let O be a cartesian pattern and C an ∞ -category with finite products. Then so is O int , and a functor F : O → C is an O -monoid if and only if its restriction F | O int is an O int -monoid.Moreover, the ∞ -category Mon O int ( C ) is precisely the full subcategory of Fun( O int , C ) consisting offunctors that are right Kan-extended from O el , so that right Kan extension gives an equivalenceMon O int ( C ) ≃ Fun( O el , C ) . Remark 2.13.
Let C be an ∞ -category with sifted colimits and finite products where the cartesianproduct preserves sifted colimits in each variable. If O is a cartesian pattern, then the full subcate-gory Mon O ( C ) is closed under sifted colimits in Fun( O , C ): given a sifted diagram φ : I → Mon O ( C )its colimit in Fun( O , C ), which is computed pointwise, satisfies(colim I φ )( O ) ∼ −→ colim I ( φ ( O ) × · · · × φ ( O n )) ≃ (cid:18) colim I φ ( O ) (cid:19) × · · · × (cid:18) colim I φ ( O n ) (cid:19) . It follows that for any morphism f : O → P of cartesian patterns, the functor f ∗ : Mon P ( C ) → Mon O ( C ) preserves sifted colimits. REE ALGEBRAS THROUGH DAY CONVOLUTION 5 Examples of Cartesian Patterns
In this section we mention some examples of cartesian patterns, and indicate where our definitionsspecialize to more familiar notions:
Example 3.1.
For the base pattern F ♭ ∗ , an F ♭ ∗ -monoid is precisely a commutative monoid (in thesense considered in [Lur17], but going back to Segal’s work [Seg74] on special Γ-spaces). Example 3.2.
Let „ op ,♭ denote the simplex category with the usual inert–active factorizationsystem (where the inert maps in „ are the subinterval inclusions and the active maps are those thatpreserve the end points) and [1] as the unique elementary object. This is a cartesian pattern usingthe map to F ∗ given by | [ n ] | = h n i , and with the map | φ | for φ : [ n ] → [ m ] in „ given by | φ | ( i ) = ( j, φ ( j − < i ≤ φ ( j ) , , otherwise . Then a „ op ,♭ -monoid is an associative monoid. Example 3.3.
The product „ n, op has a factorization system with inert and active maps definedcomponentwise. We get a cartesian pattern „ n, op ,♭ by taking ([1] , . . . , [1]) to be the unique elemen-tary object and the map to F ∗ to be given by the composite „ n, op → F n ∗ → F ∗ where the second map is the “smash product” of pointed finite sets. (This takes ( h m i , . . . , h m n i )to h Q ni =1 m i i ; see [Lur17, Notation 2.2.5.1] for a precise description.) Then a „ n, op ,♭ -monoid can bedescribed as an n -fold iterated associative monoid, or equivalently an E n -monoid by the Dunn–LurieAdditivity Theorem. Example 3.4.
If Φ is a perfect operator category in the sense of [Bar18] and Λ(Φ) is its Leinstercategory, then there is a cartesian pattern Λ(Φ) ♭ given by the inert–active factorization system of[Bar18], with the terminal object of Φ as the unique elementary object and the map to F ∗ thatinduced by the unique operator morphism to F . By choosing Φ to be the operator categories F offinite sets, O of finite ordered sets, and the cartesian product O n , this example specializes to theprevious ones. Another example is the iterated wreath product O ≀ n , whose Leinster category is Joyal’scategory ˆ op n of n -dimensional pasting diagrams. It is proved in [Bar18] that ˆ op ,♭n -monoids areequivalent to „ n, op ,♭ -monoids, and so are also equivalent to E n -monoids by the additivity theorem. Example 3.5.
Let ˜ be the dendroidal category of [MW07], with the active–inert factorizationsystem described in [Koc11, CHH18]. Then ˜ op ,♭ denotes the cartesian pattern given by thisfactorization system, with the corollas as the elementary objects and the functor to F ∗ as definedin [CH20], given by counting the number of corollas in a tree. An ˜ op ,♭ -monoid in S then describesa (pointed) one-object ∞ -operad. Example 3.6.
The category ƒ of acyclic connected finite graphs defined by Hackney, Robertsonand Yau in [HRY15] has an active–inert factorization system by [Koc16, 2.4.14] (where the activemaps are called “refinements” and the inert maps are called “convex open inclusions”). We thenwrite ƒ op ,♭ for the algebraic pattern given by this factorization system, with the graphs with exactlyone vertex as the elementary objects. The functor ƒ op ,♭ → F ♭ ∗ given by counting the number ofvertices in a graph exhibits ƒ op ,♭ as a cartesian pattern. According to [HR18], ƒ op ,♭ -monoids in S model one-object ∞ -properads. Example 3.7.
Any (symmetric) ∞ -operad O → F ∗ as in [Lur17] has an inert–active factorizationsystem where the inert morphisms are the cocartesian morphisms lying over inert morphisms in F ∗ and the active ones are those that lie over active morphisms in F ∗ . If we take O ♭ to be the algebraicpattern defined by this factorization system, with the elementary objects those that map to h i (so O el is the underlying ∞ -groupoid O ≃h i of the fibre at h i ), then the given map to F ∗ exhibits HONGYI CHU AND RUNE HAUGSENG O ♭ as a cartesian pattern. (We will discuss a generalization of this class of cartesian patterns inRemark 4.16.) Example 3.8. A generalized ∞ -operad E → F ∗ as defined in [Lur17, § ∞ -operads. If we again choose the elementaryobjects to be those that lie over h i we get a cartesian pattern E ♭ using the given functor to F ∗ .4. O -Monoidal ∞ -Categories and Algebras In this section we define O -monoidal ∞ -categories and O -algebras in them. Definition 4.1.
Let O be a cartesian pattern. An O -monoidal ∞ -category is a cocartesian fibration V ⊗ → O whose associated functor O → Cat ∞ is an O -monoid. (We will often not mention thefibration explicitly and simply say that V ⊗ is an O -monoidal ∞ -category.) Notation 4.2.
Let V ⊗ → O be an O -monoidal ∞ -category. If O is an object of O lying over h n i in F ∗ , we will often denote by O ( V , . . . , V n ) or just O ( V i ) the object of V ⊗ O that corresponds to( V , . . . , V n ) under the equivalence V ⊗ O ≃ n Y i =1 V O i given by the cocartesian pushforwards over ρ Oi . Remark 4.3. If f : O → P is a morphism of cartesian patterns, then it follows from Lemma 2.11(reinterpreted through the straightening equivalence for functors to Cat ∞ ) that base change along f takes a P -monoidal ∞ -category V ⊗ to an O -monoidal ∞ -category f ∗ V ⊗ . Remark 4.4. O -monoidal ∞ -categories are a special case of Segal O -fibrations in the terminologyof [CH19]. Definition 4.5.
Let V ⊗ be an O -monoidal ∞ -category. An O -algebra in V ⊗ is a section V ⊗ O A such that A takes inert morphisms in O to cocartesian morphisms in V ⊗ . We write Alg O ( V ) for thefull subcategory of Fun / O ( O , V ⊗ ) spanned by the O -algebras. Example 4.6.
Taking O to be F ♭ ∗ our definitions specialize to symmetric monoidal ∞ -categories andcommutative algebras as defined in [Lur17], while if we take O to be the cartesian pattern associatedto an ∞ -operad we get the notions of O -monoidal ∞ -categories and O -algebras of [Lur17]. Similarly,from „ op ,♭ we get monoidal ∞ -categories and associative algebras. Definition 4.7.
More generally, if f : O → P is a morphism of cartesian patterns and V ⊗ is a P -monoidal ∞ -category, then an O -algebra in V ⊗ is a commutative triangle O V ⊗ P Af such that A takes inert morphisms in O to cocartesian morphisms in V ⊗ . We write Alg O / P ( V ) forthe full subcategory of Fun / P ( O , V ⊗ ) spanned by the O -algebras; if P is clear from the context wewill sometimes just write Alg O ( V ). Base change along f induces a natural equivalenceAlg O / P ( V ) ≃ Alg O ( f ∗ V ) . We can view O -algebras as a special case of morphisms of cartesian patterns, using the followingcanonical pattern structure on an O -monoidal ∞ -category: REE ALGEBRAS THROUGH DAY CONVOLUTION 7
Definition 4.8.
Let π : V ⊗ → O be an O -monoidal ∞ -category. We say a morphism in V ⊗ is active if it lies over an active morphism in O , and inert if it is cocartesian and lies over an inert morphism in O . The inert and active morphisms then form a factorization system on V ⊗ by [Lur17, Proposition2.1.2.5]; we make V ⊗ an algebraic pattern using this factorization system and all the objects thatlie over elementary objects in O as elementary objects. The composite map V ⊗ → O → F ∗ thenexhibits V ⊗ as a cartesian pattern. Remark 4.9.
Let V ⊗ be an O -monoidal ∞ -category and f : P → O a morphism of cartesianpatterns. Given a commutative triangle P V ⊗ O , Ff the functor F is a morphism of cartesian patterns if and only if it preserves inert morphisms, i.e. ifand only if it is a P -algebra, since the commutativity of the diagram automatically implies that F preserves active morphisms and elementary objects. Definition 4.10. If V ⊗ and W ⊗ are O -monoidal ∞ -categories, then a lax O -monoidal functor between them is a commutative triangle V ⊗ W ⊗ O F such that F preserves inert morphisms. Equivalently, a lax O -monoidal functor is a morphism ofalgebraic patterns over O or a V ⊗ -algebra in W ⊗ over O . If the functor from V ⊗ → W ⊗ preserves all cocartesian morphisms over O we call it an O -monoidal functor . Remark 4.11.
Since Alg O ( V ) is a full subcategory of Fun / O ( O , V ⊗ ), it is not just functorial in the O -monoidal ∞ -category V ⊗ , but even 2-functorial: a lax O -monoidal functor F : V ⊗ → W ⊗ gives afunctor F ∗ : Alg O ( V ) → Alg O ( W )given by composition with F , and a natural transformation η : F → F ′ over O gives (again bycomposition) a natural transformation η ∗ : F ∗ → F ′∗ . This means, for example, that any adjunctionbetween O -monoidal ∞ -categories induces by composition an adjunction on ∞ -categories of O -algebras. Notation 4.12. If V ⊗ → O is an O -monoidal ∞ -category we’ll write V := V ⊗ × O O el . In particular,we have V ⊗ E ≃ V E for every E ∈ O el . As this notation is sometimes ambiguous, we’ll also occasionallyuse V ⊗ / el instead of V . (Note that V ≃ V ⊗ / el must be distinguished from the ∞ -groupoid V el ofelementary objects, which is the underlying ∞ -groupoid V ≃ of the ∞ -category V .) Notation 4.13. If V ⊗ is an O -monoidal ∞ -category, then we write V ⊗ / int := O int × O V ⊗ . (The ∞ -category V ⊗ / int must be distinguished from the subcategory ( V ⊗ ) int of inert morphisms in V ⊗ : V ⊗ / int contains all morphisms that lie over inert morphisms in O , while ( V ⊗ ) int contains onlythe cocartesian ones.) Lemma 4.14. If V ⊗ is an O -monoidal ∞ -category, then there is a natural equivalence Alg O int / O ( V ⊗ ) ≃ Fun / O el ( O el , V ) between O int -algebras and sections of V → O el . HONGYI CHU AND RUNE HAUGSENG
Proof.
Pulling back V ⊗ to O int we get (since all morphisms in O int are inert) an equivalenceAlg O int / O ( V ⊗ ) ≃ Fun cocart / O int ( O int , V ⊗ / int ) . By Remark 2.12 the cocartesian fibration V ⊗ int → O int corresponds to a functor O int → Cat ∞ that isright Kan extended from O el . Translating the universal property of right Kan extension along thestraightening equivalence, we get for every cocartesian fibration E → O int a natural equivalenceMap cocart / O int ( E , V ⊗ / int ) ≃ Map cocart / O el ( E | O el , V ) . This upgrades to a natural equivalence of ∞ -categoriesFun cocart / O int ( E , V ⊗ / int ) ≃ Fun cocart / O el ( E | O el , V ) , since for C ∈ Cat ∞ there is a natural equivalenceMap Cat ∞ ( C , Fun cocart / O int ( E , V ⊗ )) ≃ Map cocart / O int ( C × E , V ⊗ ) . In our case this givesFun cocart / O int ( O int , V ⊗ / int ) ≃ Fun cocart / O el ( O el , V ) ≃ Fun / O el ( O el , V ) , where the last equivalence holds since O el is an ∞ -groupoid. (cid:3) Remark 4.15.
Similarly, if f : O → P is a morphism of cartesian patterns and V ⊗ is a P -monoidal ∞ -category, we have a natural equivalenceAlg P int / O ( V ⊗ ) ≃ Fun / O el ( P el , V ) . Remark 4.16.
The notion of O -monoidal ∞ -category can be weakened to that of an O - ∞ -operad ,which is a functor p : E → O such that:(i) For every object O in E lying over O ∈ O and every inert morphism φ : O O ′ in O , thereexists a p -cocartesian morphism φ : O → O ′ lying over φ .(ii) For every object O ∈ O , the functor E O → Y i E O i , induced by the cocartesian morphisms over the inert maps ρ Oi , is an equivalence.(iii) Given O in E O , choose cocartesian morphisms ρ Oi : O → O i over ρ Oi . Then for any O ′ ∈ O and O ′ ∈ E O ′ , the commutative squareMap E ( O ′ , O ) Q i Map E ( O ′ , O i )Map O ( O ′ , O ) Q i Map O ( O ′ , O i )is cartesian.This is the same as a weak Segal fibration over O as defined in [CH19, § O - ∞ -operads to emphasize that this notion specializes to the (symmetric) ∞ -operads of [Lur17] over thepattern F ♭ ∗ , to non-symmetric (or planar) ∞ -operads (as in [GH15] and [Lur17, § „ op ,♭ , and more generally for an operator category Φ to Φ- ∞ -operads in the sense of [Bar18]for the pattern Λ(Φ) ♭ . If E is an O - ∞ -operad then it has a canonical pattern structure by [CH19,Lemma 9.4], where the inert morphisms are those that are cocartesian and lie over inert morphismsin O , and the active morphisms are those that lie over active morphisms in O ; this is a cartesianpattern via the composite E → O | – | −→ F ∗ . REE ALGEBRAS THROUGH DAY CONVOLUTION 9 Monoids as Algebras
Let O be a cartesian pattern. Our goal in this section is to prove that O -monoids in an ∞ -category C with finite products are equivalent to O -algebras in an O -monoidal ∞ -category determined bythe cartesian product in C . More precisely, we will prove the following generalization of [Lur17,Proposition 2.4.1.7]: Proposition 5.1.
Suppose C is an ∞ -category with finite products. Let C × → F ∗ be the correspond-ing cartesian symmetric monoidal ∞ -category. If O is a cartesian pattern, then there is a naturalequivalence Mon O ( C ) ≃ Alg O / F ∗ ( C × ) . Here the cartesian symmetric monoidal ∞ -category C × is defined as in [Lur17], and before weprove the proposition we need to recall this definition, which we phrase using the terminology ofalgebraic patterns: Definition 5.2.
Let Γ × (cf. [Lur17, Notation 2.4.1.2]) denote the full subcategory of F ∆ ∗ spannedby the inert morphisms in F ∗ . Let ev , ev : Γ × → F ∗ denote the functors given by evaluation at 0and 1, respectively. Lemma 5.3. ev : Γ × → F ∗ is a cartesian fibration.Proof. This follows from the uniqueness of inert–active factorizations. (cid:3)
Definition 5.4.
We can apply the construction of [Lur09, Corollary 3.2.2.12] to the cartesianfibration ev and the cocartesian fibration C × F ∗ → F ∗ to obtain a cocartesian fibration C × → F ∗ with the universal property that there is a natural equivalenceMap / F ∗ ( K, C × ) ≃ Map( K × F ∗ Γ × , C )for any ∞ -category K over F ∗ . In particular the ∞ -category C × is given fibrewise over h n i byFun(Γ ×h n i , C ). (And by the dual of [GHN17, Proposition 7.3] this is indeed the corresponding functor.)If C is an ∞ -category with finite products, we define C × to be the full subcategory of C × whoseobjects over h n i are the functors F : Γ ×h n i → C such that for every object φ : h n i → h m i , the map F ( φ ) → Q mi =1 F ( ρ i φ ) is an equivalence. Proposition 5.5 (Lurie, [Lur17, Proposition 2.4.1.5]) . If C is an ∞ -category with products thenthe restricted functor C × → F ∗ is a symmetric monoidal ∞ -category. (cid:3) Remark 5.6.
From the definition it follows that any functor g : C → D induces a natural morphismof cocartesian fibrations g × : C × → D × ; if g preserves products then this restricts to a naturalsymmetric monoidal functor g × : C × → D × . Definition 5.7.
We say a morphism h m i h m ′ ih n i h n ′ i in Γ × is inert or active if the horizontal maps in the square are inert or active, respectively. Lemma 5.8.
The inert and active morphisms determine a factorization system on Γ × .Proof. Given a morphism as above, then by the factorization system on F ♭ ∗ we get horizontal inert-active factorizations h m i h m ′′ i h m ′ ih n i h n ′′ i h n ′ i . The existence of a factorization system on Γ × is equivalent to the existence of an inert morphismindicated by the dotted arrow. The map h m ′′ i h m ′ i h n ′ i factors into h m ′′ i h q i h n ′ i .The essential uniqueness of factorizations implies that h m i h n i h n ′′ i h n ′ i coincides with h m i h m ′′ i h q i h n ′ i . Hence, there is an equivalence h n ′′ i ≃ h q i which proves the existence ofthe dotted morphism in the diagram above. (cid:3) Definition 5.9.
We give Γ × an algebraic pattern structure using this factorization system, withid h i as the only elementary object. Remark 5.10.
It is clear from the definition of Γ × that the evaluations at 0 and 1 give morphismsof algebraic patterns ev , ev : Γ × → F ♭ ∗ . Moreover, the evaluation at 1 exhibits Γ × as a cartesianpattern. Remark 5.11. If O is a cartesian pattern, then we can equip the pullback O × F ∗ Γ × over evaluationat { } with a canonical pattern structure (which according to [CH19, Corollary 5.5] gives the fibreproduct in the ∞ -category of algebraic patterns). Here a morphism in O × F ∗ Γ × that is given by O → O ′ and a commutative square | O | | O ′ |h n i h n ′ i is inert or active if O → O ′ and the horizontal maps in the square are inert or active, respectively,and the elementary objects are the pairs ( E, id h i ) with E ∈ O el . The composite O × F ∗ Γ × → Γ × ev −−→ F ∗ exhibits O × F ∗ Γ × as a cartesian pattern. Definition 5.12.
Let i : F ∗ → Γ × be the functor that takes h n i to id h n i (given by composition with∆ → ∆ ); this is fully faithful and a morphism of cartesian patterns. If O is a cartesian pattern,we define i O : O → O × F ∗ Γ × by pulling back i , so that i O takes O ∈ O to ( O, id | O | ). This is againfully faithful, and since the target is a fibre product of patterns it is also a morphism of cartesianpatterns. Remark 5.13.
An active morphism ( O, | O | h n i ) i O ( O ′ ) ≃ ( O ′ , | O ′ | == | O ′ | ) is given by anactive map O O ′ together with a commutative square | O | | O ′ |h n i | O ′ | . The uniqueness of the inert–active factorization then implies that | O | ∼ = h n i , and hence i O hasunique lifting of active morphisms in the sense of [CH19, Definition 6.1]. Lemma 5.14.
Let O be a cartesian pattern and C an ∞ -category with finite products. Compositionwith i O and right Kan extension along it restrict to an adjoint equivalence i ∗ O : Mon O × F ∗ Γ × ( C ) ∼ −→←− ∼ Mon O ( C ) : i O , ∗ . Proof.
Since i O is a morphism of cartesian patterns, the functor i ∗ O restricts to the full subcategoriesof monoids. Moreover, since i O has unique lifting of active morphisms, its right adjoint i O , ∗ alsorestricts to monoids by [CH19, Proposition 6.3], and it is fully faithful since i O is fully faithful.It remains to show that i O , ∗ is essentially surjective on monoids. Let M : O × F ∗ Γ × → C be a REE ALGEBRAS THROUGH DAY CONVOLUTION 11 monoid, and let (
O, j : | O | h n i ) be an object of O × F ∗ Γ × . We must show that the canonical map M ( O, j ) → ( i O , ∗ i ∗ O M )( O, j ) is an equivalence. But we have a commutative square M ( O, j ) ( i O , ∗ i ∗ O M )( O, j ) Q ni =1 M ( O i , id h i ) Q ni =1 ( i O , ∗ i ∗ O M )( O i , id h i ) , where the vertical maps are equivalences since M and i O , ∗ i ∗ O M are monoids. Moreover, the bottomhorizontal map is an equivalence since i O , ∗ is fully faithful and the objects ( O i , id h i ) ≃ i O ( O i ) arein the image of i O . The top horizontal map is therefore also an equivalence. (cid:3) Lemma 5.15.
Under the natural equivalence
Fun / F ∗ ( O , C × ) ≃ Fun( O × F ∗ Γ × , C ) , the full subcategory Mon O × F ∗ Γ × ( C ) is identified with Alg O / F ∗ ( C × ) .Proof. By definition, a functor F : O → C × over F ∗ lies in Alg O / F ∗ ( C × ) if and only if F factorsthrough the full subcategory C × , and F takes inert morphisms to cocartesian morphisms. We canreformulate these requirements in terms of the corresponding functor F ′ : O × F ∗ Γ × → C as thefollowing pair of conditions:(1) The map F ′ ( O, φ ) → n Y i =1 F ′ ( O, ρ i φ )is an equivalence for every object ( O, φ : | O | h n i ).(2) For every inert map ψ : O ′ O in O the morphism F ′ ( O ′ , φ | ψ | ) → F ′ ( O, φ )is an equivalence.By the definition of C × , condition (1) holds if and only if F factors through C × , while the descrip-tion of cocartesian morphisms in C × in [Lur17, Proposition 2.4.1.5.(2)] shows that F takes inertmorphisms to cocartesian morphisms if and only condition (2) holds.On the other hand, F ′ is a monoid if for every object ( O, φ ) the map F ′ ( O, φ ) → n Y i =1 F ′ ( O φ − i , id h i )is an equivalence. To see that this is equivalent to the first pair of conditions, observe that we havecommutative triangles F ′ ( O, φ ) Q ni =1 F ′ ( O, ρ i φ ) Q ni =1 F ′ ( O φ − i , id h i ) ,F ′ ( O ′ , φ | ψ | ) F ′ ( O, φ ) Q ni =1 F ′ ( O φ − i , id h i ) . First suppose conditions (1) and (2) hold. The first triangle the horizontal is an equivalence by (1).The equality ρ i φ = id h i | ρ O i | and (2) then shows that the right diagonal morphisms are equivalencesas well. Hence, the left diagonal is an equivalence, which implies that F ′ is a monoid. Conversely,if F ′ is a monoid then the left diagonal morphism of the first triangle and the right diagonal of the second triangle are clearly equivalences. Then the equivalences O ′ ( φ | ψ | ) − i ≃ O φ − i ≃ O ( ρ i φ ) − i imply that the other diagonal morphisms of the triangles are equivalences. Hence, the horizontalmorphisms are also equivalences, which gives conditions (1) and (2), respectively. (cid:3) Proof of Proposition 5.1.
Combine Lemmas 5.14 and 5.15. (cid:3)
Remark 5.16. If f : O → P is a morphism of cartesian patterns then it is clear from the proofthat the equivalence of Proposition 5.1 is natural. Thus if C is an ∞ -category with finite products,composition with f gives a commutative squareMon P ( C ) Alg P / F ∗ ( C × )Mon O ( C ) Alg O / F ∗ ( C × ) ∼ f ∗ f ∗ ∼ Moreover, if D is another ∞ -category with finite products and g : C → D is a product-preservingfunctor, then g and the symmetric monoidal functor g × : C × → D × from Remark 5.6 fit in acommutative square Mon O ( C ) Alg O / F ∗ ( C × )Mon O ( D ) Alg O / F ∗ ( D × ) , ∼ g ∗ g ×∗ ∼ which combines with composition with f to give a natural commutative cube.6. Day Convolution over Cartesian Patterns
In this section we introduce Day convolution for presheaves of spaces on O -monoidal ∞ -categories,where O is any cartesian pattern. We generalize a simple and elegant approach to Day convolutionfor presheaves due to Heine [Hei18, § Notation 6.1.
Let RFib denote the full subcategory of Cat ∆ ∞ spanned by the right fibrations. Proposition 6.2.
The functor ev : RFib → Cat ∞ , given by evaluation at ∈ ∆ , is a cartesianand cocartesian fibration, and the corresponding contravariant functor is equivalent to P( – ) := Fun(( – ) op , S ) : Cat op ∞ → d Cat ∞ . Proof.
The ∞ -category Cat ∞ has pullbacks, so the functor ev : Fun(∆ , Cat ∞ ) → Cat ∞ is acartesian fibration, with cartesian morphisms given by pullback squares. Since the pullback ofa right fibration is again a right fibration, the full subcategory RFib inherits cartesian morphismsfrom Fun(∆ , Cat ∞ ), which implies that ev : RFib → Cat ∞ is a cartesian fibration. To show thatit is also a cocartesian fibration it then suffices by [Lur09, Corollary 5.2.2.5] to check that for anymorphism f : C ′ → C in Cat ∞ , the cartesian pullback functor f ∗ : RFib C → RFib C ′ has a left adjoint. Under the straightening equivalence, this functor corresponds to the functor f ∗ : P( C ) → P( C ′ ) given by composition with f op , which indeed has a left adjoint given by left Kanextension along f op .To identify the corresponding functor we use the naturality of the straightening equivalence, asdiscussed in [GHN17, Appendix A]. By a variant of [GHN17, Corollary A.32], the pseudo-naturalityof straightening on the model category level induces a natural equivalenceRFib (–) ∼ −→ Fun((–) op , S ) . REE ALGEBRAS THROUGH DAY CONVOLUTION 13
Here the functoriality in the domain is induced by strict pullbacks on the model category level usingthe covariant model structures on slices of simplicial sets [Lur09, § , Set ∆ ) RFib of Fun(∆ , Set ∆ )spanned by the right fibrations between quasicategories in the sense of [Lur09], with the weakequivalences inherited from the Joyal model structure on Set ∆ . Evaluation at 1 gives a cartesianfibration of relative categories Fun(∆ , Set ∆ ) RFib → Set ∆ , whose associated functor from Set ∆ to relative categories induces on the ∞ -level the functor RFib (–) we considered above. In thissituation Hinich [Hin16] has shown that the functor associated to the induced cartesian fibration of ∞ -categories is that obtained from the associated functor to relative categories by inverting weakequivalences. (cid:3) Definition 6.3.
Let RSl denote the full subcategory of RFib spanned by the right fibrations givenby slice categories, i.e. right fibrations of the form C /c → C . Corollary 6.4. ev : RSl → Cat ∞ is a cocartesian fibration, and the inclusion RSl → RFib pre-serves cocartesian morphisms. The corresponding functor
Cat ∞ → Cat ∞ is the identity, and theinclusion corresponds to the Yoneda embeddings C ֒ → P( C ) .Proof. Under the straightening equivalence RFib C ≃ P( C ), the slice fibration C /c → C correspondsto the representable presheaf Map C (– , c ). Moreover, for a functor f : C → C ′ the cocartesian pushfor-ward f ! : RFib C → RFib C ′ corresponds to the functor f ! : P( C ) → P( C ′ ) given by left Kan extensionalong f op . This lives in a commutative square C P( C ) C ′ P( C ′ ) f f ! where the horizontal maps are the Yoneda embeddings. In particular, f ! takes the representablepresheaf Map C (– , c ) to Map C ′ (– , f ( c )). In terms of right fibrations, this means f ! ( C /c → C ) ≃ ( C ′ /f ( c ) → C ′ );thus RSl inherits cocartesian morphisms from RFib, and this makes ev : RSl → Cat ∞ a cocartesianfibration. The corresponding functor Cat ∞ → d Cat ∞ is equivalent to that taking C to the full sub-category of P( C ) spanned by the representable presheaves; the naturality of the Yoneda embeddingshows that this is equivalent to the identity of Cat ∞ , as required, and that the inclusion RSl ֒ → RFibcorresponds to the Yoneda embedding. (cid:3)
Corollary 6.5.
Suppose π : F → B is the cocartesian fibration corresponding to a functor F : B → Cat ∞ . Then there is a pullback square of ∞ -categories F RSl B Cat ∞ . π ev F In other words, ev : RSl → Cat ∞ is the universal cocartesian fibration.Proof. Pullback of cocartesian fibrations corresponds to composition of functors, so F ∗ RSl → B isthe cocartesian fibration for id ◦ F ≃ F . (cid:3) Our next goal is to prove an O -monoidal version of Corollary 6.5, which needs some preliminaries. Remark 6.6.
The ∞ -categories RFib and RSl are both closed under cartesian products as fullsubcategories of Fun(∆ , Cat ∞ ). These ∞ -categories therefore have cartesian products, and the functor to Cat ∞ given by evaluation at 1 in ∆ preserves products. We hence have induced sym-metric monoidal functors ev × : RFib × , RSl × → Cat ×∞ . Our next goal is to show that these functors are both cocartesian fibrations, and indeed exhibitRFib × and RSl × as Cat ×∞ -monoidal ∞ -categories. To see this we apply the following generalcriterion: Proposition 6.7.
Let O be a cartesian pattern. Suppose F : C ⊗ → D ⊗ is an O -monoidal functorbetween O -monoidal ∞ -categories such that(1) for every E ∈ O el , the functor on fibres F E : C E → D E is a cocartesian fibration,(2) for every active morphism φ : O E in O with E ∈ O el , in the commutative square Q i C O i C E Q i D O i D E , φ C ! Q i F Oi F E φ D ! the functor φ C ! takes Q i F O i -cocartesian morphisms (i.e. tuples of F O i -cocartesian morphisms)to F E -cocartesian morphisms.Then:(i) F is a cocartesian fibration that exhibits C ⊗ as a D ⊗ -monoidal ∞ -category.(ii) Given a morphism ψ : O ( D i ) → O ′ ( D ′ j ) in D ⊗ lying over φ : O → O ′ in O and an object O ( C i ) in C ⊗ over O ( D i ) , the cocartesian morphism over ψ with this domain is the composite O ( C i ) → φ ! O ( C i ) → ψ ′ ! φ ! O ( C i ) where O ( C i ) → φ ! O ( C i ) is cocartesian over φ , the map ψ factors uniquely (since F preservescocartesian morphisms) as O ( D i ) → F ( φ ! O ( C i )) ψ ′ −→ O ′ ( D ′ j ) , where ψ ′ lies in the fibre D ⊗ O ′ and φ ! O ( C i ) → ψ ′ ! φ ! O ( C i ) is F O ′ -cocartesian over ψ ′ .(iii) If f : P → D ⊗ is a morphism of cartesian patterns, we have a natural cartesian square Alg P / D ⊗ ( C ⊗ ) Alg P / O ( C ⊗ ) { f } Alg P / O ( D ⊗ ) Proof.
To prove that F is a cocartesian fibration we use the criterion of (the dual of) [HMS19,Lemma A.1.8], which requires us to check that(a) for every O ∈ O , the functor F O : C ⊗ O → D ⊗ O is a cocartesian fibration,(b) for every morphism φ : O → O ′ in O , in the commutative square C ⊗ O C ⊗ O ′ D ⊗ O D ⊗ O ′ , φ C ! F O F O ′ φ D ! the functor φ C ! takes F O -cocartesian morphisms to F O ′ -cocartesian morphisms.Condition (a) is clear, since F O is equivalent to the product Y i F O i : Y i C O i → Y i D O i , REE ALGEBRAS THROUGH DAY CONVOLUTION 15 where each F O i is by assumption a cocartesian fibration. Moreover, a morphism in C ⊗ O is F O -cocartesian if and only if under the equivalence with Q i C O i it corresponds to a tuple of F O i -cocartesian morphisms. If φ is an inert morphism in O then φ C ! corresponds to a projection tosome factors in this product, hence condition (b) is immediate for inert morphisms. Using thefactorization system we can then reduce condition (b) to the case of an active morphism O E with E ∈ O el , where it holds by assumption. The description of the cocartesian morphisms in theproof of [HMS19, Lemma A.1.8] also gives (ii).To see that F exhibits C ⊗ as D ⊗ -monoidal, observe that for O ∈ O the commutative square C ⊗ O Q i C O i D ⊗ O Q i D O i ∼ F O F Oi ∼ is cartesian, since the horizontal maps are equivalences. For any O ( D i ) ∈ D ⊗ O we therefore have anequivalence on fibres C ⊗ O ( D i ) ∼ −→ Y i C D i , as required.To prove (iii), observe that we have a pullback squareFun / D ⊗ ( P , C ⊗ ) Fun / O ( P , C ⊗ ) { f } Fun / O ( P , D ⊗ ) . This restricts to the full subcategories of algebras because (ii) implies that a functor from P → C ⊗ over D ⊗ is an algebra if and only if the underlying functor over O is an algebra (since every inertmorphism in C ⊗ is F -cocartesian over an inert morphisms in D ⊗ ). (cid:3) Corollary 6.8.
Suppose π : E → B is a cocartesian fibration such that(1) E and B have finite products,(2) π preserves these,(3) if e i → e ′ i ( i = 1 , ) are π -cocartesian morphisms in E then e × e → e ′ × e ′ is again π -cocartesian.Then the induced symmetric monoidal functor π × : E × → B × is a cocartesian fibration that exhibits E × as B × -monoidal. (cid:3) Lemma 6.9.
Suppose E → A and F → B are right fibrations corresponding to functors E : A op → S , F : B op → S . Then the right fibration E × F → A × B corresponds to the functor E × F : A op × B op → S (taking ( a, b ) to E ( a ) × F ( b ) ).Proof. Consider the right fibrations A × E , F × B → A × B . These are the pullbacks of E and F along the projections from A × B to A and B , respectively.Since pullbacks of right fibrations correspond to compositions of functors, the associated functorsare therefore A op × B op → A op E −→ S , A op × B op → B op F −→ S , respectively. Now we can identify E × F with ( E × B ) × A × B ( A × F ), which is a product of rightfibrations over A × B . Since straightening preserves limits, the corresponding functor is the product E × F as required. (cid:3) Corollary 6.10.
The functors ev × : RSl × , RFib × → Cat ×∞ are cocartesian fibrations that exhibit their domains as Cat ×∞ -monoidal ∞ -categories. The inclusion RSl × ֒ → RFib × is Cat ×∞ -monoidal (i.e. preserves cocartesian morphisms over Cat ×∞ ).Proof. The only non-obvious condition in Corollary 6.8 is that products of cocartesian morphismsin RFib are again cocartesian (since the cocartesian morphisms in RSl are inherited from RFib, itis enough to consider the case of RFib).Suppose E → A and F → B are right fibrations corresponding to functors E : A op → S , F : B op → S . Then the product E × F → A × B corresponds to E × F by Lemma 6.9. Given α : A → A ′ and β : B → B ′ , the cocartesian pushforward ( α × β ) ! ( E × F ) corresponds to the left Kan extension of E × F along ( α × β ) op , given by( x, y ) ∈ ( A ′ × B ′ ) op colim ( a,b ) ∈ ( A x/ × B y / ) op E ( a ) × F ( b ) . Since the product in S commutes with colimits in each variable, this is equivalent to( x, y ) ∈ ( A ′ × B ′ ) op (cid:18) colim a ∈ ( A x/ ) op E ( a ) (cid:19) × (cid:18) colim b ∈ ( B y/ ) op F ( b ) (cid:19) which by Lemma 6.9 is the functor corresponding to α ! E × β ! F , as required.For the last claim note that cocartesian morphisms in RSl are inherited from RFib, therefore, thedescription of cocartesian morphisms in Proposition 6.7(ii) shows that the inclusion RSl × ֒ → RFib × preserves cocartesian morphisms over Cat ×∞ . (cid:3) Remark 6.11.
Using Proposition 6.7(ii) we can describe the cocartesian morphisms in RFib × asfollows: Given a morphism ( C , . . . , C n ) → D in Cat ×∞ , which corresponds to a functor Φ : C ×· · · × C n → D , and an object of RFib × over ( C , . . . , C n ), which we can identify with a family ofpresheaves F i : C op i → S , ( i = 1 , . . . , n ), the cocartesian morphism over Φ takes this to the left Kanextension Φ ! ( Q i F i ) : D op → S along Φ op of the product Y i F i : Y i C op i ( F i ) −−→ S × n × −→ S . Proposition 6.12.
Let O be a cartesian pattern, and suppose π : C ⊗ → O is an O -monoidal ∞ -category, corresponding to an O -monoid M in Cat ∞ . Let A : O → Cat ×∞ be the corresponding O -algebra under the equivalence of Proposition 5.1. Then there is a pullback square C ⊗ RSl × O Cat ×∞ . ev × A Remark 6.13.
We can interpret this as exhibiting the pullback O × F ∗ RSl × → O × F ∗ Cat ×∞ as theuniversal cocartesian fibration of O -monoidal ∞ -categories. Proof of Proposition 6.12.
By Corollary 6.5 we have a pullback square C ⊗ RSl O Cat ∞ , MM where M is by assumption an O -monoid. The functor M takes an object X ≃ O ( X , . . . , X n ) ∈ C ⊗ to the slice ( C ⊗ O ) /X . But the equivalence C ⊗ O ≃ Q ni =1 C O i induces an equivalence( C ⊗ O ) /X ≃ n Y i =1 C O i /X i . REE ALGEBRAS THROUGH DAY CONVOLUTION 17
This shows that the functor M is a C ⊗ -monoid. By the naturality of the equivalence betweenmonoids and algebras, as discussed in Remark 5.16, our pullback square therefore corresponds to acommutative square C ⊗ RSl × O Cat ×∞ , AA where A is the C ⊗ -algebra corresponding to M . It then remains to verify that this square is cartesian.Here the vertical maps are cocartesian fibrations, and we first observe that A preserves co-cartesian morphisms. Since A is an algebra it suffices to check this in the case of a cocartesianmorphism φ : O ( C , . . . , C n ) → C ′ over an active morphism φ : O E with E ∈ O el . Here C ′ ≃ M ( φ )( O ( C , . . . , C n )), and A ( φ ) is by construction the morphism( C O /C , . . . , C O n /C n ) → C E/C ′ corresponding to the functor M ( φ ) : Y i C O i /C i ≃ C ⊗ O/O ( C ,...,C n ) → C E/C ′ . By definition of M as a pullback it preserves cocartesian morphisms, so this is a cocartesian mor-phism in RSl. The description of cocartesian morphisms in RSl × in Remark 6.11 now implies that A ( φ ) is therefore cocartesian in RSl × , as required.To prove that the square is cartesian it now suffices see this it induces equivalences on fibres overall O ∈ O . But since A preserves cocartesian morphisms we have for O ∈ O a commutative square C ⊗ O RSl × A ( O ) Q i C O i Q i RSl M ( O i ) ∼ ∼ where the vertical maps are equivalences since C ⊗ is O -monoidal and RSl × is Cat ×∞ -monoidal, whilethe bottom horizontal map is an equivalence since C ⊗ is pulled back from RSl along M . (cid:3) Definition 6.14.
Suppose C ⊗ is an O -monoidal ∞ -category. By Proposition 5.1 this correspondsto an O -algebra C : O → Cat ×∞ . The (contravariant) Day convolution of C ⊗ is the O -monoidal ∞ -category given by the pullback P O ( C ) ⊗ RFib × O Cat ×∞ . C Remark 6.15.
Using Remark 6.11 we can describe the cocartesian morphisms in P O ( C ) ⊗ : given φ : O → E in O active with E ∈ O el and F i ∈ P( C O i ), we take the left Kan extension along φ op! : Y C op O i ≃ ( C ⊗ O ) op → C op E of the product Y i F i : Y i C op O i ( F i ) −−→ Y i S × −→ S . Proposition 6.16.
The Yoneda embedding gives a natural O -monoidal functor C ⊗ → P O ( C ) ⊗ . Proof.
By Corollary 6.10 the inclusion RSl ֒ → RFib induces a Cat ×∞ -monoidal functor RSl × ֒ → RFib × . Pulling this back along the O -algebra C : O → Cat ×∞ corresponding to C ⊗ we get usingProposition 6.12 an O -monoidal functor C ⊗ → P O ( C ) ⊗ . Over E ∈ O el it follows from Corollary 6.4that this functor is given by the Yoneda embedding C E ֒ → P( C E ). (cid:3) Notation 6.17.
Suppose C ⊗ → O is an O -monoidal ∞ -category, corresponding to an O -monoid M : O → Cat ∞ . Since (–) op is an automorphism of Cat ∞ , the composite O M −→ Cat ∞ (–) op −−−→ Cat ∞ is also an O -monoid. We write C op , ⊗ → O for the corresponding cocartesian fibration. (Note that ifwe write C ⊗ → O op for the cartesian fibration for M , then C op , ⊗ ≃ ( C ⊗ ) op .) To avoid confusion wewill avoid the notation C op for the pullback of C op , ⊗ to O el (since this is not the opposite ∞ -categoryof C , but the fibrewise opposite) and instead write C op , ⊗ / el .We can now prove the universal property for mapping into the Day convolution: Proposition 6.18.
Suppose C ⊗ → O is an O -monoidal ∞ -category, then there is an equivalence Alg O (P O ( C ) ⊗ ) ≃ Mon C op , ⊗ ( S ) . Proof.
Let M : O → Cat ∞ be the monoid corresponding to C ⊗ , and let A : O → Cat ×∞ be the corre-sponding algebra. Using the definition of P O ( C ) ⊗ as a pullback we then have natural equivalencesAlg O (P O ( C ) ⊗ ) ≃ Alg O / Cat ×∞ (RFib × ) (by pullback) ≃ { A } × Alg O / F ∗ (Cat ×∞ ) Alg O / F ∗ (RFib × ) (by Proposition 6.7(iii)) ≃ { M } × Mon O (Cat ∞ ) Mon O (RFib) , (by Proposition 5.1)where the right-hand side is the full subcategory of Fun / Cat ∞ ( O , RFib) spanned by the monoids.Since the cartesian fibration ev : RFib → Cat ∞ corresponds to the functor Fun((–) op , S ), by[GHN17, Proposition 7.3], we have an equivalenceFun / Cat ∞ ( O , RFib) ≃ Fun( O × Cat ∞ E , S ) , where E → Cat ∞ is the cocartesian fibration corresponding to the functor op : Cat ∞ → Cat ∞ . Bydefinition, the pullback O × Cat ∞ E is precisely C op , ⊗ → O . We have therefore identified Alg O (P O ( C ⊗ ))with a full subcategory of Fun( C op , ⊗ , S ), and we need to check that this is precisely the full subcat-egory of C op , ⊗ -monoids.Under the equivalence of [GHN17, Proposition 7.3], a functor φ : C op , ⊗ → S corresponds to thefunctor Φ : O → RFib that takes O ∈ O to the right fibration for the presheaf Φ | C ⊗ O : ( C ⊗ O ) op → S .We then observe that this gives an O -monoid in RFib precisely when Φ is a C op , ⊗ -monoid, usingthe commutative squares Φ( O ) Q i Φ( O i ) C ⊗ O Q i C O i . ∼ Here the vertical maps are right fibration, so that the top horizontal map is an equivalence if andonly if the square is cartesian, which is equivalent to the map on fibres being an equivalence forevery O ( C i ) ∈ C ⊗ O . The map on fibres we can identify with φ ( O ( C i )) → Y i φ ( C i ) , so all of these are equivalences precisely when φ is a C op , ⊗ -monoid. (cid:3) REE ALGEBRAS THROUGH DAY CONVOLUTION 19
Corollary 6.19.
Let C ⊗ be a small O -monoidal ∞ -category. We say a C op , ⊗ -monoid M : C op , ⊗ → S is representable if for every E ∈ O el , the restriction M | C op E : C op E ≃ ( C op , ⊗ ) E → S is a representable presheaf. There is a natural equivalence Alg O ( C ) ≃ Mon rep C op , ⊗ ( S ) . Proof.
The O -monoidal inclusion C ⊗ ֒ → P O ( C ) ⊗ identifies Alg O ( C ) with the full subcategory ofAlg O (P O ( C )) spanned by the O -algebras A such that A ( E ) ∈ P( C E ) is representable for every E ∈ O el . This full subcategory is identified with Mon rep C op , ⊗ ( S ) under the equivalence of Proposition 6.18. (cid:3) Lemma 6.20.
Suppose f : O → P is a morphism of cartesian patterns and C ⊗ → P is a P -monoidal ∞ -category. Then there is a natural equivalence P O ( f ∗ C ) ⊗ ≃ f ∗ P P ( C ) ⊗ of O -monoidal ∞ -categories.Proof. By definition we have a commutative diagram f ∗ P P ( C ) ⊗ P P ( C ) ⊗ RFib × O P
Cat ×∞ , f C where C is the algebra corresponding to the P -monoidal ∞ -category C ⊗ , and both squares arecartesian. Then the composite square is also cartesian. On the other hand, by Remark 5.16 thecomposite C ◦ f is the O -algebra corresponding to f ∗ C ⊗ , and so the pullback of RFib × along C ◦ f is by definition P O ( f ∗ C ) ⊗ . (cid:3) Corollary 6.21.
Let f : O → P be a morphism of cartesian pattern. We have natural equivalencesof ∞ -categories Alg O / P (P P ( C ) ⊗ ) ≃ Mon f ∗ C op , ⊗ ( S ) . Proof.
By Lemma 6.20 pulling back along f gives natural equivalencesAlg O / P (P P ( C ) ⊗ ) ≃ Alg O ( f ∗ P P ( C ) ⊗ ) ≃ Alg O (P O ( f ∗ C ) ⊗ ) . Since we also have ( f ∗ C ) op , ⊗ ≃ f ∗ ( C op , ⊗ ) the result now follows from Proposition 6.18. (cid:3) Remark 6.22.
As a special case, for the inclusion O int → O we get a commutative square ofequivalences Alg O int / O (P O ( C ) ⊗ ) Mon C op , ⊗ / int ( S )Fun / O el ( O el , P O ( C )) Fun( C op , ⊗ / el , S ) ∼∼ ∼∼ where the top horizontal map is an equivalence by Corollary 6.21, the right vertical map by Re-mark 2.12, the left vertical map by Lemma 4.14, and the bottom horizontal map by a trivial versionof Corollary 6.21 or by [GHN17, Proposition 7.3].We will next prove that every O -monoidal functor from a small O -monoidal ∞ -category extendsto the Day convolution, provided the target is compatible with small colimits in the following sense: Definition 6.23. If K is some class of ∞ -categories, we say that an O -monoidal ∞ -category C ⊗ → O is compatible with K -colimits if the ∞ -categories C E for E ∈ O el have K -shaped colimits, and forevery active map φ : O → E in O with E ∈ O el , the functor φ ! : Y i C O i ≃ C ⊗ O → C E preserves K -shaped colimits in each variable. If K is the class of all small ∞ -categories, we say that C ⊗ is compatible with (small) colimits or cocontinuously O -monoidal . Remark 6.24.
For any small O -monoidal ∞ -category C ⊗ , the Day convolution P O ( C ) ⊗ is compat-ible with small colimits. This is easy to see using the description of cocartesian morphisms in termsof products and left Kan extensions, since products in S preserve colimits in each variable. Definition 6.25.
Suppose C ⊗ and D ⊗ are O -monoidal ∞ -categories that are compatible with smallcolimits. We say that an O -monoidal functor F : C ⊗ → D ⊗ is cocontinuous if the underlying functors F E : C E → D E preserve small colimits for E ∈ O el . Proposition 6.26.
Let C ⊗ be a small O -monoidal ∞ -category and V ⊗ be an O -monoidal ∞ -categorycompatible with small colimits. Then every O -monoidal functor F : C ⊗ → V ⊗ induces a cocontinuous O -monoidal functor F ! : P O ( C ) ⊗ → V ⊗ such that the composite C ⊗ y ⊗ −−→ P O ( C ) ⊗ F ! −→ V ⊗ is equivalent to F , and F ! ,E : P( C E ) → V E for E ∈ O el is the unique cocontinuous functor extending F E along the Yoneda embedding of C E . If in addition V ⊗ is locally small, then F ! has a lax O -monoidal right adjoint F ∗ , given over E ∈ O el by the restricted Yoneda embedding F ∗ E : V E → P( V E ) → P( C E ) . Proof.
The O -monoidal functor F : C ⊗ → V ⊗ corresponds under the equivalence of Proposition 5.1to a morphism of O -algebras in d Cat ×∞ , i.e. a natural transformation φ : O × ∆ → d Cat ×∞ over F ∗ . Wecan pull back the cocartesian fibration [ RFib × → d Cat ×∞ along this to obtain a cocartesian fibration E + → O × ∆ . Let E be the full subcategory of E + containing those objects over 0 that lie in P O ( C ) ⊗ and those objects over 1 that lie in V ⊗ . We claim that the restricted projection E → O × ∆ isagain cocartesian (but the inclusion into E + does not preserve all cocartesian morphisms). Givenan object Φ ∈ P( C E ), which we can write as a small colimit colim x ∈ I y ( φ ( x )) of representablepresheaves, the cocartesian morphism in E + over (id E , →
1) takes Φ ∈ P( C E ) to the colimitcolim x ∈ I y ( F φ ( x )) computed in (large) presheaves on V E . Considering maps in b P( V E ) from thisto representable presheaves, we see there is an initial one, given by the same colimit computed inthe ∞ -category V E . This gives a cocartesian morphism in E over (id E , → V ⊗ with small colimits we see easily that E is a cocartesianfibration. Unstraightening this over ∆ we get a commutative diagram P O ( C ) ⊗ V ⊗ O F ! where F ! preserves cocartesian morphisms, as required. To get the right adjoint we apply thecriterion of [HMS19, Lemma A.1.10] to see that the composite E → O × ∆ → ∆ is also a cartesianfibration, and the cartesian morphisms lie over equivalences on O . Since we already know that thecomposition E → O is a cocartesian fibration, the only thing to check is that for every O ∈ O thecocartesian fibration E O → ∆ is a cartesian fibration. By identifying this cocartesian fibrationwith the functor P O ( C ) ⊗ O → V ⊗ O , it suffices to show that it has a right adjoint. Since this functor isequivalent to Q i P( C O i ) → Q i V O i and products of adjoints are adjoints, we only need to see that REE ALGEBRAS THROUGH DAY CONVOLUTION 21 each component F O i , ! : P( C O i ) → V O i has a right adjoint. If V O i is locally small then F O i , ! has aright adjoint F ∗ O i : V O i → P( V O i ) → P( C O i ) given by the composite of the Yoneda embedding andthe precomposition with F O i . This gives a commutative diagram V ⊗ P O ( C ) ⊗ O , F ∗ and it only remains to show that F ∗ preserves inert morphisms. Given an inert morphism φ : O O ′ we want to show that the canonical natural transformation F ∗ O ′ φ ! → φ ! F ∗ O , which arises as the mate transformation of the squareP O ( C ) ⊗ O V ⊗ O P O ( C ) ⊗ O ′ V ⊗ O ′ , φ ! F O, ! φ ! F O ′ , ! is an equivalence. We can identify this with the square Q i P( C O i ) Q i V O i Q j P( C O ′ j ) Q j V O ′ j , Q i F Oi, ! Q j F O ′ j, ! where the vertical maps are given by projections to the same subset of factors in the product. Itis then clear that the mate square also commutes, since the right adjoint of Q i F ! ,O i is the product Q i F ∗ O i . (cid:3) Remark 6.27.
The extension of F : C ⊗ → V ⊗ to P O ( C ) ⊗ is in fact unique. Since we do not needthis universal property we will only give a sketch of the argument: Given a cocontinuous O -monoidalfunctor Φ : P O ( C ) ⊗ → V ⊗ , we can compose this with the Yoneda embedding C ⊗ → P O ( C ) ⊗ ; thisdata we can interpret as a 2-simplex in the ∞ -category of large O -monoidal ∞ -categories, whichcorresponds to a 2-simplex of algebras O × ∆ → d Cat ×∞ . We can pull back [ RFib × along this to obtain a cocartesian fibration E + → O × ∆ . Then we considerthe full subcategory E whose objects over 0 are those in P O ( C ) ⊗ , whose objects over 1 are those inP O ( C ) ⊗ (but now viewed inside b P O (P O ( C )) ⊗ ), and whose objects over 2 are those in V ⊗ . As before,we can check that E → O × ∆ is a cocartesian fibration; its fibre over ∆ { , } corresponds to theextension (Φ | C ⊗ ) ! , its fibre over ∆ { , } to Φ, and using the universal property for mapping out ofpresheaves its fibre over ∆ { , } is an O -monoidal equivalence between two versions of P O ( C ) ⊗ O -Monoidal Localizations and Presentability of Algebras In this section we first discuss O -monoidal localizations and then consider presentably O -monoidal ∞ -categories, in the following sense: Definition 7.1.
We say an O -monoidal ∞ -category V ⊗ is presentably O -monoidal if it is compatiblewith small colimits and the ∞ -categories V E for E ∈ O el are all presentable. Our main result is that every presentably O -monoidal ∞ -category is an O -monoidal localizationof a Day convolution O -monoidal structure on a small full subcategory. We apply this to show thatif V is presentably O -monoidal then the ∞ -category Alg O ( V ) is presentable.We begin by proving with a general existence result for O -monoidal adjunctions, in the followingsense: Definition 7.2.
Consider a commutative triangle
C DB . Gp q
We say that G has a left adjoint relative to B if G has a left adjoint F : D → C and the unit map d → GF d maps to an equivalence in B for all d ∈ D . Remark 7.3. If F is a left adjoint relative to B as above, then it follows that pF ( d ) ≃ qGF ( d ) ≃ q ( d ), so that F is a functor over B . Moreover, the counit map F Gc → c also lies over an equivalencein B : applying p is the same as applying qG , and the map qGF Gc → qGc is an equivalence sincethe composite qGc → qGF Gc → qGc is the identity by one of the adjunction equivalences, and the first unit map is an equivalence byassumption. Thus G has a left adjoint relative to B if and only if it has a left adjoint in the( ∞ , ∞ -categories over B . Definition 7.4.
Suppose G : C ⊗ → D ⊗ is a lax O -monoidal functor. We say that G has a O -monoidal left adjoint if it has a left adjoint relative to O and this is lax O -monoidal. Remark 7.5.
By definition, G has an O -monoidal left adjoint if and only if it has a left adjointin the ( ∞ , O -monoidal ∞ -categories and lax O -monoidal functors. It is not hard toshow that if G has a left adjoint F relative to O that is a lax O -monoidal functor, then F is in fact O -monoidal. Proposition 7.6.
Consider a commutative triangle C ⊗ D ⊗ O , p G q where p and q are O -monoidal ∞ -categories and G is lax O -monoidal. Suppose(1) G E : C E → D E admits a left adjoint F E for all E ∈ O el ,(2) for every active map φ : O → E in O with E ∈ O el , the natural transformation F E ◦ φ C ! → φ D ! ◦ Y i F O i of functors Q i C O i → D E , is an equivalence.Then the functor G admits a left adjoint F relative to O , and this is an O -monoidal functor.Proof. This is the O -monoidal analogue of [Lur17, Corollary 7.3.2.12] and follows from the criterionfor existence of relative left adjoints in [Lur17, Proposition 7.3.2.11]. To apply this we must showthe following conditions hold:(a) For every O ∈ O , the functor G O : C ⊗ O → D ⊗ O admits a left adjoint F O .(b) for every morphism φ : O → O ′ in O the natural transfromation F O ′ ◦ φ C ! → φ D ! ◦ F O is an equivalence. REE ALGEBRAS THROUGH DAY CONVOLUTION 23
Condition (a) follows from (1) since the functor G O is equivalent to the product Q i G O i : Q i C O i → Q i D O i , which has left adjoint Q i F O i . Condition (b) is obvious for inert maps φ (since then φ C ! and φ D ! are both projections unto the same collection of factors in a product), so using the factorizationsystem it is enough to consider φ active. But then we can write φ C ! as the product Q i φ C i, ! where φ i : O i O ′ i comes from the factorization O O ′ O i O ′ i , ρ O ′ i φ i where we know the natural transformation for φ i is an equivalence by assumption (2). The map inquestion is therefore a product of maps we know are equivalences. Condition (b) is also equivalentto F : D ⊗ → C ⊗ preserving cocartesian edges, so F is an O -monoidal functor. (cid:3) Remark 7.7.
For any morphism of cartesian patterns f : P → O , a relative adjunction between O -monoidal ∞ -categories as in Proposition 7.6 induces via composition an adjunction on ∞ -categoriesof algebras, F ∗ : Alg P / O ( D ) ⇄ Alg P / O ( C ) : G ∗ . This follows from the 2-functoriality of Alg P / O (–) as discussed in Remark 4.11. Definition 7.8. An O -monoidal localization is an O -monoidal functor V ⊗ → U ⊗ that has a rightadjoint relative to O which is lax O -monoidal and fully faithful. Remark 7.9.
Let L : V ⊗ → U ⊗ be an O -monoidal localization with right adjoint i : U ⊗ ֒ → V ⊗ . Forany morphism of cartesian patterns f : P → O , we get an induced localization of ∞ -categories ofalgebras: as in Remark 7.7 we have an induced adjunction L ∗ : Alg P / O ( V ) ⇄ Alg P / O ( U ) : i ∗ , given by composition with L and i , and the equivalence L ∗ i ∗ ≃ ( Li ) ∗ ≃ id implies that i ∗ is fullyfaithful. Since a P -algebra A : O → V ⊗ factors through the full subcategory U ⊗ if and only if A ( E ) ∈ U E for every E ∈ O el , we see that Alg P / O ( U ) is the full subcategory of Alg P / O ( V ) spannedby the algebras with this property. We can interpret this as the commutative squareAlg P / O ( U ) Alg P / O ( V )Fun P el / O el ( U ) Fun P el / O el ( V )being cartesian. Notation 7.10.
Suppose V ⊗ is an O -monoidal ∞ -category. Given a collection of full subcategories U E ⊆ V E for E ∈ O el , the full subcategory U ⊗ ⊆ V ⊗ generated by ( U E ) E ∈ O el is that spanned bythe objects over O ∈ O that lie in the full subcategory of V ⊗ O that is identified with Q i U O i underthe equivalence V ⊗ O ≃ Q i V O i . (Note that in general U ⊗ is not an O -monoidal ∞ -category, thoughit is an O - ∞ -operad.) Corollary 7.11.
Let V ⊗ be an O -monoidal ∞ -category, and suppose given full subcategories U E ⊆ V E for all E ∈ O el such that(1) each inclusion i E : U E ֒ → V E has a left adjoint L E ,(2) for every active morphism φ : O E in O with E ∈ O el , the natural map φ V ! O ( X , . . . , X O n ) → φ V ! O ( L O X , . . . , L O n X n ) is taken to an equivalence by L E .If U ⊗ ⊆ V ⊗ is the full subcategory generated by ( U E ) E ∈ O el , then:(i) The restricted functor U ⊗ → O is an O -monoidal ∞ -category. (ii) The inclusion i : U ⊗ ֒ → V ⊗ is lax O -monoidal.(iii) The lax monoidal functor i exhibits U ⊗ as an O -monoidal localization. In other words, thefunctor i has a left adjoint L : V ⊗ → U ⊗ relative to O , and this is an O -monoidal functor.Proof. For O ∈ O , let L O : V ⊗ O → U ⊗ O denote the functor corresponding to the product Q i L O i ,which is left adjoint to the inclusion i O . We claim that for φ : O → O ′ in O and O ( U i ) ∈ U ⊗ O , themorphism O ( U i ) → L O ′ φ V ! O ( U i ) is cocartesian. It is easy to see that it is locally cocartesian, andthen condition (2) implies that these locally cocartesian morphisms compose. This proves (i). If φ is inert, then we do not have to apply L O ′ (since φ V ! just projects to some factors in a product),so the inclusion i preserves inert morphisms, which gives (ii). This means all the conditions ofProposition 7.6 hold for i , which gives (iii). (cid:3) Our next goal is to show that every presentably O -monoidal ∞ -category can be described as an O -monoidal localization of a Day convolution. We start by briefly discussing the more general caseof accessibly O -monoidal ∞ -categories: Definition 7.12.
We say an O -monoidal ∞ -category V ⊗ is accesibly O -monoidal if the ∞ -categories V E for E ∈ O el are all accessible, and for every active map φ : O → E with E ∈ O el , the functor φ V ! : Q i V O i → V E is accessible. We say that O is κ -accessibly O -monoidal for some regular cardinal κ if the ∞ -categories V E are all κ -accessible and for every active map φ : O → E with E ∈ O el , thefunctor φ V ! : Q i V O i → V E preserves κ -filtered colimits. We say that V ⊗ is κ -presentably O -monoidal for some regular cardinal κ if V ⊗ is both presentably and κ -accessibly O -monoidal. Remark 7.13. If V ⊗ is accessibly (presentably) O -monoidal, then we can always choose a regularcardinal κ such that V ⊗ is κ -accessibly ( κ -presentably) O -monoidal. Proposition 7.14.
Suppose V ⊗ is an accessibly O -monoidal ∞ -category. Then there exists a regularcardinal κ such that V ⊗ is κ -accessibly O -monoidal, the full subcategory V κ, ⊗ generated by thecollection ( V κE ) E ∈ O el of κ -compact objects is an O -monoidal ∞ -category, and the inclusion V κ, ⊗ ֒ → V ⊗ is O -monoidal.Proof. We can choose a regular cardinal λ such that V E is λ -accessible for each E ∈ O el and thefunctor φ V ! : Q i V O i → V E preserves λ -filtered colimits in each variable. We can then choose aregular cardinal κ ≫ λ such that for every such active map φ , we have φ V ! Y i V λO i ! ⊆ V κE . By [CH20, Lemma 2.6.11], any object of V κE is the colimit of a κ -small λ -filtered diagram in V λE .Since φ V ! preserves λ -filtered colimits in each variable, for O ( v i ) ∈ V κ, ⊗ O we can write φ V ! ( O ( v i )) as a κ -small colimit of κ -compact objects, and hence this object is also κ -compact by [Lur09, Corollary5.3.4.15]. The full subcategory V κ, ⊗ therefore inherits cocartesian morphisms from V ⊗ , which meansthat it is an O -monoidal ∞ -category and the inclusion into V ⊗ is an O -monoidal functor. (cid:3) Corollary 7.15.
Suppose V ⊗ is a presentably O -monoidal ∞ -category. Then there exists a regularcardinal κ such that V ⊗ is an O -monoidal localization of P O ( V κ ) ⊗ .Proof. By Proposition 7.14 we can choose a regular cardinal κ such that the full subcategory V κ, ⊗ isan O -monoidal ∞ -category and the inclusion i : V κ, ⊗ ֒ → V ⊗ is O -monoidal. Since V ⊗ is compatiblewith small colimits, by Proposition 6.26 there then exists a cocontinuous O -monoidal functor L : P O ( V κ ) ⊗ → V ⊗ extending i along the Yoneda embedding V κ, ⊗ ֒ → P O ( V κ ) ⊗ , and this has a lax O -monoidal rightadjoint R : V ⊗ → P O ( V κ ) ⊗ . It remains to show that R is fully faithful. This amounts to showingthat for φ : O → E in O active with E ∈ O el and O ( V i ) ∈ V ⊗ O , the natural map L E φ P O ( V κ )! R O O ( V i ) → φ V ! O ( V i ) REE ALGEBRAS THROUGH DAY CONVOLUTION 25 is an equivalence. This follows since it is true when O ( V i ) lies in V κ, ⊗ O and all the functors preserve κ -filtered colimits in each variable (for R O this is true since it is equivalent to the product of therestricted Yoneda embeddings V O i ֒ → P( V κO i ), which tautologically preserve κ -filtered colimits). (cid:3) Remark 7.16.
Corollary 7.15 says in particular that for any presentably O -monoidal ∞ -category V ⊗ there exists a small O -monoidal ∞ -category C ⊗ and an O -monoidal localizationP O ( C ) ⊗ L −→ V ⊗ . Our next goal is to obtain another description of localizations of Day convolutions, in terms oflocalizing at classes of maps compatible with the O -monoidal structure: Notation 7.17.
Let C be a small ∞ -category and S a set of maps in P( C ). We write P S ( C ) for thefull subcategory of P( C ) spanned by the S -local objects, i.e. the objects Φ such that Map P( C ) (– , Φ)takes the morphisms in S to equivalences. Definition 7.18.
Let C ⊗ be a small O -monoidal ∞ -category. A collection S = ( S E ) E ∈ O el of setsof morphisms S E in C E is compatible with the O -monoidal structure if for every active morphism φ : O E in O with E ∈ O el , the functor φ P O ( C )! : Y i P( C O i ) ≃ P O ( C ) ⊗ O → P( C E )takes a morphism (id , . . . , id , s, id , . . . , id) with s ∈ S O i into the strongly saturated class S E gen-erated by S E . We then write P O , S ( C ) ⊗ for the full subcategory generated by the collection of fullsubcategories P S E ( C E ) of S E -local objects.By Corollary 7.11 we get an O -monoidal localization of P O ( C ) ⊗ : Corollary 7.19.
Let C ⊗ be a small O -monoidal ∞ -category and S a collection of sets of morphismscompatible with the O -monoidal structure. Then there is an O -monoidal localization P O ( C ) ⊗ L S −→ P O , S ( C ) ⊗ left adjoint to the inclusion P O , S ( C ) ⊗ ֒ → P O ( C ) ⊗ .Proof. It suffices to verify the two conditions in Corollary 7.11. Since S E is a set for every E ∈ O el ,the inclusion P S E ( C E ) ֒ → P( C E ) has a left adjoint L S ,E , which exhibits P S E ( C E ) as the localizationat the strongly saturated class of maps generated by S E . Hence, the first condition is satisfied, andfor the second condition we observe that the compatibility of the maps in S with the O -monoidalstructure implies that for every active map φ : O E , the map φ P O ( C )! O ( X , . . . , X i , . . . , X n ) → φ P O ( C )! O ( X , . . . , L S ,O i X i , . . . , X n ) lies in S E . In particular, the map φ P O ( C )! O ( X , . . . , X n ) → φ P O ( C )! O ( L S ,O X , . . . , L S ,O n X n ) is taken to an equivalence by L S ,E , which is the second conditionof Corollary 7.11. (cid:3) Remark 7.20.
Let us say an O -monoidal localization is accessible if the component of the fullyfaithful right adjoint at each object of O el is an accessible functor. Then it follows from the classi-fication of accessible localizations of presheaf ∞ -categories in [Lur09, § O -monoidal localization of P O ( C ) ⊗ is of the form P O , S ( C ) ⊗ for some collection of sets of morphisms S compatible with the O -monoidal structure. Remark 7.21.
Suppose V ⊗ is a presentably O -monoidal ∞ -category and κ is a regular cardinalsuch that V ⊗ is κ -presentably O -monoidal and the full subcategory V κ, ⊗ is O -monoidal. Then V ⊗ is equivalent to P O , S κ ( V κ ) ⊗ where S κE consists of the mapscolim I y ( φ ) → y (colim I φ )in P( V κE ) where φ : I → V κE ranges over a set of representatives of κ -small colimit diagrams. Definition 7.22.
Let C ⊗ be a small O -monoidal ∞ -category and S a collection of sets of morphismscompatible with the O -monoidal structure on P O ( C ) ⊗ . We say that a C op , ⊗ -monoid M : C op , ⊗ → S is S -local if for every E ∈ O el the restriction M E : C op E ≃ ( C op , ⊗ ) E → S is S E -local. We write Mon S C op , ⊗ ( S ) for the full subcategory of Mon C op , ⊗ ( S ) spanned by the S -localmonoids. Proposition 7.23.
Let C ⊗ and S be as in Definition 7.22. Then there is a natural equivalence Alg O (P O , S ( C )) ≃ Mon S C op , ⊗ ( S ) . Proof.
By Remark 7.9 the inclusion P O , S ( C ) ⊗ ֒ → P O ( C ) ⊗ identifies Alg O (P O , S ( C )) with the fullsubcategory of Alg O (P O ( C )) spanned by the algebras A such that A ( E ) lies in P S E ( C E ) for every E ∈ O el . Under the equivalence Alg O (P O ( C )) ≃ Mon C op , ⊗ ( S )of Proposition 6.18 this full subcategory is identified with Mon S C op , ⊗ ( S ). (cid:3) Remark 7.24.
Proposition 7.23 implies that the ∞ -category Alg O (P O , S ( C )) is equivalent to thefull subcategory of Fun( C op , ⊗ , S ) spanned by objects that are local with respect to a set of maps.Thus Alg O (P O , S ( C )) is an accessible localization of a presheaf ∞ -category and so is in particulara presentable ∞ -category. Since every presentably O -monoidal ∞ -category is equivalent to one ofthis form by Corollary 7.15, we have proved the following: Corollary 7.25.
Suppose V ⊗ is a presentably O -monoidal ∞ -category. Then the ∞ -category Alg O ( V ) is presentable. (cid:3) Extendability and Free Algebras
In this section we recall the notion of extendability for a morphism f : O → P of cartesian patterns,and show that if V is a presentably P -monoidal ∞ -category, then the left adjoint f ! : Alg O ( V ) → Alg P ( V )to the functor given by composition with f can be described by an explicit colimit formula. Inparticular, if O int → O is extendable (in which case we just say that O is extendable) then we getan explicit formula for free O -algebras. Definition 8.1.
A morphism f : O → P of cartesian patterns has unique lifting of inert morphisms if for every inert morphism φ : f ( O ) P in P there is a unique lift to an inert morphism ψ : O O ′ in O such that f ( ψ ) ≃ φ . In other words, the induced map of ∞ -groupoids( O int O/ ) ≃ → ( P int f ( O ) / ) ≃ is an equivalence. Remark 8.2. If f : O → P has unique lifting of inert morphisms, then by [CH19, Corollary 7.4]we can define a functor P int → Cat ∞ that takes P ∈ P to O act /P and an inert morphism α : P P ′ to the functor α ! : O act /P → O act /P ′ that takes ( O, f ( O ) P ) to ( O ′ , f ( O ′ ) P ′ ) obtained from theinert-active factorization of f ( O ) P P ′ as f ( O ) f ( O ′ ) P ′ using the unique lifting of theinert part. Definition 8.3.
A morphism f : O → P of cartesian patterns is extendable if(1) f has unique lifting of inert morphisms,(2) for P ∈ P over h n i , the functor O act /P → n Y i =1 O act /P i taking ( O, φ : f ( O ) → P ) to ( ρ Pi, ! ( O, φ )) i is cofinal. REE ALGEBRAS THROUGH DAY CONVOLUTION 27
Remark 8.4.
The general definition of an extendable morphism in [CH19, Definition 7.7] has athird condition, but this is automatic in the case of cartesian patterns: Namely, given an activemorphism φ : f ( O ) P , we can use the unique lifting of inert morphisms to define a functor P el , op P/ → Cat ∞ taking α : P → E to O el α ! O/ . If O el ( φ ) → P el P/ denotes the corresponding cartesianfibration then there is a functor O el ( φ ) → O el O/ that takes ( α, α ! O → E ′ ) to O → α ! O → E ′ . Thecondition is that this functor should induce an equivalencelim O el O/ F → lim O el ( φ ) F for every functor F : O el → S . However, if f is a morphism of cartesian patterns then this functoris necessarily an equivalence: if h n i = | O | and h m i = | P | then O el O/ and P el P/ are isomorphic tothe discrete sets { ρ Oi : i = 1 , . . . , n } and { ρ Pi : i = 1 , . . . , m } , while O el ρ Pi, ! O/ is isomorphic to the set | φ | − ( i ). Thus O el ( φ ) is the set of pairs { ( i, j ) : 1 ≤ i ≤ m, j ∈ | φ | − ( i ) } and the map to O el O/ is theobvious isomorphism of this with { , . . . , n } (implied by φ being active). Example 8.5. If f : O → P is an extendable morphism of cartesian patterns then for any commu-tative square C ⊗ D ⊗ O P , Ff where C ⊗ is an O -monoidal ∞ -category, D ⊗ is a P -monoidal ∞ -category, and f preserves cocartesianmorphisms, then the morphism F is extendable. This is a special case of [CH19, Proposition 9.5].In particular, for any P -monoidal ∞ -category D ⊗ , in the pullback square f ∗ D ⊗ D ⊗ O P , ¯ ff the morphism ¯ f : f ∗ D ⊗ → D ⊗ is extendable. Proposition 8.6.
Suppose f : O → P is an extendable morphism of cartesian patterns. Then thefunctor f ∗ : Mon P ( S ) → Mon O ( S ) has a left adjoint f ! , given by left Kan extension along f , whichsatisfies f ! M ( P ) ≃ colim O ∈ O act /P M ( O ) . Proof.
This is a special case of [CH19, Proposition 7.13]. We give a brief sketch of the proof, as itis particularly simple in the case of cartesian patterns. Since f ∗ : Fun( P , S ) → Fun( O , S ) has a leftadjoint f ! given by left Kan extension, it suffices to show that if M is an O -monoid, then the leftKan extension f ! M is an O -monoid. We have natural equivalences f ! M ( P ) ≃ colim O ∈ O /P M ( O ) ≃ colim O ∈ O act /P M ( O ) (Definition 8.3(1) and [CH19, 7.2]) ≃ colim ( O i ) ∈ Q i O act /Pi Y i M ( O i ) (Definition 8.3(2)) ≃ Y i colim O i ∈ O act /Pi M ( O i ) ( S cartesian closed) ≃ Y i f ! M ( P i ) , as required. (cid:3) Remark 8.7.
The same result is true more generally for monoids in any ∞ -category where thecartesian product commutes with colimits indexed by the ∞ -categories O act /P . Definition 8.8.
Let O be a cartesian pattern. We write Act O ( O ) for the ∞ -groupoid of activemorphisms to O in O . We say O is extendable if the functorAct O ( O ) → Y i Act O ( O i ) , taking φ : O ′ O to the morphism ρ Oi, ! φ given by the inert-active factorization O ′ Oρ Oi, ! O ′ O i , φ ρ Oi ρ Oi, ! φ is an equivalence. Remark 8.9.
Since the equivalences are precisely the morphisms that are both active and inert, wecan identify Act O ( O ) with ( O int ) act /O . Thus O is extendable if and only if the inclusion O int → O is anextendable morphism of cartesian patterns (since unique lifting of inert morphisms is tautologicalin this case). Corollary 8.10.
Suppose O is an extendable cartesian pattern. Then the functor U O : Mon O ( S ) → Fun( O el , S ) given by restriction to O el has a left adjoint F O given by right Kan extension along O el ֒ → O int followed by left Kan extension along O int → O . This satisfies F O (Φ)( O ) ≃ colim O ′ O ∈ Act O ( O ) Y i Φ( O ′ i ) . Proof.
Combine Proposition 8.6 with Remark 2.12. (cid:3)
Example 8.11.
Let us say a cartesian pattern O is strongly extendable if the functor O act /O → n Y i =1 O act /O i given by ( ρ Oi, ! ) i is an equivalence. If O is a strongly extendable cartesian pattern, then for anymorphism of O - ∞ -operads (i.e. weak Segal O -fibrations in the terminology of [CH19]) E FO , f the morphism f is extendable by [CH19, Corollary 9.16]. In particular, any O - ∞ -operad is anextendable cartesian pattern.We now want to extend these descriptions of left adjoints from monoids to algebras, starting withthe special case of Day convolution: Proposition 8.12.
Suppose C is a small P -monoidal ∞ -category, and f : O → P is an extendablemorphism of cartesian patterns. Then the functor f ∗ : Alg P (P P ( C ) ⊗ ) → Alg O / P (P P ( C ) ⊗ ) has a left adjoint f ! , which for P ∈ P el satisfies ( f ! A )( P ) ≃ colim ( O,φ : f ( O ) P ) ∈ O act /P φ ! A ( O ) . REE ALGEBRAS THROUGH DAY CONVOLUTION 29
Proof.
Consider the pullback square f ∗ C op , ⊗ C op , ⊗ O P , ¯ ff where the morphism ¯ f : f ∗ C op , ⊗ → C op , ⊗ is an extendable morphism of cartesian patterns by Ex-ample 8.5. From Proposition 8.6 we therefore get a left adjoint¯ f ! : Mon f ∗ C op , ⊗ ( S ) → Mon C op , ⊗ ( S ) , given by left Kan extension; for X ∈ C op , ⊗ and M ∈ Mon f ∗ C op , ⊗ ( S ) this satisfies¯ f ! M ( X ) ≃ colim ( Y, ¯ f ( Y ) → X ) ∈ ( f ∗ C op , ⊗ ) act /X M ( Y ) . If X lies over P ∈ P , we see from the proof of [CH19, Proposition 9.5] that the canonical projection( f ∗ C op , ⊗ ) act /X → O act /P is a cocartesian fibration, whose fibre at ( O, f ( O ) P ) is( f ∗ C op , ⊗ ) O × C op , ⊗ P C op , ⊗ P/X ≃ C op , ⊗ f ( O ) × C op , ⊗ P C op , ⊗ P/X ≃ C op , ⊗ f ( O ) /X . Thus we can rewrite the formula for ¯ f ! M ( X ) as¯ f ! M ( X ) ≃ colim ( O,f ( O ) φ P ) ∈ O act /P colim ( Y,φ ! Y → X ) ∈ C op , ⊗ f ( O ) /X M ( Y ) , where we are omitting notation for the equivalence ( f ∗ C ⊗ ) O ≃ C ⊗ f ( O ) . Here we can identifycolim ( Y,φ ! Y → X ) ∈ C op , ⊗ f ( O ) /X M ( Y ) with the value at X of the left Kan extension of M | C op , ⊗ f ( O ) along φ ! : C op , ⊗ f ( O ) → C op , ⊗ P , which is the cocartesian pushforward in P P ( C ⊗ ). Under the natural equiva-lencesMon C op , ⊗ ( S ) ≃ Alg P (P P ( C ) ⊗ ) , Mon f ∗ C op , ⊗ ( S ) ≃ Alg O (P O ( f ∗ C ) ⊗ ) ≃ Alg O / P (P P ( C ) ⊗ )this formula therefore corresponds to the one above. (cid:3) Corollary 8.13.
Suppose V is a presentably P -monoidal ∞ -category, and f : O → P is an extendablemorphism of cartesian patterns. Then the functor f ∗ : Alg P ( V ) → Alg O / P ( V ) has a left adjoint f ! , which for P ∈ P el satisfies ( f ! A )( P ) ≃ colim ( O,φ : f ( O ) P ) ∈ O act /P φ ! A ( O ) . Proof.
Since V is presentably P -monoidal, by Corollary 7.15 there exists a small P -monoidal ∞ -category C and a P -monoidal localization L : P P ( C ) ⊗ → V ⊗ , with a fully faithful lax P -monoidal right adjoint i : V ⊗ → P P ( C ) ⊗ . From Remark 7.9 we then havea commutative square Alg P ( V ) Alg O / P ( V )Alg P (P P ( C ) ⊗ ) Alg O / P (P P ( C ) ⊗ ) , f ∗ V i ∗ i ∗ f ∗ where the vertical functors are both fully faithful, with left adjoints given by L ∗ . It follows that wehave a commutative square of left adjointsAlg O / P (P P ( C ) ⊗ ) Alg P (P P ( C ) ⊗ )Alg O / P ( V ) Alg P ( V ) . f ! L ∗ L ∗ f V , ! Since the right adjoint i ∗ is fully faithful, we have id ∼ −→ i ∗ L ∗ and hence f ! ∼ −→ i ∗ L ∗ f ! ≃ i ∗ f V , ! L ∗ which implies that f V , ! ≃ L ∗ f ! i ∗ by adjointness. where the left adjoint f V , ! is the composite L ∗ f ! i ∗ .Since L preserves colimits and cocartesian morphisms, this implies that f V , ! satisfies f V , ! A ( P ) ≃ L colim ( O,φ : f ( O ) P ) ∈ O act /P φ ! i ( A ( O )) ! ≃ colim ( O,φ : f ( O ) P ) ∈ O act /P φ ! A ( O ) , as required. (cid:3) Corollary 8.14.
Suppose O is an extendable cartesian pattern, and V is a presentably O -monoidal ∞ -category. Then the restriction U O : Alg O ( V ) → Alg O int / O ( V ) ≃ Fun / O el ( O el , V ) has a left adjoint F O : Fun / O el ( O el , V ) → Alg O ( V ) , which for Φ : O el → V and E ∈ O el is given by F O Φ( E ) ≃ colim ( φ : O E ) ∈ Act O ( E ) φ ! (Φ( O ) , . . . , Φ( O n )) . Moreover, the adjunction F O ⊣ U O is monadic.Proof. The existence of the left adjoint follows from Corollary 8.13 applied to the map O int → O (together with the equivalence of Lemma 4.14). To see the adjunction is monadic we apply theBarr–Beck theorem for ∞ -categories, [Lur17, Theorem 4.7.3.5]. We then need to show that U O detects equivalences, which is clear, and that U O -split simplicial objects have colimits and these arepreserved by U O . Suppose therefore that we have a U O -split simplicial diagram φ : „ op → Alg O ( V ).Since V ⊗ is presentably O -monoidal, it is an O -monoidal localization of a Day convolution P O ( C ) ⊗ ,which gives a commutative diagramAlg O ( V ) Mon C op , ⊗ ( S )Fun / O el ( O el , V ) Fun( C op , ⊗ / el , S ) . U O U ′ O Here the functor U ′ O is a monadic right adjoint by [CH19, Corollary 8.2] and so the colimit of the U ′ O -split simplicial diagram that is the image of φ exists and is preserved by U ′ O . (Alternatively,this follows from the description of sifted colimits of monoids in Remark 2.13.) But by Remark 7.9the commutative square above is cartesian, hence the colimit in Mon C op , ⊗ ( S ) actually lies in the fullsubcategory Alg O ( V ). Thus the U O -split simplicial diagram φ has a colimit and this is preserved by U O , as required. (cid:3) Remark 8.15.
We can remove the presentability condition in Corollary 8.14: since a small O -monoidal ∞ -category C ⊗ is always a full O -monoidal subcategory of the presentably O -monoidal ∞ -category P O ( C ) ⊗ , we can embed any large O -monoidal ∞ -category V ⊗ in a presentably O -monoidal ∞ -category in a larger universe. Moreover, we can do so in a way that preserves small colimits,in which case we see that the left adjoint from Corollary 8.14 restricts to the full subcategory of O -algebras in V ⊗ provided the O -monoidal structure is compatible with colimits of shape Act O ( E )for E ∈ O el . REE ALGEBRAS THROUGH DAY CONVOLUTION 31
Remark 8.16. If O is an extendable cartesian pattern, then the formula for the free O -monoidmonad on Fun( O el , S ) shows that this is an analytic monad in the sense of [GHK17], and hencecorresponds by the results of that paper to an ∞ -operad (in the sense of a not necessarily completedendroidal Segal space). We expect that this observation can be strengthened: there should be acanonical morphism O → O opd of cartesian patterns where O opd is a symmetric ∞ -operad, suchthat Mon O ( S ) ∼ −→ Mon O opd ( S ) and O el → O elopd is an epimorphism (i.e. is surjective on π ). We hopeto address this question elsewhere.9. Examples of Extendability
In this section we will give some examples of extendable patterns and morphisms, and spell outwhat our results from the previous section amount to in these examples.
Example 9.1.
The pattern F ♭ ∗ is (strongly) extendable: The category F act ∗ can be identified withthe category F of (unpointed) finite sets, so the desired equivalence F act ∗ / h n i → n Y i =1 F act ∗ / h i is the (“straightening”) equivalence F / n → n Y i =1 F between sets over n := { , . . . , n } and families of sets indexed by n , given by taking fibres at i ∈ n . The groupoid Act F ∗ ( h i ) is equivalent to the groupoid F ≃ of finite sets and bijections, i.e. ` ∞ n =0 B Σ n , and we recover the expected formula for free commutative algebras in a presentablysymmetric monoidal ∞ -category V ⊗ : U F ∗ F F ∗ ( V ) ≃ colim φ : h n i→h i∈ Act F ∗ ( h i ) φ ! ( V, . . . , V ) ≃ ∞ a n =0 V ⊗ nh Σ n . Example 9.2.
By Example 8.11, every morphism f : O → O ′ of symmetric ∞ -operads is extendable.If V ⊗ is a presentably O ′ -monoidal ∞ -category, we recover the formula for the operadic left Kanextension f ! : Alg O / O ′ ( V ) → Alg O ′ ( V ) from [Lur17]: for X ∈ O ′h i and A ∈ Alg O / O ′ ( V ), we have f ! A ( X ) ≃ colim ( O,φ ) ∈ O act /X f ( φ ) ! A ( O ) . In particular, if V ⊗ is a symmetric monoidal ∞ -category, then we have f ! A ( X ) ≃ colim ( O,φ ) ∈ O act /X | φ | ! A ( O ) ≃ colim ( O,φ ) ∈ O act /X O i A ( O i ) . Example 9.3.
As a special case of the previous example, every symmetric ∞ -operad O is anextendable cartesian pattern, and our results recover the expected formula for free O -algebras in apresentably symmetric monoidal ∞ -category V ⊗ : the forgetful functor U O : Alg O ( V ) → Fun( O el , V )has a left adjoint F O , which for E ∈ O el satisfies F O Φ( E ) ≃ colim O φ −→ E ∈ Act O ( E ) O i Φ( O i ) . If O el := O ≃h i ≃ ∗ we can define O ( n ) to be the fibre of Act O ( ∗ ) → F ≃ at the point of B Σ n (withits canonical Σ n -action), and then rewrite the formula in the more familiar form F O V ≃ ∞ a n =0 O ( n ) ⊗ Σ n V ⊗ n . Remark 9.4.
The formula in the previous example does not agree with that given in [Lur17, § O from the ∞ -operad O int containing only the inert morphismsin O , Lurie considers the extension from the ∞ -operad O × F ∗ F int ∗ containing those morphisms thatmap to inert morphisms in F ∗ (but are not necessarily cocartesian). The difference is that O × F ∗ F int ∗ remembers all the unary operations in O , while O int remembers only the invertible ones. Example 9.5.
Suppose f : O → P is a morphism of generalized symmetric ∞ -operads (in the senseof [Lur17, § F ♮ ∗ in the terminology of [CH19]). Thiscertainly has unique lifting of inert morphisms, and so is extendable as a morphism of cartesianpatterns if and only if for every P ∈ P over h n i , the functor O act /P → n Y i =1 O act /P i is cofinal. In particular, a generalized ∞ -operad O is extendable if and only ifAct O ( O ) → n Y i =1 Act O ( O i )is an equivalence for O ∈ O over h n i . By [CH19, Proposition 9.15], we do have an equivalencebetween Act O ( O ) and the iterated fibre productAct O ( O ) → Act O ( O ) × Act O ( σ ! O ) · · · × Act O ( σ ! O ) Act O ( O n ) , where σ denotes the unique map h n i → h i . Since the only active map to h i in F ∗ is the identity, for X ∈ O h i the ∞ -groupoid Act O ( X ) is equivalent to O ≃h i /X . If O h i is an ∞ -groupoid then Act O ( X )is therefore contractible for all X ∈ O h i . This shows that a generalized symmetric ∞ -operad O such that O h i is an ∞ -groupoid is always extendable. More generally, if f : O → P is a morphismof generalized ∞ -operads such that O h i and P h i are ∞ -groupoids and f h i : O h i → P h i is anequivalence, then f is extendable, since we have O act0 × P act0 P act0 /P ≃ ∗ for every P ∈ P . Example 9.6.
The pattern „ op ,♭ is strongly extendable: We can identify the category „ op , act withthe category O of finite ordered sets; then the desired equivalence( „ op , act ) / [ n ] → n Y i =1 ( „ op , act ) / [1] becomes the obvious equivalence O / n → n Y i =1 O / that takes an ordered set over n to its fibres at the points of n . Since every object of „ op has aunique active map to [1], the groupoid Act „ op ([1]) is isomorphic to the set { , , . . . } . If V ⊗ is apresentably „ op -monoidal ∞ -category we get the expected formula for free associative algebras: T „ op ( V ) ≃ colim φ : [ n ] → [1] ∈ Act „ op ([1]) φ ! ( V, . . . , V ) ≃ ∞ a n =0 V ⊗ n . We also get analogues of Examples 9.2, 9.3 and 9.5: • every morphism of non-symmetric ∞ -operads is extendable, • every non-symmetric ∞ -operad is extendable, • every generalized non-symmetric ∞ -operad whose fibre at [0] is an ∞ -groupoid is extendable, Note, however, that a more general morphism between generalized ∞ -operads whose fibres at h i are ∞ -groupoidsmay still fail to be extendable . REE ALGEBRAS THROUGH DAY CONVOLUTION 33 • every morphism of generalized non-symmetric ∞ -operads whose fibres at [0] are ∞ -groupoidsand whose restriction to these is an equivalence, is extendable. Morita Equivalences
In this section we use our results on extendable cartesian patterns to give a condition for amorphism of cartesian patterns to give an equivalence of ∞ -categories of algebras, i.e. to be a Morita equivalence in the following sense:
Definition 10.1.
We say that a morphism of cartesian patterns f : O → P is a Morita equivalence if for every P -monoidal ∞ -category V ⊗ the functor f ∗ : Alg P ( V ) → Alg O / P ( V ) , given by composition with f , is an equivalence. Remark 10.2. If f is a Morita equivalence then as a special case (taking V ⊗ to be Cat ×∞ ) we havethat pullback along f gives an equivalence between P -monoidal and O -monoidal ∞ -categories.Our discussion of free algebras leads to a checkable criterion for a morphism of extendable carte-sian patterns to be a Morita equivalence: Proposition 10.3.
Suppose O and P are extendable cartesian patterns and f : O → P is a morphismof cartesian patterns such that(1) f el : O el → P el is an equivalence of ∞ -groupoids,(2) for every E ∈ O el the functor Act O ( E ) → Act P ( f ( E )) , induced by f , is an equivalence of ∞ -groupoids.Then f is a Morita equivalence.Proof. We first consider the case where V ⊗ is presentably P -monoidal. Then we have a commutativediagram Alg P ( V ) Alg O / P ( V )Fun / P el ( P el , V ) Fun / O el ( O el , f ∗ V ) f ∗ U P U O f el , ∗ where the vertical maps are monadic right adjoints by Corollary 8.14 and the bottom horizontalmap is an equivalence by assumption (1). Using [Lur17, Corollary 4.7.3.16] we see that f ∗ is anequivalence if and only if for every ξ ∈ Fun / P el ( P el , V ) the natural map F O f el , ∗ ξ → f ∗ F P ξ is an equivalence. Since U P detects equivalences, it suffices to show that the induced map( F O f el , ∗ ξ )( E ) → ( F P ξ )( f ( E ))is an equivalence for all E ∈ P el . Since O and P are extendable we have colimit formulas for F O and F P , which identify this map with the mapcolim α : O → E ∈ Act O ( E ) α ! ξ ( f ( O )) → colim β : P → f ( E ) ∈ Act P ( f ( E )) β ! ξ ( P ) , which is the natural map of colimits arising from the morphism Act O ( E ) → Act P ( f ( E )) inducedby f . This is an equivalence by assumption (2). Thus f ∗ is an equivalence for every presentably P -monoidal ∞ -category. Since we can embed any P -monoidal ∞ -category fully faithfully in apresentably P -monoidal one (possibly after passing to a larger universe) this completes the proof. (cid:3) The existence of operadic left Kan extensions in this case was used in [GH15].
Remark 10.4. If f : O → P is a morphism between extendable cartesian patterns such that f el : O el → P el is an equivalence, then this proof in the case of S shows that condition (2) isalso necessary for f to be a Morita equivalence.In some cases this criterion can be used to identify the ∞ -operad corresponding to a cartesianpattern without using the highly technical machinery of approximations from [Lur17, § Example 10.5 (Associative algebras) . Let Ass → F ∗ denote the (symmetric) associative ∞ -operad.As in [Lur17, Remark 4.1.1.4] we can think of this as a category whose objects are the pointed finitesets h n i ∈ F ∗ , with a morphism h n i → h m i given by a morphism φ : h n i → h m i in F ∗ together withlinear orderings ≤ i of the preimages φ − ( i ), 1 ≤ i ≤ m . The composite of ( φ, ≤ i ) : h n i → h m i and( ψ, ≤ ′ j ) : h m i → h k i is given by the composite ψφ in F ∗ with the ordering ≤ ′′ t of ( ψφ ) − ( t ) given by i ≤ ′′ t i ′ ⇐⇒ φ ( i ) ≤ ′ t φ ( i ′ ) and i ≤ s i ′ if s = φ ( i ) = φ ( i ′ ) . There is a functor cut : „ op → Ass that takes [ n ] ∈ „ op to h n i and a morphism φ : [ m ] → [ n ] in „ to the morphism h n i → h m i given by i ( j, φ ( j − < i ≤ φ ( j ) , , if no such j existswith the linear ordering of cut( φ ) − ( j ) that given by identifying this with { i : φ ( j − < i ≤ φ ( j ) } .It is easy to see that cut : „ op → Ass is a morphism of cartesian patterns, and we claim that it is aMorita equivalence: Both patterns are extendable by Examples 9.3 and 9.6, with „ op , el ≃ Ass el ≃ ∗ .Moreover, Act „ op ([1]) is the discrete set { [ n ] → [1] : n = 0 , , . . . } while Act Ass ( h i ) can be identifiedwith the disjoint union over n of the contractible groupoid of linear orderings of { , . . . , n } . Theconditions of Proposition 10.3 therefore hold, and so we get for any (Ass-)monoidal ∞ -category V ⊗ an equivalence Alg „ op / Ass ( V ) ∼ −→ Alg
Ass ( V ) . Example 10.6 (Bimodules) . Let „ op ,♭/ [1] denote the category „ op / [1] := ( „ / [1] ) op with the inert/activefactorization system lifted from „ op (along the left fibration „ op / [1] → „ op ) and the three maps[1] → [1] as elementary objects. We can think of a morphism [ n ] → [1] as a sequence ( i , . . . , i n )with 0 ≤ i ≤ · · · ≤ i n ≤
1; then the elementary objects are (0 , , , → F ∗ denote the (symmetric) bimodule operad (whose algebras are given by a pair ofassociative algebras and a bimodule between them). This can be described (cf. [Lur17, Notation4.3.1.5]) as a category where • objects are lists ( h n i , ( a , b ) , . . . , ( a n , b n )) where 0 ≤ a i ≤ b i ≤ • a morphism ( h n i , ( a , b ) , . . . , ( a n , b n )) → ( h m i , ( a ′ , b ′ ) , . . . , ( a ′ m , b ′ m )) is given by a morphism( φ, ≤ i ) : h n i → h m i in Ass such that for j = 1 , . . . , m , if φ − ( j ) = { i < j i < j · · · < j i k ), then a ′ j = a i ≤ b i = a i ≤ · · · ≤ a i k ≤ b i k = b ′ j . We then define a functor „ op / [1] cut ′ −−→ Bimod by • cut ′ ( i , . . . , i n ) = ( h n i , ( i , i ) , . . . , ( i n − , i n )) , • for a morphism φ : ( i , . . . , i n ) → ( j , . . . , j m ), which is given by φ : [ m ] → [ n ] in „ such that j s = i φ ( s ) , we set cut ′ ( φ ) = cut( φ ), which satisfies the required condition.The functor cut ′ then fits in a commutative square „ op / [1] Bimod „ op Ass , cut ′ cutREE ALGEBRAS THROUGH DAY CONVOLUTION 35 and is a morphism of cartesian patterns (since the factorization systems are lifted from those on „ op and Ass). The pattern „ op ,♭/ [1] is extendable, e.g. by the non-symmetric analogue of Exam-ple 9.5, while Bimod is extendable by Example 9.3. We have „ op , el / [1] = { (0 , , (0 , , (1 , } whichis isomorphic to Bimod el = { ( h i , (0 , , ( h i , (0 , , ( h i , (1 , ∞ -groupoids Act „ op / [1] ((0 , „ op / [1] ((1 , N (with n corresponding to the unique activemorphisms (0 , . . . , (0 ,
0) and (1 , . . . , (1 , n ]),while Act „ op / [1] ((0 , N × N (with ( n, m ) corresponding to the unique active map(0 , . . . , , , . . . , → (0 ,
1) with ( n + 1) 0’s and ( m + 1) 1’s). On the other hand, the ∞ -groupoidAct Bimod (( h i , ( i, j )) we can describe as a coproduct of contractible groupoids indexed by the setAct „ op / [1] ( i, j )). Hence Proposition 10.3 applies, and so we get for any Bimod-monoidal ∞ -category V ⊗ an equivalence Alg „ op / [1] / Bimod ( V ) ∼ −→ Alg
Bimod ( V ) . Example 10.7 (Modules over commutative algebras) . Let F ♭ ∗ , h i / denote the slice category F ∗ , h i / with the inert/active factorization system lifted from F ∗ (along the left fibration F ∗ , h i / → F ∗ ) withthe two maps h i → h i as elementary objects; then F ♭ ∗ , h i / is a cartesian pattern. We can alsothink of the objects as pairs ( h n i , i ) with i ∈ h n i , with a morphism ( h n i , i ) → ( h m i , j ) given by amorphism φ : h n i → h m i in F ∗ such that φ ( i ) = j . The cartesian pattern F ♭ ∗ , h i / is extendable: The ∞ -groupoid Act F ∗ , h i / ( h n i , i ) we can describe as the groupoid of pairs ( φ : h m i → h n i , j ∈ φ − ( i ))with φ active, and so in the commutative squareAct F ∗ , h i / ( h n i , i ) Q nj =1 Act F ∗ , h i / ( h i , ρ j ( i ))Act F ∗ ( h n i ) Q nj =1 Act F ∗ ( h i ) , ∼ the map on fibres over each active map φ ∈ Act F ∗ ( h n i ) is an isomorphism; this is therefore a pullbacksquare, so the top horizontal morphism is an equivalence, as required.We let CMod → F ∗ denote the (symmetric) ∞ -operad whose algebras are a commutative algebratogether with a module over it. This can be described as a category with • objects lists ( h n i , i , . . . , i n ) with i s ∈ { , } (with ( h i ,
0) representing the algebra and ( h i , • a morphism ( h n i , i , . . . , i n ) → ( h m i , j , . . . , j m ) is given by a morphism φ : h n i → h m i in F ∗ such that for all s = 1 , . . . , m , we have X t ∈ φ − ( s ) i t = j s . We can define a functor µ : F ∗ , h i → CMod given on objects by µ ( h n i , i ) = ( h n i , δ i , . . . , δ ni ) , where δ ji = 1 if j = i , and 0 otherwise. Given a morphism ( h n i , i ) → ( h m i , j ) over φ : h n i → h m i ,we assign to it the morphism µ ( h n i , i ) → µ ( h m i , j ) over φ , which indeed exists. It is clear that µ isa morphism of cartesian patterns, and identifies F ∗ , h i / with the full subcategory of CMod spannedby the objects with a most one copy of 1. We claim that µ is a Morita equivalence: F el ∗ , h i / andCMod el are both the 2-element set containing ( h i , i ) ( i = 0 , h i , i )in CMod are in the image of µ , so thatAct F ∗ , h i / ( h i , i ) ≃ Act
CMod ( h i , i ) . Since CMod is extendable by Example 9.3, we can apply Proposition 10.3 to get for any CMod-monoidal ∞ -category V ⊗ an equivalenceAlg F ∗ , h i / / CMod ( V ) ∼ −→ Alg
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