aa r X i v : . [ m a t h . C T ] A ug Enriched -categories I: enriched presheaves John D. BermanAugust 27, 2020
Abstract
This is the first of a series of papers on enriched -categories,seeking to reduce enriched higher category theory to the higher algebraof presentable -categories, which is better understood and can beapproached via universal properties.In this paper, we introduce enriched presheaves on an enriched -category. We prove analogues of most familiar properties of presheaves.For example, we compute limits and colimits of presheaves, prove thatall presheaves are colimits of representable presheaves, and prove aversion of the Yoneda lemma. Contents A -model for higher algebra . . . . . . . . . . . . . . . . 19 This work was supported by a National Science Foundation Postdoctoral Fellowshipunder grant 1803089. The A -model for enriched categories 34 There has been explosive progress in higher category theory in the lastdecade, largely made possible by Lurie’s books Higher Topos Theory [9] andHigher Algebra [10]. Although the foundations of the subject are combi-natorial and notoriously technical, a toolbox of techniques involving pre-sentable -categories allows us to reduce many problems to universal prop-erties which better agree with intuition from ordinary category theory.If C is an -category, the -category of presheaves PSh p C q “ Fun p C op , Top q is presentable, in the sense that it admits all limits and colimits, and by theYoneda lemma, C is the full subcategory of PSh p C q spanned by representablepresheaves.Moreover, PSh p C q is freely generated by representables under colimits.For example, there are equivalences of -categoriesFun p C , D q – Fun L ‹ p PSh p C q , PSh p D qq , where Fun L ‹ denotes colimit-preserving functors which send representables torepresentables. In other words, the -category Cat of -categories embedsas a full subcategory of Pr L ‹ (presentable -categories with some distin-guished objects) via the assignment C ÞÑ PSh p C q .In this way, any question about -categories can be reduced to a questionabout presentable -categories. There are many reasons one may wish tomake this reduction. For example, presentable -categories can often bymodeled by model categories.Another example comes from the higher algebra of presentable -categories(introduced in [10] Section 4.8.1; also see [5]). If C and D are presentable2 -categories, there is a presentable -category C b D satisfying the univer-sal property: A colimit-preserving functor C b D Ñ E classifies a functor C ˆ D Ñ E which preserves colimits in either variable independently.In this way, Pr L is a symmetric monoidal -category. To give some C P Pr L an algebra structure with respect to this external tensor product b is precisely to endow C with a closed monoidal operation. Hence, the higheralgebra of Pr L is a powerful tool for constructing monoidal operations; forexample, Lurie uses it to construct a symmetric monoidal smash product ofspectra, which was considered a difficult problem in homotopy theory, evenafter it was first solved by Elmendorf-Kriz-Mandell-May [4].Our goal in this series of papers is study enriched higher category theory byreducing to the theory of presentable -categories. Hopefully, this will allowenriched -categories to be used more effectively in subjects that rely onthem, such as secondary or iterated algebraic K-theory (see [8]) or algebraicK-theory of analytic rings (see [11]).Enriched higher category theory was developed by Gepner-Haugseng [6],and a form of the enriched Yoneda lemma has been proven by Hinich [7], butthe theory so far is highly technical and has been difficult to use. Nonethe-less, it is conjectured by experts that: Conjecture 1.1. If V is a presentable, closed monoidal -category, the -category Cat V of V -enriched categories embeds as a full subcategory ofRMod V p Pr L q ‹ , the -category of presentable right V -modules with some dis-tinguished objects. In other words, if C is a V -enriched category, there should be an -categoryPSh V p C q of V -enriched presheaves. Moreover, PSh V p C q should be not onlypresentable but also right tensored over V via a functor b : PSh V p C q ˆ V Ñ PSh V p C q , given roughly by the formula p F b A qp X q “ F p X q b A .The conjecture asserts that PSh V p C q contains all of the information ofthe enriched category C . This would completely reduce enriched highercategory theory to the higher algebra of presentable -categories, whichhas been well-studied over the past decade.This is a series of papers aimed towards proving the conjecture. Inthis first paper, we introduce and study the -categories PSh V p C q of en-riched presheaves. We think of such an enriched presheaf informally as acontravariant V -enriched functor C op Ñ V . We prove:3 heorem 1.2. If V is presentable and closed monoidal, and C is a V -enriched category, then PSh V p C q is a presentable -category. Theorem 1.3. If F P PSh V p C q and A P V , let F b A denote the presheafroughly given by p F b A qp X q “ F p X q b A . This b makes PSh V p C q a pre-sentable right V -module. In order to understand a presentable -category, we generally ask two ques-tions: How can we compute limits and colimits? What is a set of objectswhich generates the -category under colimits? We completely answer boththese questions: Corollary 1.4.
The limit and colimit of F : I Ñ PSh V p C q are given by p lim PSh V p C q F qp X q “ lim V p F X q and p colim PSh V p C q F qp X q “ colim V p F X q . Corollary 1.5.
As a presentable right V -module, PSh V p C q is generatedby the representable presheaves. That is, every presheaf is a colimit ofpresheaves of the form rep X b A , where X P C and A P V . It is illustrative to compare to the ordinary theory of -categories, whichare categories enriched in the -category Top of spaces. In this case, wehave PSh p C q “ Fun p C op , Top q . However, it is also known that if M is anypresentable -category, Fun p C op , M q – PSh p C q b M .We will prove an enriched version of this equivalence by constructing an -category PSh V p C ; M q of presheaves with values in M . This constructionmakes sense provided that M is a left V -module. Theorem 1.6. If M is a presentable left V -module, thenPSh V p C ; M q – PSh V p C q b V M . We have already announced that PSh V p C q is generated by representablepresheaves. In ordinary category theory, something much stronger is true:PSh p C q is freely generated by representable presheaves, in the sense thatthere are equivalences of -categories Fun L p PSh p C q , D q – Fun p C , D q .Our last main theorem will be a generalization of this statement to en-riched category theory. In order to make the statement, we introduce an-other variant on the enriched presheaf construction.That is, we introduce an -category coPSh V p C q of V -enriched copresheaves ,which are roughly covariant functors C Ñ V . Theorem 1.7. If V is presentable and closed monoidal, PSh V p C q is dualiz-able as a right V -module in Pr L , and its dual is coPSh V p C q . L RMod V p PSh V p C q , D q – coPSh V p C ; D q , provided D is a presentable right V -module. Remark 1.8. If C is a V -enriched category, there are two natural candidatesfor the -category of V -enriched functors C Ñ D . On one hand, V -enrichedcategories form an p8 , q -category Cat V , so therefore we have an -categoryFun V p C , D q of V -enriched functors whenever D is a V -enriched category.On the other hand, we will define an -category of V -enriched copresheavescoPSh V p C ; D q any time D is an -category right tensored over V (a right V -module).It is important to recognize that, although these two constructions conveyessentially the same information, one is defined for V -enriched categories D and the other for right V -module -categories D . If V is presentableand closed monoidal, Gepner-Haugseng have shown that presentable right V -modules D are canonically V -enriched ([6] 7.4.13), and in this case weconjecture the two constructions are equivalent. Our results rest on Gepner-Haugseng’s operadic model for enriched -categories [6]. If V is a monoidal -category, we expect a V -enriched cate-gory to consist of a set S of objects, along with: • objects C p X, Y q P V for each X, Y P S ; • composition morphisms C p X, Y q b C p Y, Z q Ñ C p X, Z q ; • identity morphisms 1 Ñ C p X, X q ; • and coherences relating these.This data is encoded in an -operad Assoc S (which depends on the set S ),so that a V -enriched category can be identified with an Assoc S -algebra in V .We define Assoc S in Definition 3.1 and Remark 3.2; note that the definitionis fairly concrete, as Assoc S is the nerve of a 1-category.A V -enriched presheaf on C should consist of the following in additionto the above data: • objects F p X q P V for each X P S ; • morphisms C p X, Y q b F p Y q Ñ F p X q ;5 and coherences relating these.This data is also encoded in an -operad, which we call LM S (Definition3.11). That is, an LM S -algebra in V should be regarded as a pair p C ; F q ,where C is a V -enriched category, and F is a V -enriched presheaf on C .We write Cat V S “ Alg
Assoc S p V q and PSh V S “ Alg LM S p V q . There is aninclusion Assoc S Ď LM S , inducing a forgetful functor θ : PSh V S Ñ Cat V S . If C is a V -enriched category, we may define PSh V p C q to be the fiber θ ´ p C q . Example 1.9.
We will always keep in mind the example | S | “ . In thiscase, Assoc S is the usual associative operad Assoc, and LM S is the leftmodule operad LM. In other words, an enriched category C with one object X can be identified with the endomorphism algebra C p X, X q . A presheaf onthis enriched category can be identified with a left module over C p X, X q .In Higher Algebra Chapter 4 [10], Lurie studies -categories of left mod-ules, which are -categories of presheaves on enriched categories with oneobject. This paper should be viewed as a generalization of that chapter tothe case | S | ą . Defining PSh V p C q as the fiber θ ´ p C q agrees with our intuition, but it is noteasy to prove anything about these fibers.In order to prove our main results, we will need a second model forenriched category theory, which is more explicit although less formally well-behaved. To do this, we construct -preoperads ∆ op { S and ∆ op { S ˆ ∆ , where∆ is the diagram category 0 Ñ
1, and ∆ { S is the category of finite,nonempty, totally ordered sets equipped with a function to S . That is,∆ { S “ ∆ ˆ Set
Set { S . Remark 1.10.
To describe ∆ op { S and ∆ op { S ˆ ∆ as -preoperads, we alsoneed to equip them with maps to the commutative operad. See Sections 4.1,respectively 4.2. Definition 1.11. An A - V -enriched category (with set S of objects) is a ∆ op { S -algebra in V . An A - V -enriched presheaf is a ∆ op { S ˆ ∆ -algebra in V . We will prove that this A -model is equivalent to the Gepner-Haugsengmodel (Corollaries 4.5 and 4.12) by utilizing Lurie’s theory of approxima-tions to -operads (which is reviewed in Section 2.3).Because ∆ op { S and ∆ op { S ˆ ∆ are -preoperads rather than -operads, the A -model is not as formally well-behaved as the operadic model. However,it has two benefits. 6he first benefit is that ∆ op { S is easier to construct than Assoc S . Thesecond, more significant, benefit is the very close relationship between ∆ op { S and ∆ op { S ˆ ∆ .That is, an A -enriched presheaf can be regarded as a map F Ñ F offunctors ∆ op { S Ñ ş V , where ş V is the -operad associated to V (see Section1.4 notation (4)). It turns out that F is just the underlying A -enrichedcategory. Therefore, an A -enriched presheaf on C can be regarded as asingle functor ∆ op { S Ñ ş V satisfying properties.This provides us with a new model for PSh V p C q , which is more technicalbut also more concrete than its definition as a fiber of θ : PSh V S Ñ Cat V S .This is Proposition 4.15.We will use this new model to conclude Theorem 1.2 and Corollary 1.4.We will also prove (Corollary 4.21) that PSh V p C q is functorial in C , in thesense of a functor PSh V p´q : Cat V S Ñ Pr L . Example 1.12.
To get a feeling for the A -model, suppose that V is carte-sian monoidal (monoidal under its product). Then V -enriched categories C (with set S of objects) can be identified with functors C A : ∆ op { S Ñ V , in thatthere is a full subcategory inclusion Cat V S Ď Fun p ∆ op { S , V q . For X, Y P S , let r X, Y s P ∆ op { S denote the function r s Ñ S which sends ÞÑ X and ÞÑ Y .Then C A r X, Y s is the mapping object C p X, Y q P V .When | S | “ , then the enriched category C carries the same informationas a V -algebra A “ C p X, X q , and C A is a simplicial object of V – thesimplicial object given by the Milnor construction ([10] 4.1.2.4): C A “ ¨ ¨ ¨ A ˆ / / / / / / A / / / / o o o o . o o Given such a C A : ∆ op { S Ñ V , a presheaf on C can be identified with a secondfunctor F A : ∆ op { S Ñ V along with a natural transformation F A Ñ C A .We think of this natural transformation as a functor ∆ op { S ˆ ∆ Ñ V .In the case | S | “ , such a presheaf is a left X -module M , and it iscompletely specified by a map of simplicial objects: F A “ (cid:15) (cid:15) ¨ ¨ ¨ X ˆ ˆ M (cid:15) (cid:15) / / / / / / X ˆ M (cid:15) (cid:15) / / / / o o o o M (cid:15) (cid:15) o o C A “ ¨ ¨ ¨ X ˆ / / / / / / X / / / / o o o o . o o he downward maps are just projection away from M ; all the informationabout the algebra and module structures is carried in the horizontal maps.In general, A -enriched categories and presheaves can be thought of asMilnor constructions like this, but indexed by the thickened simplex category ∆ op { S instead of ∆ op . We could represent C A pictorially as a simplicial objectof V where the n -simplices are given by an S ˆ ¨ ¨ ¨ ˆ S -hypercube of entries ¨ ¨ ¨ ˜ C p X ,X qb C p X ,X q ¨¨¨ C p X ,X qb C p X n ,X n q ... ... C p X n ,X n qb C p X ,X q ¨¨¨ C p X n ,X n qb C p X n ,X n q ¸ / / / / / / ˜ C p X ,X q ... C p X n ,X n q ¸ / / / / o o o o . o o Obviously such a diagram is cumbersome; it is included as a mnemonic, butit won’t appear in this paper again.
Our discussion so far concerns the presheaves on a V -enriched category C with values in V (roughly, V -enriched functors C op Ñ V ). There are twovariants on this construction.First, an enriched copresheaf on C is roughly a V -enriched functor C Ñ V .The theory of enriched copresheaves is completely parallel to the theory ofenriched presheaves; we introduce an -operad RM S , and a copresheaf isan RM S -algebra.Let V rev denote the -category V with its reverse monoidal operationdescribed by X b rev Y “ Y b X . Then C op is naturally a V rev -enrichedcategory (Example 5.3) andcoPSh V p C q – PSh V rev p C op q (Remark 5.7). Therefore, everything we prove about presheaves is also trueabout copresheaves. Example 1.13. If V is symmetric monoidal, then V rev – V as monoidal -categories, so coPSh V p C q – PSh V p C op q . This is not true in general. Second, a V -enriched copresheaf on C with values in M is roughly a V -enriched functor C op Ñ M . We write PSh V p C ; M q for the -categorythereof.It turns out that this construction makes sense whenever M is left ten-sored over V (a left V -module in Cat). In this case, we may regard the pair p V ; M q as an LM-algebra in Cat, or an LM-monoidal -category, and we de-fine a presheaf with values in M to be an LM S -algebra in this LM-monoidal -category. 8n Sections 5 and 6, we take advantage of these constructions to studythe interplay of presheaves (left actions) with copresheaves (right actions).The key is a pairing x´ , ´y : LM S ˆ RM T Ñ Assoc S > T introduced in Section 5.2. Careful analysis of this pairing allows us to: • construct the right V -action on PSh V p C q in Section 5.2 (Theorem 1.3)and prove (Theorem 1.6)PSh V p C ; M q – PSh V p C q b V M ; • construct the Yoneda embedding Y P coPSh V p C ; PSh V p C qq in Sec-tion 6 and prove that it induces a duality between presheaves andcopresheaves (Theorem 1.7). Remark 1.14.
Since a copresheaf is like an enriched functor, Y can beregarded informally as an enriched functor C Ñ PSh V p C q . In this sense, Y really is the Yoneda embedding.On the other hand, by Theorems 1.6 and 1.7 together,coPSh V p C ; PSh V p C qq – PSh V p C q b V coPSh V p C q . In this sense, we may regard Y as an element of PSh V p C q b V coPSh V p C q .It is precisely the element which exhibits the duality between PSh V p C q andcoPSh V p C q . We study the -category Cat V S whose • objects are V -enriched categories C with set S of objects; • morphisms are V -enriched functors F : C Ñ D which act as the identityon the set S of objects, F p X q “ X .This construction is functorial in the set S , Cat V ´ : Set op Ñ Cat, and it ispossible to construct an -category Cat V of all V -enriched categories bytaking a sort of oplax colimit over Cat V ´ . This is due to Gepner-Haugseng[6], although the idea is older (of first studying enriched categories with afixed set of objects and then extrapolating). See [2] for the formulation asan oplax colimit. 9n this paper, we make a systematic study of PSh V p C q for any fixed V -enriched category C , and we will describe the functoriality as C varies withinCat V S (see Section 4.3). However, we will not review the construction ofCat V , and we will not make any comparison of PSh V p C q and PSh V p D q when C and D have different sets of objects.The reader should expect such results in a future paper of this series.They do not appear here because they use substantially different techniques(centered around oplax colimits) than the results of this paper (which centeraround A -algebras, the pairing x´ , ´y : LM S ˆ RM T Ñ Assoc S > T , and theBarr-Beck Theorem). Nonetheless, it is instructive to keep some of theexpected results in mind. These include Conjecture 1.1 as well as:1. We expect a functor PSh V p´q : Cat V Ñ RMod V p Pr L q ; that is, forany V -enriched functor F : C Ñ D , we expect to have an adjunction F ˚ : PSh V p C q Ô PSh V p D q : F ˚ with F ˚ a V -module functor;2. If V is symmetric monoidal, we expect the functor PSh V p´q to besymmetric monoidal; that is, we expectPSh V p C b D q – PSh V p C q b V PSh V p D q ;3. If Fun V denotes the -category of V -enriched functors, we expectstatements of the formPSh V p C ; M q – Fun V p C op , M q , coPSh V p C ; M q – Fun V p C , M q . Remark 1.15.
Gepner-Haugseng construct Cat V in a slightly different, butequivalent way. That is, they construct an -operad Assoc S for each space(or -groupoid) S , interpreting an Assoc S -algebra as an enriched categorywith space S of objects. When S is discrete, this recovers our Assoc S .Then Cat V ´ is even functorial Top op Ñ Cat, and Gepner-Haugseng defineCat V to be the oplax colimit over Top op .However, regardless of whether we take the oplax colimit over Set op orTop op , we obtain equivalent -categories Cat V by [6] Theorem 5.3.17. To summarize, we might use the term ‘enriched category’ for an enrichedcategory with a set of objects, or ‘enriched -category’ for an enriched cate-gory with a space of objects. Then there is no difference between the theoriesof V -enriched categories and V -enriched -categories. Thanks to Aaron Mazel-Gee for pointing out this distinction. he reason we choose S to vary over sets, not spaces, is very simple: If S is a set, Assoc S is (the nerve of ) an ordinary category, which makes theentire theory pleasantly concrete. We begin in Section 2 with some review of marked -categories and -operads. (Our notation for -operads differs in some respects from Lurie’s,so this section should not be skipped.)In Section 3, we review enriched categories and introduce enriched presheaves.This section should be fairly accessible. We end this section by introducingthe representable presheaves rep X . We prove a key technical result aboutthem (Theorem 3.25), which is best understood through the following corol-lary: Corollary 3.27. If F P PSh V p C ; M q , Map p rep X b M, F q – Map p M, F p X qq . If we take M “ V , this is a kind of enriched Yoneda lemma, in the sense thatMap p rep X b´ , F q , which is a priori a presheaf on V , is just the representablepresheaf described by F p X q .The last three sections are more technical. Section 4 introduces the new A -model for enriched higher category theory, which we prove is equivalentto Gepner-Haugseng’s operadic model. We prove Theorem 1.2 and Corollary1.4.In Section 5, we construct the right V -module structure on PSh V p C q and prove Theorem 1.3, Corollary 1.5, and Theorem 1.6. The hard part isthe construction of the V -action; the equivalence is an application of theBarr-Beck Theorem.In Section 6, we construct the Yoneda embedding as a copresheaf Y incoPSh V p C ; PSh V p C qq , and we prove Theorem 1.7. By this point, we will havedone most of the work already, so this section is short.The casual reader who wishes to skim the paper without understandingthe proofs is recommended to read Sections 2.1-2.2, 3, 4.3, and the sectionintroductions. The more serious reader should follow the same prescriptionbefore returning to Section 2.3 and reading from there to fill in the gaps. The author thanks the National Science Foundation for his time as a post-doctoral fellow, as well as his postdoctoral mentor Andrew Blumberg, who11as taken an active interest in this project. He also thanks Rok Gregoricand Rune Haugseng for helpful conversations.Enriched categories with one object are just associative algebras, andpresheaves over them are just left modules. Chapter 4 of Lurie’s book HigherAlgebra contains all of our results already in the case of enriched categorieswith one object, and most of the proofs generalize directly. Therefore, theauthor also thanks Jacob Lurie for many proof techniques, and for doingmost of the hard work already.
Because of the heavy reliance on Higher Algebra, we consistently cite it asHA rather than [10], and cite Higher Topos Theory as HTT rather than [9].We will follow Lurie’s notation in most cases, with the following exceptions:1. We will write Cat (not Cat ) for the -category of -categories, sim-ilarly dropping notationally anywhere it won’t introduce confusion.2. If F : C Ñ Cat is a functor, we use ş F Ñ C for the associatedcocartesian fibration.3. We use notation like O rather than Lurie’s O b for -operads.4. We regard a monoidal -category as a functor C : Assoc Ñ Cat, whereAssoc is the associative operad. We write the associated -operad ş C rather than Lurie’s C b , because it is the associated cocartesianfibration ş C Ñ Assoc.In other respects, we follow Lurie’s notation. For example: • We distinguish between Cat, the large -category of small -categories,and y Cat, the very large -category of large -categories. • We denote by Pr L the subcategory of y Cat spanned by the presentable -categories, along with functors that preserve small colimits (equiv-alently, admit right adjoints).Pr L has a symmetric monoidal operation b , and an algebra in Pr L carries the same data as a presentable, closed monoidal -category.12 Preliminaries
In this section we review some preliminaries, beginning with marked -categories in Section 2.1. This material follows [2].The rest of the section is on -operads, following HA (Higher Algebra[10]). Section 2.2 reviews the foundations of -operads and -preoperads.Then Section 2.3 is on operadic approximation, and 2.4 is on the A -modelfor algebras and modules, an extended example of operadic approximation.These last two sections are used crucially in Section 4 to describe limitsand colimits in -categories of presheaves. However, the casual reader woulddo better to read Section 3 first, and only then return to 2.3 and 2.4. Definition 2.1. A marked -category is an -category C along with aspecified collection of morphisms, such that: • all equivalences are marked; • given two equivalent morphisms f – g , f is marked if and only if g is; • any composite of marked morphisms is marked.If C , D are marked -categories, a functor F : C Ñ D is marked if it sendsmarked morphisms to marked morphisms. There is an ( ,2)-category of marked -categories and marked functors,which we denote Cat : . Remark 2.2.
There are many ways to construct Cat : : • Let Cat : denote the 2-category of marked 1-categories, marked func-tors, and (all) natural equivalences. A marked -category can be spec-ified by a marking on the homotopy category, so Cat : “ Cat ˆ Cat Cat : .See [2] 2.2 or [10] 4.1.7.1. • Cat : is equivalent to the full subcategory of Fun p ∆ , Cat q spanned bythose functors C mk Ñ C which exhibit C mk as a subcategory of markedmorphisms. See [1] 1.14. • Cat : can be described in terms of a model structure on marked simpli-cial sets.
13e will often be interested in multiple markings on the same -category,so we will use notation like C : , C , C , . . . to denote markings on C .The following reference table provides examples of marked -categoriesand establishes notation that we will use throughout the paper. Notation Marked morphisms Comments C all morphisms the sharp marking C equivalences the flat marking C p -(co)cartesian morphisms for a given (co)cartesianfibration p : C Ñ DC § inert morphisms for -operads and similar;see Definition 2.7 C ; totally inert morphisms technical (Warning 2.20) C ! left inert morphisms technical (Definition 5.11) Example 2.3. If p : C Ñ D is a cocartesian (respectively cartesian) fibrationand D : is marked, there is an induced marking C : , in which f is marked ifand only if f is p -cocartesian (respectively p -cartesian) and p p f q is marked.If D has the flat marking, the induced marking on C is the flat marking C . If D has the sharp marking, the induced marking on C is the natu-ral marking C : the marked morphisms are precisely the p -cocartesian mor-phisms. If C : , D : P Cat : , we will write Fun : p C : , D : q for the full subcategory ofFun p C , D q spanned by marked functors .If C : P Cat : , there is a universal functor C Ñ | C : | which sends eachmarked morphism to an equivalence [2]. Roughly, | C : | is obtained from C by localizing (adjoining formal inverses) at all the marked morphisms. Example 2.4. If C is any -category, then | C | – C , and | C | is the geomet-ric realization , or the -groupoid built by formally inverting all morphisms. The -category Cat : admits all small limits and colimits ([2] 2.5). The limit(respectively colimit) of a diagram of marked -categories F : I Ñ Cat : isthe limit (colimit) of the underlying diagram of -categories, marked via: • A morphism φ of colim p F q is marked if there exists i P I such that F p i q Ñ colim p F q sends some marked morphism of F p i q to φ ; • A morphism φ of lim p F q is marked if for all i P I , lim p F q Ñ F p i q sends φ to a marked morphism of F p i q . In principle, Fun : p C : , D : q is itself naturally a marked category; marked morphisms arenatural transformations that send each object of C to a marked morphism of D . However,we will never use this marking. xample 2.5. If A : Ñ C : and B : Ñ C : are marked functors, then thepullback A ˆ C B inherits a natural marking: a morphism is marked if andonly if the projections to A and B are each marked. With this marking, A ˆ C B is the pullback in Cat : . If S is a set, we denote by S ` “ S >t˚u the pointed set obtained by adjoininga new basepoint. For each integer n , we also denote • x n y ˝ “ t , . . . , n u , an object of the category Fin of finite sets; • x n y “ x n y ˝` , an object of the category Fin ˚ of finite pointed sets; • r n s “ t ă ă ¨ ¨ ¨ ă n u , an object of the simplex category ∆ of finite,nonempty, totally ordered sets. Definition 2.6.
We say that a function of pointed sets f : S ` Ñ T ` is inert if | f ´ p t q| “ for all t P T . Notice there is no condition on f ´ p˚q .We denote by Comm § the category of finite pointed sets, marked by inertmorphisms . Definition 2.7 (HA 2.1.4.2) . An -preoperad is an -category equippedwith a functor to Comm.If p : O Ñ Comm is an -preoperad, we say a morphism of O is inert if it is p -cocartesian and its image in Comm is inert. We will always regard -preoperads as marked by their inert morphisms, writing O § P Cat : .A morphism of -preoperads is a functor over Comm which sends inertmorphisms to inert morphisms. That is, the -category POp is the fullsubcategory of Cat :{ Comm § spanned by those -categories over Comm whichare marked by their inert morphisms. Warning 2.8.
This definition is slightly stronger than Lurie’s. He definesan -preoperad to be any marked -category over Comm § , while we requirethat it carries the canonical inert marking. Before we continue, note that there are inert morphisms ρ i : x n y Ñ x y foreach 1 ď i ď n , defined by ρ i p j q “ i “ j and ˚ otherwise.If p : O Ñ Comm is an -preoperad, we write O n for the fiber over x n y ,which is a pullback O ˆ Comm tx n yu . We use § by analogy to , because the inert marking on an -operad is typicallyclosely related to the natural marking on a cocartesian fibration. efinition 2.9 (HA 2.1.1.10,14) . An -preoperad p : O Ñ Comm is an -operad if:1. For every X P O and inert morphism f : p p X q Ñ Y in Comm, there isan inert morphism (equivalently, a p -cocartesian morphism) X Ñ ¯ Y in O lifting f ; this implies that an inert morphism ρ : x n y Ñ x m y induces a functor ρ ˚ : O n Ñ O m .2. The functor O n Ñ O ˆ n induced by the inerts ρ i : x n y Ñ x y is anequivalence of -categories.3. For every X, Y P O and f : p p X q Ñ p p Y q , let Map f O p X, Y q be theunion of connected components of Map O p X, Y q lying over f . Choosean inert Y Ñ Y i lying over each inert ρ i : p p Y q Ñ x y . ThenMap f O p X, Y q Ñ ź i Map ρ i f O p X, Y i q is an equivalence.The -operads form a full subcategory Op Ď POp.
The terminal object of Op is Comm § itself, which we regard as the commu-tative -operad (hence the notation Comm).We also have the equally important associative -operad Assoc: Anobject of Assoc is a finite pointed set. A morphism is a basepoint-preservingfunction f : S ` Ñ T ` , equipped with the data of total orderings on f ´ p t q for each t P T . (Note there is no extra data on f ´ p˚q .) Definition 2.10 (HA 2.1.2.13) . If O is an -operad, an O -monoidal -category is a functor V : O Ñ Cat such that ş V § p ÝÑ O § Ñ Comm § exhibitsthe cocartesian fibration ş V § as an -operad.In this case, we call p a cocartesian fibration of -operads .A lax O -monoidal functor V Ñ V is a map of -operads ş V Ñ ş V over O , and an O -monoidal functor is a lax O -monoidal functor which alsosends p -cocartesian morphisms to p -cocartesian morphisms. Definition 2.11.
Suppose that O § is an -operad, V is an O -monoidal -category, and O § is an -preoperad over O § . An O -algebra in V is a mapof -preoperads over O § : O § ! ! ❇❇❇❇❇❇❇❇ / / ❴❴❴❴❴❴❴❴ ş V § } } ④④④④④④④ O § . here is an -category of algebras Alg O { O p V q “ Fun :{ O § p O § , ş V § q . We record a small lemma for later use:
Lemma 2.12.
For any -operad O , | O § | is contractible.Proof. The claim is equivalent to: The functor Fun : p O § , C q Ñ C , given byevaluation at the terminal object H , is an equivalence.Suppose F : O Ñ C is a functor which sends inert morphisms to equiv-alences. For any morphism f : X Ñ Y in O , F sends the inert morphisms Y Ñ H and X f ÝÑ Y Ñ H to equivalences, so F p f q is an equivalence.Therefore, Fun : p O § , C q “ Fun : p O , C q – Fun p| O | , C q . Since O has a termi-nal object, the geometric realization is contractible, and this completes theproof.We will also be crucially interested in the left module -operad LM (HA4.2.1). It has the defining property that an LM-algebra in a monoidal -category V is a pair p A, M q , where A is an algebra in V , and M is a left A -module. We will give a formal definition in Section 4, as a special case ofa more general construction LM S . Example 2.13.
An LM-monoidal -category may be regarded as a pair p V ; M q , where V is a monoidal -category and M is an -category lefttensored over V . Suppose p V ; M q is such an LM-monoidal -category. An LM-algebra in p V ; M q consists of a pair p A, M q , where A is an algebra in V , and M is aleft A -module in M . We writeLMod p V ; M q “ Alg LM { LM p V ; M q . There is a canonical forgetful functor θ : LMod p V ; M q Ñ Alg p V q , given by θ p A, M q “ A . Therefore, we may define: Definition 2.14. If A P Alg p V q , the -category of left A -modules in M isLMod A p M q “ LMod p V ; M q ˆ Alg p V q t A u . Our goal is to generalize many of the nice properties satisfied by LMod A p M q to -categories of presheaves. However, there is a major obstacle: Thedefinition of LMod A p M q is fairly unnatural. For example, if V and M are both presentable satisfying mild conditions, then LMod A p M q is alsopresentable, but this is not at all clear from the definition.17urie solves this problem in [10] Chapter 4 by giving an alternative A -model for algebras and left modules, which is equivalent to the operadicmodel discussed above. Then he identifies LMod A p M q with an -categoryof marked functors.The A -model relies heavily on the theory of operadic approximations,which we review first. Suppose O is an -operad. We will often want to construct a simpler -preoperad I which has the same algebras as O ; that is, such that there is afunctor f : I Ñ O which induces an equivalence Alg O p V q Ñ Alg I p V q for eachmonoidal -category V . We do this using Lurie’s theory of approximationsto -operads (HA 2.3.3). Definition 2.15.
A morphism f : S ` Ñ T ` in Comm is called active if f ´ p˚q “ t˚u (HA 2.1.2.1-3). If p : O Ñ Comm is an -operad, a morphismin O is called active if its image in Comm is active. Definition 2.16 (HA 2.3.3.6) . Suppose f : I Ñ O is a map of -preoperads,and O is an -operad. We say f is an approximation to O if:1. For every X P I and inert morphism φ : p p X q Ñ x y in Comm, thereis an inert morphism X Ñ Y in I lifting φ .2. For every Y P I and active morphism φ : X Ñ f p Y q in O , there is an f -cartesian morphism ¯ X Ñ Y lifting φ .We say f is a strong approximation to O if it is an approximation to O and f : I Ñ O is an equivalence between the fibers over x y P Comm.
Theorem 2.17 (HA 2.3.3.23) . Let O be an -operad, V a monoidal -category, and f : I Ñ O a strong approximation to O . Then the map f ˚ : Alg O p V q Ñ Alg I p V q induced by composition with f is an equivalence of -categories. We are motivated by two examples of operadic approximations.
Example 2.18 (HA 4.1.2.11) . There is a strong approximation to Assoc,Cut : ∆ op Ñ Assoc. A morphism f ˚ : r n s Ñ r m s in ∆ op is inert if and onlyif the associated morphism f ˚ : r m s Ñ r n s in ∆ embeds r m s as a convexsubset t i ă i ` ă ¨ ¨ ¨ ă i ` m u Ď r n s . xample 2.19 (HA 4.2.2.8) . There is a strong approximation to LM,LCut : ∆ op ˆ ∆ Ñ LM, which fits into a commutative square ∆ op ˆ t u Cut / / (cid:15) (cid:15) Assoc (cid:15) (cid:15) ∆ op ˆ ∆ LCut / / LM . A morphism pr n s f ˚ ÝÑ r m s , i Ñ j q in ∆ op ˆ ∆ is inert if and only if f ˚ isinert in ∆ op and either: • f ˚ p m q “ n ; • or j “ . We will delay explicit constructions of Cut and LCut until Section 5, asspecial cases of more general constructions Cut S and LCut S . Warning 2.20.
From the inert marking on ∆ op ˆ ∆ , notice that the twoembeddings ∆ op Ď ∆ op ˆ ∆ induce two different markings on ∆ op : • The embedding t u Ď ∆ induces the inert marking ∆ op § ; • The embedding t u Ď ∆ induces the totally inert marking ∆ op ; , wherea morphism f ˚ : r n s Ñ r m s of ∆ op is totally inert if it is inert and f ˚ p m q “ n . A -model for higher algebra We will end this section by reviewing the A -model for algebras and leftmodules.For a monoidal -category V , the composite ∆ op Cut ÝÝÑ
Assoc V ÝÑ Cat isthe simplicial -category BV given by the Milnor construction (HA 4.1.2.4): ¨ ¨ ¨ V ˆ / / / / / / V / / / / o o o o ˚ . o o Definition 2.21. An A -algebra of V is a section of ş BV p ÝÑ ∆ op whichsends inert morphisms to p -cocartesian morphisms, and we write A Alg p V q “ Fun :{ ∆ op § p ∆ op § , ş BV § q . By Example 2.18 and Theorem 2.17, composition with Cut : ∆ op Ñ Associnduces an equivalence Alg p V q Ñ A Alg p V q .19 emark 2.22. The preceding material follows HA 4.1.3, where Lurie refersto ş BV Ñ ∆ op as the planar -operad (which he writes V f Ñ ∆ op ) asso-ciated to the -operad ş V Ñ Assoc (which he writes V b Ñ Assoc).
Now suppose we also have a left V -module -category M ; that is, thepair p V ; M q is an LM-monoidal -category as in Example 2.13. Denote by B p V ; M q the composite ∆ op ˆ ∆ ÝÝÝÑ LM p V ; M q ÝÝÝÝÑ
Cat. For concreteness, B p V ; M q may be regarded as the morphism of simplicial -categories: BV ˙ M “ q (cid:15) (cid:15) ¨ ¨ ¨ V ˆ ˆ M (cid:15) (cid:15) / / / / / / V ˆ M (cid:15) (cid:15) / / / / o o o o M (cid:15) (cid:15) o o BV “ ¨ ¨ ¨ V ˆ / / / / / / V / / / / o o o o ˚ . o o The downward maps are simply projection away from M ; the module struc-ture is recorded by the horizontal maps in the top row. The bottom simpli-cial -category is just BV (which is in particular independent of M ) becausethe square of Example 2.19 commutes. Definition 2.23. A left A -module is a section of ş B p V ; M q p ÝÑ ∆ op ˆ ∆ which sends inert morphisms to p -cocartesian morphisms, and we write A LMod p V ; M q “ Fun :{ ∆ op ˆ ∆ § p ∆ op ˆ ∆ § , ş B p V ; M q § q . By Example 2.19, composition with LCut : ∆ op ˆ ∆ Ñ LM induces anequivalence LMod p V ; M q Ñ A LMod p V ; M q .If A P A Alg p V q , then we may define A LMod A p M q just as in we didin the operadic model: Definition 2.24. If p V ; M q is an LM-monoidal -category and A P A Alg p V q ,then A LMod A p M q “ A LMod p V ; M q ˆ A Alg p V q t A u . The benefit of the A -model is that we can also give a much more explicitmodel for A LMod A p M q than this last definition. In order to do so, weneed a bit more notation.Applying the Grothendieck construction to the map of simplicial -categories ş BV ˙ M Ñ ş BV , we have a functor q over ∆ op , which sends p -cocartesian morphisms to p -cocartesian morphisms: ş BV ˙ M q / / p % % ❑❑❑❑❑❑❑❑❑ ş BV p | | ①①①①①①①① ∆ op emark 2.25. This discussion follows HA 4.2.2, where Lurie uses the no-tation M f “ ş BV ˙ M . Therefore, he writes M f Ñ V f where we write ş BV ˙ M Ñ ş BV . Remark 2.26.
Our discussion so far is model-independent, but everythingis fairly concrete if we work with quasicategories. Suppose we are given acocartesian fibration of quasicategories ş p V ; M q Ñ LM which realizes M asa left V -module. Pullback along the two inclusions ∆ op Ď ∆ op ˆ ∆ LCut
ÝÝÝÑ
LMyields quasicategory models for ş BV ˙ M and ş BV .In this case, the functor q : ş BV ˙ M Ñ ş BV is a categorical fibration(HA 4.2.2.19) and a locally cocartesian fibration (HA 4.2.2.20). Remark 2.27.
We will be interested in the following markings (with nota-tion as in the triangle above): • The marking ş BV § by inert morphisms, or morphisms which are p -cocartesian and lie over inert morphisms in ∆ op ; • The marking ş BV ˙ M ! by locally q -cocartesian morphisms (we usethe notation ! to emphasize this is not the expected marking by p -cocartesian morphisms); • The marking ş BV ˙ M ; by totally inert morphisms, or morphismswhich are p -cocartesian and lie over totally inert morphisms in ∆ op (see Warning 2.20). We end with three important propositions describing left module -categories. Proposition 2.28 (HA 4.2.2.19) . If V is a monoidal -category, M is aleft V -module -category, and A is an A -algebra of V (that is, a markedsection of ş BV § Ñ ∆ op § ), then A LMod A p M q – Fun :{ ş BV § p ∆ op ; , ş BV ˙ M ; q . Proof.
See Proposition 4.15 for the proof of a more general result (or justsee HA 4.2.2.19 for this one).
Proposition 2.29 (HA 4.8.4.12) . If V is a monoidal -category, M , N are left V -module -categories, and Fun LMod V p M , N q is the -category of V -linear functors (HA 4.6.2.7), thenFun LMod V p M , N q – Fun :{ ş BV p ş BV ˙ M ! , ş BV ˙ N ! q . orollary 2.30. A LMod A p Cat q is equivalent as an p8 , q -category to thefull subcategory of Cat :{ ş BV spanned by ş BV ˙ M ! as M varies over left V -modules. Actually, we haven’t defined LMod A p Cat q as an p8 , q -category, but Propo-sition 2.29 implies that there is an equivalence of -categories, and thatit would promote to an equivalence of p8 , q -categories given any sensibledefinition of LMod A p Cat q as an p8 , q -category. On the other hand, wemight simply take Corollary 2.30 as our definition of LMod A p Cat q as an p8 , q -category.For the next proposition, note that we can define right modules in parallelwith left modules, using an -operad RM in place of LM. We still have astrong approximation ∆ op ˆ ∆ Ñ RM, and an RM-monoidal -categoryis a pair p V ; N q exhibiting N as a right V -module -category. In this case,the associated functor B p V ; N q : ∆ op ˆ ∆ Ñ Cat may be identified with amorphism of simplicial -categories N ¸ BV “ q (cid:15) (cid:15) ¨ ¨ ¨ N ˆ V ˆ (cid:15) (cid:15) / / / / / / N ˆ V (cid:15) (cid:15) / / / / o o o o N (cid:15) (cid:15) o o BV “ ¨ ¨ ¨ V ˆ / / / / / / V / / / / o o o o ˚ . o o Proposition 2.31 (HA 4.8.4.3) . If V is a monoidal -category, N , M areright (respectively left) V -module -categories, and N b V M is the tensorproduct of V -modules (HA 4.4), let r be the canonical cocartesian fibration p ş N ¸ BV q ˆ ş BV p ş BV ˙ M q Ñ ∆ op . Then N b V M – |p ş N ¸ BV q ˆ ş BV p ş BV ˙ M q | , where the marking is by r -cocartesian edges. Remark 2.32.
According to Proposition 2.31, a functor F : N b V M Ñ Z is classified by a functor ¯ F : p ş N ¸ BV ˆ ş BV p ş BV ˙ M q Ñ Z . HA 4.8.4.3actually asserts more.Note that the fiber of p ş N ¸ BV ˆ ş BV p ş BV ˙ M q Ñ ∆ op over r n s is N ˆ V ˆ n ˆ M . If the restriction of ¯ F to N ˆ V ˆ n ˆ M Ñ Z preservessmall colimits for each n , then F specializes to a colimit-preserving functor N b L V M Ñ Z , where b L V is a relative tensor product taken in Pr L . The operadic model for enriched categories
Fix a set S . Gepner-Haugseng [6] construct an -operad Assoc S (theycall it O S ) with the universal property: If V is a monoidal -category, anAssoc S -algebra in V is a V -enriched category with set S of objects. We writeCat V S “ Alg
Assoc S { Assoc p V q . A V -enriched presheaf on C P Cat V S is then something like a contravariant, V -enriched functor from C to V . However, this is not a suitable definitionat this point, because of two obstacles: • V does not belong to the same -category Cat V S , as it doesn’t have set S of objects; • V is not itself V -enriched unless it is closed monoidal; even then, it isnot obvious how to construct the self-enrichment on V .The first obstacle is not serious; Gepner-Haugseng construct an -categoryCat V of all V -enriched categories. However, we will delay this constructionuntil a future paper. In any case, it is not easy to understand enrichedfunctor -categories in Cat V , so we prefer to avoid this approach.The second obstacle is more serious. While it is possible to construct theself-enrichment on V (see [6] Section 7), the construction is rather obscureand not easy to use.Instead, we will construct an -operad LM S with a natural embeddingAssoc S Ď LM S . If C is a V -enriched category, which is an Assoc S -algebrain V , then an enriched presheaf on C is a lift to an LM S -algebra. In otherwords, if we write PSh V S “ Alg LM S { Assoc p V q , we can regard an object of PSh V S as a pair p C , F q , where C P Cat V S and F isan enriched presheaf on C . The inclusion Assoc S Ď LM S induces a forgetfulfunctor θ : PSh V S Ñ Cat V S , and we definePSh V p C q “ θ ´ p C q , the -category of enriched presheaves on C .In fact, we always work in a more general situation. If M is an -category left tensored over V , the pair p V , M q is itself an LM-algebra in Cat,or an LM-monoidal -category. We will define PSh V ; M S “ Alg LM S { LM p V ; M q ,which we interpret as the -category of pairs p C , F q , where C is a V -enriched23ategory, and F is an enriched presheaf with values in M . This should bethought of informally as an enriched, contravariant functor from C to M .In Section 3.1, we review the construction of enriched categories, due toGepner-Haugseng [6]. Then in Section 3.2, we introduce enriched presheaves.Finally, in Section 3.3, we introduce two closely related constructions:If X P C , a presheaf on C can be evaluated at X , and evaluation at X isfunctorial ev X : PSh V p C ; M q Ñ M . Also for each X , there is a representablepresheaf rep X P PSh V p C q given informally by rep X p Y q “ C p Y, X q . In ourmain result of this section (Theorem 3.25), we prove that rep X is a freepresheaf. In other words: Corollary 3.29.
The functor ev X : PSh V p C ; M q Ñ M has a left adjointdescribed by rep X b ´ : M Ñ PSh V p C ; M q . The presheaf rep X b M is giveninformally by p rep X b M qp Y q “ C p Y, X q b M . This can be regarded as a form of the enriched Yoneda lemma:
Corollary 3.27. If F P PSh V p C q and A P V , thenMap p rep X b A, F q – Map p A, F p X qq . Definition 3.1.
Fix a set S . An object of Assoc S is a finite pointed set E ` along with two functions s, t : E Ñ S . A morphism is a basepoint-preservingfunction f : E ` Ñ E along with a total ordering of f ´ p e q for each e P E ,such that:1. If f ´ p e q is empty, then s p e q “ t p e q ;2. If f ´ p e q “ t e ă e ă . . . ă e n u nonempty, then s p e q “ s p e q , t p e n q “ t p e q , and t p e i q “ s p e i ` q for each ď i ă n .We say f is inert if | f ´ p e q| “ for all e P E , which makes Assoc § S marked. Remark 3.2.
We make sense of the definition as so: We regard an objectof Assoc S as a directed graph Γ with set S of vertices and set E of edges.Each edge e P E has source vertex s p e q and target vertex t p e q .A morphism f : Γ Ñ Γ is a way of transforming Γ into Γ by means ofthe following three operations:1. deleting some edges – these are the edges for which f p e q “ ˚ ; . adding some loops (edges from a vertex to itself ) – this corresponds tocondition (1) above;3. deleting edges e , . . . , e n which form a path from s “ s p e q to t “ t p e n q ,and replacing them by a single edge from s to t – this corresponds tocondition (2) above.This description should also make clear how to compose morphisms.An inert morphism is one that involves only the operation (1), and anactive morphism is one that involves only the operations (2)-(3). Remark 3.3.
In [3], we call the active morphisms in Assoc S bypass oper-ations , and we study the symmetric monoidal envelope of Assoc S , which wecall Bypass S . Remark 3.4.
There is an evident forgetful functor Assoc S Ñ Assoc. Itsends inert morphisms to inert morphisms, and it is an isomorphism when | S | “ (in this case the functions s, t : E Ñ S carry no information, andthe two conditions of Definition 3.1 are satisfied for free). Suppose there are two graphs Γ , Γ P Assoc S . Then there is a new graphΓ b Γ given by disjoint union of the sets of edges in Γ and Γ . This b endowsAssoc S with a monoidal structure.The unit of b is H , the graph with no edges. Also, given X, Y P S , let p X, Y q P
Assoc S denote the graph with a single edge from X to Y .Every object of Assoc S can be written uniquely (up to permutation) inthe form p X , Y q b ¨ ¨ ¨ b p X n , Y n q . There are also morphisms correspondingto the three operations of Remark 3.2:1. p X, Y q Ñ H for each
X, Y P S ;2. H Ñ p
X, X q for each X P S ;3. p X, Y q b p
Y, Z q Ñ p
X, Z q for each X, Y, Z P S .Morphisms of type (1) are inert. Morphisms of type (2) and (3) will describeidentity morphisms and composition in enriched categories. Proposition 3.5.
The composite p : Assoc § S Ñ Assoc § Ñ Comm § is an -operad, and Assoc § S Ñ Assoc § is a map of -operads.Proof. First we check that the inert morphisms in Assoc S are p -cocartesian,so that the inert marking we have described is the correct inert marking foran -operad. Let f : Γ Ñ Γ be the inert morphism in question, so that25 ´ induces an inclusion of graphs Γ Ď Γ (in the sense of an inclusion ofedge sets, not a morphism in Assoc S ). Assume we have a map h : Γ Ñ Γ ,and a commutative triangle p p Γ q f (cid:15) (cid:15) h / / p p Γ q .p p Γ q u ; ; ✈✈✈✈✈ Then the restriction of h along Γ Ď Γ describes a map ¯ u : Γ Ñ Γ (theonly possible lift of u to Assoc S ), precisely because the triangle commutes.This is what it means for f to be p -cocartesian. Hence Assoc § S Ñ Comm § is an -preoperad.Now we need to check conditions (1)-(3) of Definition 2.9.(1) Given Γ P Assoc S and an inert morphism f : p p Γ q Ñ T ` , f ´ exhibits T as a subset of the set of edges of Γ. Let Γ be the subgraph spanned bythose edges. Then f lifts to an inert morphism ¯ f : Γ Ñ Γ .(2) If O “ Assoc S , each O n is just the (discrete) set p S q n , where anelement p X, Y q P S is interpreted as an edge from X to Y . It follows that O n Ñ O ˆ n is an equivalence.(3) A morphism Γ Ñ Γ in Assoc S is determined only by the underlyingmorphism f : E ` Ñ E and data and conditions on each fiber f ´ p e q .Therefore Assoc § S is an -operad. The forgetful functor Assoc S Ñ Associs compatible with the forgetful functors to Comm. It sends inerts to inerts,because in each case a morphism is inert if and only if it lies over an inertmorphism in Comm, so it is a map of -operads. Definition 3.6. If V is a monoidal -category and S a set, a V -enrichedcategory with set S of objects is an Assoc S -algebra in V , and we writeCat V S “ Alg
Assoc S { Assoc p V q . Example 3.7.
Suppose | S | “ . Then Assoc S – Assoc as in Remark3.4. Therefore, V -enriched categories with one object can be identified withassociative algebras in V . Unpacking the definition, a V -enriched category with set S of objects is amap of -operads C : Assoc § S Ñ ş V § . Such a functor C comes with thefollowing data, plus coherences: • ‘hom’ objects C p X, Y q P ş V “ V ;26 ‘identity’ morphisms 1 Ñ C p X, X q ; • ‘composition’ morphisms C p X, Y q b C p Y, Z q Ñ C p X, Z q .This is exactly the classical structure of an enriched category.A morphism C Ñ D in Cat V S consists of maps C p X, Y q Ñ D p X, Y q . Inother words, it is an enriched functor F : C Ñ D which acts as the identityon objects: F p X q “ X for all X P S . Example 3.8.
Let denote the contractible -category ˚ with its uniquemonoidal structure. The corresponding functor Assoc Ñ Cat is the constantfunctor with value ˚ , so the associated cocartesian fibration is the identityAssoc Ñ Assoc. Therefore, Cat S “ Fun :{ Assoc p Assoc S , Assoc q – ˚ is con-tractible for each S .In summary, there is a unique (up to equivalence) -enriched category S with set S of objects. We may regard it as having S p X, Y q “ for each X, Y P S , where P V is the monoidal unit. Example 3.9.
For any monoidal -category V , there is a unique monoidalfunctor Ñ V . By pushing forward the Assoc S -algebra S of the last exam-ple, we obtain a V -enriched category S . It has the property S p X, Y q “ forall X, Y P S . This is the trivial V -enriched category with set S of objects . Remark 3.10. If V is presentable and closed monoidal, then Cat V S “ Alg
Assoc S p V q is also presentable by HA 3.2.3.5. Definition 3.11.
Suppose S ` is a pointed set and Γ P Assoc S ` . Call Γleft-modular if s p e q ‰ ˚ for all edges e in Γ .If S is a set, LM S is the full subcategory of Assoc S ` spanned by leftmodular graphs.A morphism of LM S is inert if it is inert in Assoc S ` . In other words, we may think of the objects of LM S as graphs on vertex-set S ` , such that no edges have source ˚ .Hence there are two kinds of edges of LM S : those of the form p X, Y q ,and those of the form p Z, ˚q . The edges that do not involve ˚ span the fullsubcategory Assoc S , so that there are inclusions of marked categoriesAssoc § S Ď LM § S Ď Assoc § S ` , and LM S inherits from Assoc S ` the monoidal operation b .27 xample 3.12. If | S | “ , then LM S – LM, the left module operad.
For any function S Ñ T , the induced functor Assoc S ` Ñ Assoc T ` restrictsto LM S Ñ LM T . In particular, there are canonical functors LM S Ñ LM foreach S which forget the labelings of the vertices of a graph (except for thedistinguished vertex ˚ ). Proposition 3.13.
The composite LM § S Ñ LM § Ñ Comm § is an -operad. The proof is exactly like Proposition 3.5.Notice that the -operads Assoc S and LM S have the property: A mor-phism is inert if and only if it lies over an inert morphism in Comm. There-fore, all of the functors in the following diagram are -operad maps (where S Ñ T is a function):Assoc S (cid:31) (cid:127) / / (cid:15) (cid:15) LM S (cid:31) (cid:127) / / (cid:15) (cid:15) Assoc S ` (cid:15) (cid:15) Assoc T (cid:31) (cid:127) / / LM T (cid:31) (cid:127) / / Assoc T ` . Definition 3.14.
Suppose p V ; M q is an LM-monoidal -category; that is, V is a monoidal -category and M is a left V -module -category. A V -enriched presheaf with underlying set S and values in M is an LM S -algebrain p V ; M q , and we writePSh V ; M S “ Alg LM S { LM p V ; M q . In order to compare to Cat V , we will need a small lemma: Lemma 3.15.
With notation as before, Cat V – Alg
Assoc S { LM p V ; M q . We will henceforth abuse notation by defining Cat V “ Alg
Assoc S { LM p V ; M q any time we are working with a pair p V ; M q . Proof.
There is a pullback ş V / / (cid:15) (cid:15) ş p V ; M q (cid:15) (cid:15) Assoc (cid:31) (cid:127) / / LM . which is compatible with inert morphisms, in the sense that a morphismin ş V is inert if and only if its images in ş p V ; M q and Assoc are inert.Therefore, an -operad map Assoc S Ñ ş V over Assoc records the samedata as an -operad map Assoc S Ñ ş p V ; M q over LM.28ow let’s unpack the definition. By the lemma, the inclusion Assoc S Ď LM S induces a forgetful functor θ : PSh V ; M S Ñ Cat V S . In total, the data of an enriched presheaf consists of the restriction to Assoc S ,which is an enriched category C , with the following data and coherences: • objects F p X, ˚q P M for each X P S ; • morphisms C p X, Y q b F p Y, ˚q Ñ F p X, ˚q for each X, Y P S .We generally write just F p X q instead of F p X, ˚q . We read (2) as a methodfor turning hypothetical ‘morphisms’ X Ñ Y in C into maps F p Y q Ñ F p X q in M . This data is something like a ‘functor’ C op Ñ M , and it is for thisreason we call an LM S -algebra a presheaf . Example 3.16 (Presheaves with values in V ) . Suppose V is a monoidal -category. LM is an -operad over Assoc via LM Ď Assoc x y Ñ Assoc.The composite LM Ñ Assoc V ÝÑ Cat describes an LM-monoidal -category p V ; V q ; that is, V acts on itself by tensoring on the left. We writePSh V S “ PSh V ; V S , understanding by convention that we are taking presheaves with values in V .By construction, ş p V ; V q – ş V ˆ Assoc
LM, and this pullback is compatiblewith inert morphisms. Just as in the proof of the last lemma, we concludePSh V S – Alg LM S { Assoc p V q . We have constructed the -category PSh V ; M S of pairs p C ; F q where C is anenriched category and F is a presheaf. However, we will typically be moreinterested in fixing C and studying the -category PSh V p C ; M q of presheaveson C : Definition 3.17. If C is a V -enriched category with set S of objects, thenthe -category of enriched presheaves on C with values in M isPSh V p C ; M q “ PSh V ; M S ˆ Cat V S t C u . When M “ V with its canonical left action on itself, we will also writePSh V p C q “ PSh V p C ; V q . V p C ; M q depends on the set S although it is not specifiednotationally. (In fact, S is part of the data of C , so in this sense it is implicitin the notation.) Example 3.18.
If the enriched category C has a single object, let A be theassociated algebra (the endomorphism algebra of the single object). In thiscase, PSh V p C ; M q “ LMod A p M q by definition (HA 4.2.1.13) Example 3.19.
We will prove later: • (Proposition 3.30) If S is the trivial V -enriched category with set S of objects (Example 3.9) then PSh V p S ; M q – M ; • (Corollary 3.31) If is the trivial monoidal -category, PSh M S – M . Remark 3.20.
All of our definitions so far are model-independent, but be-fore we move on, we should say a word about the quasicategory model. Sup-pose we are given a cocartesian fibration ş p V ; M q Ñ LM of quasicategoriesexhibiting M as a left V -module -category. Then the constructionsCat V S “ Fun :{ LM § p Assoc § S , ş p V ; M q § q PSh V ; M S “ Fun :{ LM § p LM § S , ş p V ; M q § q model Cat V S and PSh V ; M S concretely as quasicategories.Since the inclusion Assoc S Ñ LM S is a categorical fibration, the forgetfulfunctor θ : PSh V ; M S Ñ Cat V S is also a categorical fibration, so PSh V p C ; M q may be modeled as the literal fiber θ ´ p C q , or the quasicategory of liftsAssoc § S (cid:15) (cid:15) C / / ş V § (cid:15) (cid:15) LM § S / / ❴❴❴ ş p V ; M q § . Suppose F is a V -enriched presheaf with values in M . That is, F is a mapof -operads F : LM S Ñ ş p V ; M q for some set S . If X P S , then F p X, ˚q is an object of ş p V ; M q lying over p , ˚q P LM t u . Since the fiber over p , ˚q is just M , F p X, ˚q is an object of M .30 efinition 3.21. As above, evaluation at p X, ˚q P LM S inducesev X : PSh V ; M S Ñ M . We call this functor evaluation at X and also write F p X q “ ev X p F q .We may also restrict to the fibers PSh V p C ; M q Ď PSh V ; M S , yielding func-tors ev X : PSh V p C ; M q Ñ M for each enriched category C . We will now construct the representable presheaves. For any X P S , there isa function π X : S ` Ñ S given by π X p˚q “ X and π X p Y q “ Y . This inducesan operad map π X ˚ : LM S Ď Assoc S ` Ñ Assoc S , and composition with π X ˚ induces rep X : Cat V S Ñ PSh V S .The inclusion Assoc S Ñ LM S is a section of π X ˚ , so rep X p C q is a presheafon C . Definition 3.22. If C is a V -enriched category with set S of objects and X P S , then rep X p C q is the representable presheaf at X . When C is clearfrom context, we will just write rep X . Notice that rep X p Y q “ C p Y, X q by construction. The representable presheavesare free , in the following sense. Definition 3.23.
Consider p C , F q P PSh V ; M S , so that C is an enriched cate-gory and F a presheaf on C . If X P S and M P M , we say that a morphism λ : M Ñ F p X q exhibits F as freely generated by M at X if for all Y P S ,the map C p Y, X q b M Ñ C p Y, X q b F p X q Ñ F p Y q is an equivalence. Example 3.24.
Let be the unit of the monoidal structure on V . Thenrep X P PSh V p C ; V q is freely generated by at X . Theorem 3.25.
For any C P Cat V S , X P C , and M P M , there exists apresheaf F P PSh V p C ; M q which is freely generated by M at X . Moreover,for any p D , G q P PSh V ; M S , composition with λ induces an equivalenceMap PSh V ; M S pp C , F q , p D , G qq Ñ Map
Cat V S p C , D q ˆ Map M p M, G p X qq . This is the main result of this section, but it is not the theorem itself thatinterests us so much as its corollaries. We record those corollaries beforeproving the theorem.
Corollary 3.26. If C P Cat V S , X P S , and M P M , then there existsa presheaf in PSh V p C ; M q which is freely generated by M at X , and it isessentially unique in the following sense:If F , F are two such presheaves, there is an equivalence F Ñ F which is compatible with the maps λ i : M Ñ F i p X q and is unique up tohomotopy.
31e refer to the presheaf freely generated by M at X as rep X b M , notationjustified by the formula p rep X b M qp Y q – C p Y, X q b M – rep X p Y q b M . Corollary 3.27 (Enriched Yoneda lemma, weak form) . If p C , F q P PSh V ; M S , X P S , and M P M , then λ : M Ñ p rep X b M qp X q induces an equivalenceMap PSh V p C ; M q p rep X b M, F q Ñ Map M p M, F p X qq . Example 3.28. If M “ V and M “ is the monoidal unit, then the lastcorollary asserts Map p rep X , F q – Map p , F p X qq . Corollary 3.29.
The functor ev X : PSh V p C ; M q Ñ M has a left adjointdescribed by rep X b ´ : M Ñ PSh V p C ; M q . We can use this last corollary to prove the claims of Example 3.19:
Proposition 3.30.
Let S be the trivial V -enriched category of Example 3.9.For any X P S , the functor ev X : PSh V p˚ X ; M q Ñ M is an equivalence.Proof. Suppose that F is a presheaf on ˚ X . For each X, Y P S , we havestructure maps c X,Y : F p Y q “ S p X, Y q b F p Y q Ñ F p X q . We claim c X,Y is an equivalence.Since F p X q id ÝÑ S p X, X q b F p X q c X,X
ÝÝÝÑ F p X q is the identity, c X,X is theidentity on F p X q . Consider the commutative square1 S p X, Y q b S p Y, X q b F p X q c Y,X / / S p X, Y q b F p Y q c X,Y (cid:15) (cid:15) S p X, X q b F p X q c X,X / / F p X q , it follows that c X,Y and c Y,X are inverse to each other, hence are equiva-lences.In other words, the map 1 S p Y, X q b F p X q Ñ F p Y q is an equivalence forall Y , which means that the identity F p X q Ñ F p X q exhibits F as freelygenerated by F p X q at X , so F – rep X b F p X q . In other words, the unitof the adjunction ev X : PSh V p˚ X ; M q Ô M : rep X b ´ is an equivalence.On the other hand, the counit p rep X b M qp X q Ñ M is also an equivalencebecause rep X p X q – X and rep X b ´ is a pair of inverse functors, and ev X is anequivalence. 32 orollary 3.31. Let be the trivial monoidal -category as in Example3.8. Then any -category M has a (unique) trivial action of . By Example3.8, Cat S is contractible. By Proposition 3.30, ev X : PSh p S ; M q Ñ M isan equivalence. Therefore θ : PSh M S Ñ Cat S is equivalent to the functor M Ñ ˚ ; that is, we have an equivalenceev X : PSh M S Ñ M . The rest of this section constitutes a proof of Theorem 3.25, following HA4.2.4.2. The proof is a straightforward application of HA 3.1.3 (free algebrasover -operads), but in order to apply it, we will need to introduce somenotation. None of the notation will reappear in this paper. Define LM S,X to be the subcategory of LM S spanned by: • graphs Γ P LM S such that every edge which terminates at ˚ originatesat X ; • morphisms f : Γ Ñ Γ such that for any edge e P Γ, f p e q terminatesat ˚ if and only if e terminates at ˚ .Let Triv denote the trivial -operad of HA 2.1.1.20 (the subcategory ofComm spanned by all objects and inert morphisms), and b the coproductof -operads. These satisfy the universal properties Alg Triv p V q – V andAlg O b O p V q – Alg O p V q ˆ Alg O p V q by HA 2.1.3.5, respectively 2.2.3.6.Consider the full subcategories of LM S,X spanned by edges of the form p X, ˚q , respectively edges which do not terminate at ˚ . These subcategoriesare isomorphic to Triv, respectively Assoc S , and the induced map of -operads Assoc S b Triv Ñ LM S,X is an isomorphism by construction of b (HA 2.2.3.3). Therefore, the inclusion LM S,X Ď LM S induces a functor ξ : PSh V ; M S “ Alg LM S { LM p V ; M q Ñ Alg LM S,X { LM p V ; M q – Cat V S ˆ M defined by ξ p C , F q “ p C , F p X qq . Now we can formulate: Lemma 3.32.
Given algebras p C , M q P Cat V S ˆ M – Alg LM S,X { LM p V ; M q and p ¯ C , F q P PSh V ; M S “ Alg LM S { LM p V ; M q , as well as a morphism λ : p C , M q Ñ p ¯ C , F p X qq “ ξ p ¯ C , F q , the following are equivalent:1. λ exhibits p ¯ C , F q as the free LM S -algebra generated by p C , M q in thesense of HA 3.1.3.1;2. λ : C Ñ ¯ C is an equivalence and λ : M Ñ F p X q exhibits F as freelygenerated by M at X in the sense of Definition 3.23. roof. We recall HA Definition 3.1.3.1 for reference (the case O “ LM, A “ LM S,X , and B “ LM S ).If Γ P LM S , define A act { Γ “ LM S,X ˆ LM S p LM act S q { Γ , where LM act S is thesubcategory of LM S spanned by all objects and active morphisms.Then (1) asserts that the induced map α Γ : p A act { Γ q Ź Ñ ş p V ; M q is anoperadic colimit diagram for all graphs Γ with a single edge. To unpackthis, we split into two cases:(a) If Γ “ p A, B q for any A, B P S , then Γ P A act { Γ is terminal. Therefore, α Γ is an operadic colimit diagram if and only if λ : C p A, B q Ñ ¯ C p A, B q is an equivalence.(b) If Γ “ p A, ˚q for any A P S , then p A, X qbp X, ˚q P A act { Γ is terminal, so α Γ is an operadic colimit diagram if and only if λ : C p A, X q b M Ñ F p X q is an equivalence.Hence, the statements (1) and (2) unpack to the same conditions. Proof of Theorem 3.25.
In light of the lemma, HA 3.1.3.3 asserts that freepresheaves exist (noting, as in the proof of the lemma, that A act { Γ has aterminal object for each Γ P LM S with a single edge). The equivalence inthe theorem statement is a restatement of HA 3.1.3.2. A -model for enriched categories Recall that a functor p : A Ñ B is called a presentable fibration if either ofthe following equivalent conditions hold (HTT 5.5.3.3): • p is cartesian and the functor B op Ñ y Cat factors through Pr R Ď y Cat; • p is cocartesian and the functor B Ñ y Cat factors through Pr L Ď y Cat.
Definition 4.1. If V is a monoidal -category and M is a left V -module,we say that p V ; M q is a presentable pair if the equivalent conditions hold: • V and M are both presentable, V is closed monoidal ( V b V Ñ V pre-serves colimits independently in each variable), and M is a presentableleft V -module ( V ˆ M Ñ M preserves colimits independently in eachvariable); • The pair p V ; M q , which is a priori an LM-algebra in Cat, restricts toan LM-algebra in Pr L . Theorem 1.2. If p V ; M q is a presentable pair and C is V -enriched, thenPSh V p C ; M q is presentable. We will also describe how to compute limits and colimits in PSh V p C ; M q : Corollary 4.20. If p V ; M q is a presentable pair and C is V -enriched:1. A functor p : K Ÿ Ñ PSh V p C ; M q is a limit diagram if and only ifev X p : K Ÿ Ñ M is a limit diagram for all X P C ;2. A functor p : K Ź Ñ PSh V p C ; M q is a colimit diagram if and only ifev X p : K Ź Ñ M is a colimit diagram for all X P C ; These and other important structural results appear in Section 4.3. Theirproofs rely on the A -model for enriched categories and presheaves, techni-cal tools which we first introduce in Sections 4.1 and 4.2.We recommend that the reader begin by skimming the results in Section4.3, and then return to Sections 4.1 and 4.2 before attempting to understandthe proofs. Define ∆ { S “ ∆ ˆ Set
Set { S . In this way, an object of ∆ { S is simply afunction ℓ : r n s Ñ S . For ease of exposition, we will refer to this as theobject X “ t X ă ¨ ¨ ¨ ă X n u when ℓ p i q “ X i . We think of X as theordered set r n s “ t ă ¨ ¨ ¨ ă n u along with a labeling of each element in S .A morphism X Ñ Y is an order-preserving function f : r n s Ñ r m s whichpreserves the labeling, Y f p i q “ X i .We begin by introducing a strong approximation Cut S : ∆ op { S Ñ Assoc S generalizing Example 2.18 (which corresponds to the case | S | “ r n s “ t ă ¨ ¨ ¨ ă n u as a directedgraph on the set t , . . . , n u :0 / / / / ¨ ¨ ¨ / / n, and therefore as an object of Assoc r n s . If ℓ : r n s Ñ S is an object of ∆ { S ,push forward along ℓ ˚ : Assoc r n s Ñ Assoc S to a graph Cut S p ℓ q P Assoc S .Concretely, if X “ t X ă ¨ ¨ ¨ ă X n u P ∆ { S , thenCut S p X q “ p X , X q b p X , X q b ¨ ¨ ¨ b p X n ´ , X n q . Y “ t Y ă ¨ ¨ ¨ ă Y m u and f : X Ñ Y is a map in ∆ { S , then there isan induced map f ˚ : Cut S p Y q Ñ Cut S p X q as follows: f ˚ sends the edge p Y i ´ , Y i q to the edge p X j ´ , X j q if f p j ´ q ă i ď f p j q . If there is no such j , f ˚ sends p Y i ´ , Y i q to the basepoint. The total ordering on r n s inducestotal orderings on the fibers of f ˚ , and f ˚ satisfies the two properties ofDefinition 3.1, so that we have a functorCut S : ∆ op { S Ñ Assoc S . The composite ∆ op { S Cut S ÝÝÝÑ
Assoc S Ñ Comm makes ∆ op { S an -preoperad. Remark 4.2.
Recall (Example 2.18) that a morphism r n s Ñ r m s of ∆ op isinert if it embeds r m s as a convex subset t i ă ¨ ¨ ¨ ă i ` m u Ď r n s . Then amorphism of ∆ op { S is inert if and only if the underlying morphism in ∆ op isinert.In particular, we can verify any such morphism is p -cocartesian, just asin the proof of Proposition 3.5. Proposition 4.3.
The functor Cut S : ∆ op { S Ñ Assoc S is a strong approxi-mation to the -operad Assoc S .Proof. Both ∆ op { S and Assoc S have the property: A morphism is inert if andonly if it lies over an inert morphism in Comm. Therefore, Cut S sends inertmorphisms to inert morphisms, so it is a morphism of -preoperads.We need to check the two conditions of Definition 2.16 to prove that Cut S is an approximation. Since the fibers of ∆ op { S and Assoc S over x y P Commare each equivalent to the (discrete) set S , it is clear that Cut S is a strongapproximation if and only if it is an approximation.(1) Note that p : ∆ op { S Ñ Comm is given by p p X ă ¨ ¨ ¨ ă X n q “ x n y ,where i P x n y corresponds to the edge p X i ´ , X i q in Cut S p X ă ¨ ¨ ¨ ă X n q .Given an inert morphism φ : t , . . . , n ´ u Ñ x y , let i “ φ ´ p q , and let t X i ă X i ` u Ñ t X ă ¨ ¨ ¨ ă X n u be the natural inclusion. This describesan inert morphism t X ă ¨ ¨ ¨ ă X n u Ñ t X i ă X i ` u in ∆ op { S lifting φ , soCut S satisfies Definition 2.16(1).(2) Fix an active morphism φ : Γ Ñ p Y , Y qb¨ ¨ ¨bp Y m ´ , Y m q in Assoc S .That is, φ transforms the graph Γ into p Y , Y qb¨ ¨ ¨bp Y m ´ , Y m q via repeatedapplication of the two moves: • p A, B q b p
B, C q Ñ p
A, C q ; • H Ñ p
A, A q . 36ecall that the morphism φ includes the data of total orderings on thefibers φ ´ p Y i , Y i ` q . Taken together, these induce a total ordering on theedges of Γ such that they form a path from Y to Y m . Hence we may writeΓ “ p X , X q b ¨ ¨ ¨ b p X n ´ , X n q where X “ Y and X n “ Y m .Define 0 “ k ď ¨ ¨ ¨ ď k m “ n such that φ p X j , X j ` q “ p Y i , Y i ` q whenever k i ď j ă k i ` . That is, p X k i , X k i ` q b ¨ ¨ ¨ b p X k i ` ´ , X k i ` q is the fiber φ ´ p X i , X i ` q , or the fiber is empty when k i “ k i ` . Since φ ´ p X i , X i ` q must form a path from X i to X i ` , we conclude X k i “ Y i forall i . Therefore, the indices k , . . . , k m describe a morphism¯ φ : t X ă ¨ ¨ ¨ ă X n u Ñ t Y ă ¨ ¨ ¨ ă Y m u of ∆ op { S lifting φ . By construction, φ is universal among such lifts of φ , whichis to say it is f -cartesian.Thus Cut S satisfies Definition 2.16(2), which completes the proof. Definition 4.4. If V is a monoidal -category, an A - V -enriched category with set S of objects is a ∆ op { S -algebra in V , and they form an -category A Cat V S “ Alg ∆ op { S { Assoc p V q . Applying Theorem 2.17, we find:
Corollary 4.5.
Cut S induces an equivalence Cat V S Ñ A Cat V S . Remark 4.6. If V is a monoidal -category, recall from Section 2.4 thatwe can regard V either as a functor V : Assoc Ñ Cat, with associated -operad ş V Ñ Assoc, or as a functor BV : ∆ op Ñ Cat, with associated planar -operad ş BV Ñ ∆ op . By construction we have a pullback ş BV / / (cid:15) (cid:15) ş V (cid:15) (cid:15) ∆ op Cut / / Assoc , so A Cat V S – Fun :{ ∆ op p ∆ op § { S , ş BV § q . We have just shown that ∆ op { S is a strong approximation to the -operadAssoc S , which means that we can identify V -enriched categories with ∆ op { S -algebras in V . 37ow we will show that ∆ op { S ˆ ∆ is a strong approximation to the -operad LM S . First, we construct the functor LCut S : ∆ op { S ˆ ∆ Ñ LM S .This will closely parallel the construction of Cut S in Section 4.1.Given r n s P ∆, consider the following graphs on t , . . . , n u ` : r n s “ r Ñ Ñ ¨ ¨ ¨ Ñ n Ñ ˚sr n s “ r Ñ Ñ ¨ ¨ ¨ Ñ n s , which are objects of LM r n s . Suppose r n s ℓ ÝÑ S is an object of ∆ { S , inducing ℓ ˚ : LM r n s Ñ LM S . We write LCut S p ℓ q “ ℓ ˚ r n s and LCut S p ℓ q “ ℓ ˚ r n s ,which are left-modular graphs on the set S ` . Exactly as in Section 4.1, wehave functors LCut S , LCut S : ∆ op { S Ñ LM S . Remark 4.7.
The functor LCut S is essentially the same as Cut S ; that is,it factors ∆ op { S Cut S ÝÝÝÑ
Assoc S Ď LM S . There are inert morphisms r n s Ñ r n s which send the edge n Ñ ˚ tothe basepoint, and act as the identity function on the other edges. Theseassemble into a natural transformation LCut S Ñ LCut S ; that is, a functor∆ Ñ Fun p ∆ op { S , LM S q . There is a corresponding functorLCut S : ∆ op { S ˆ ∆ Ñ LM S . The composite with LM S Ñ Comm makes ∆ op { S ˆ ∆ an -preoperad. Remark 4.8.
Recall (Example 2.19) that a morphism pr n s f ÝÑ r m s , i Ñ j q of ∆ op ˆ ∆ is inert if f is inert in ∆ op and either: • f p m q “ n ; • or j “ .A morphism of ∆ op { S ˆ ∆ is inert if and only if the underlying morphism in ∆ op ˆ ∆ is inert. Proposition 4.9.
The functor LCut S : ∆ op { S ˆ ∆ Ñ LM S is a strong ap-proximation to the -operad LM S .Proof. The proof is just like Proposition 4.3; see also HA 4.2.2.8.38 efinition 4.10.
Suppose p V ; M q is an LM-monoidal -category; that is, V is a monoidal -category, and M is a left V -module -category.An A - V -enriched presheaf with set S of objects is a ∆ op { S ˆ ∆ -algebrain p V ; M q , and they form an -category A PSh V ; M S “ Alg ∆ op { S ˆ ∆ { LM p V ; M q . As in Lemma 3.15, A Cat V S – Alg ∆ op { S { LM p V ; M q , so composition with theinclusion ∆ op { S ˆ t u Ñ ∆ op { S ˆ ∆ induces a forgetful functor θ : A PSh V ; M S Ñ A Cat V S . Definition 4.11. If C P A Cat V S , then the -category of A - V -enrichedpresheaves on C with values in M is the fiber A PSh V p C ; M q “ A PSh V ; M S ˆ A Cat V S t C u . Corollary 4.12.
Let C be a V -enriched category with set S of objects, and ¯ C the corresponding A - V -enriched category. Then composition with LCut S induces equivalences PSh V ; M S Ñ A PSh V ; M S , PSh V p C ; M q Ñ A PSh V p ¯ C ; M q . Proof.
The first equivalence follows from Proposition 4.9 by Theorem 2.17.In fact, we have a commutative squarePSh V ; M S / / (cid:15) (cid:15) A PSh V ; M S (cid:15) (cid:15) Cat V S / / A Cat V S , where the horizontal maps are equivalences, and so the second equivalencefollows by taking fibers.Suppose V is a monoidal -category and M is a left V -module. Recallfrom Section 2.4 that we can regard the pair p V ; M q as either a functor p V ; M q : LM Ñ Cat with corresponding -operad ş p V ; M q Ñ LM, or as39 functor B p V ; M q : ∆ op ˆ ∆ Ñ Cat with corresponding planar -operad ş B p V ; M q Ñ ∆ op ˆ ∆ . By construction we have a pullback ş B p V ; M q / / (cid:15) (cid:15) ş p V ; M q (cid:15) (cid:15) ∆ op ˆ ∆ / / LM , so A PSh V ; M S – Fun :{ ∆ op ˆ ∆ p ∆ op { S ˆ ∆ § , ş B p V ; M q § q . Remark 4.13.
Although everything we have said in this section is model-independent, now we will say a word about quasicategories, paralleling Re-mark 3.20 for the operadic model. Suppose we are given a cocartesian fi-bration of quasicategories ş B p V ; M q Ñ ∆ op ˆ ∆ exhibiting M as a left V -module -category in the A -sense. Then we have the following explicitquasicategory models: A Cat V S “ Fun :{ ∆ op ˆ ∆ p ∆ op § { S , ş B p V ; M q § q ; A PSh V ; M S “ Fun :{ ∆ op ˆ ∆ p ∆ op { S ˆ ∆ § , ş B p V ; M q § q . ∆ op { S lies over ∆ op ˆ ∆ , as always, by ∆ op { S Ñ ∆ op ˆ t u Ď ∆ op ˆ ∆ . Sincethe inclusion ∆ op { S Ñ ∆ op { S ˆ ∆ is a categorical fibration, the forgetful functor θ : A PSh V ; M S Ñ A Cat V S is also a categorical fibration, so A PSh V p C ; M q may be modeled as the literal fiber θ ´ p C q , which is to say the quasicategoryof lifts ∆ op § { S C / / (cid:15) (cid:15) ş BV § (cid:15) (cid:15) ∆ op { S ˆ ∆ § F / / ❴❴❴ ş B p V ; M q § . Unlike in the operadic model, we can unpack this last diagram further. Re-call (following Remark 2.22) that B p V ; M q : ∆ op ˆ ∆ Ñ Cat can be identi-fied with a morphism BV ˙ M Ñ BV of simplicial -categories. Applyingthe Grothendieck construction, we have a diagram ş BV ˙ M q / / p % % ❑❑❑❑❑❑❑❑❑❑ ş BV p | | ①①①①①①①① ∆ op , q sends p -cocartesian morphisms to p -cocartesian morphisms. Alsorecall the marking ş BV ˙ M ; by totally inert morphisms (Warning 2.20 andRemark 2.27). Definition 4.14.
Say that a morphism of ∆ op { S is totally inert if the underly-ing morphism of ∆ op is totally inert, and write ∆ op ;{ S for this marking. Equiv-alently, a morphism is totally inert if its image under ∆ op { S ˆ t u Ď ∆ op { S ˆ ∆ is inert. Proposition 4.15.
For any pair p V ; M q and set S , A PSh V ; M S is equivalentto the full subcategory of Fun : p ∆ op ;{ S , ş BV ˙ M ; q spanned by functors suchthat ∆ op { S Ñ ş BV ˙ M Ñ ş BV is an A -algebra.If C P A Cat V S , then A PSh V p C ; M q – Fun :{ ş BV § p ∆ op ;{ S , ş BV ˙ M ; q . Proof.
The proof follows HA 4.2.2.19 (stated earlier as Proposition 2.28),which is the case | S | “ ş B p V ; M q i is the fiber of ş B p V ; M q Ñ ∆ op ˆ ∆ over ∆ op ˆ t i u , then: • ş B p V ; M q § “ ş BV ˙ M ; ; • ş B p V ; M q § “ ş BV § ; • If X i P ş B p V ; M q i are two objects both lying over Y P ∆ op { S , and f : X Ñ X is a morphism lying over p Y, q Ñ p Y, q P ∆ op S ˆ ∆ (theidentity map on Y ), then f is inert if and only if the induced functor q p X q Ñ X is an equivalence.Therefore, giving a lift F as in the square∆ op § { S C / / (cid:15) (cid:15) ş BV § (cid:15) (cid:15) ∆ op { S ˆ ∆ § F / / ❴❴❴ ş B p V ; M q § . is the same as giving a marked functor F : ∆ op ;{ S Ñ ş BV ˙ M ; , as well as anatural transformation F Ñ C sends each Y P ∆ op { S to an inert morphism of B p V ; M q lying over p Y, q Ñ p Y, q ; in other words, C should factor C “ q F .The proposition follows from this description.41 .3 Limits and colimits of presheaves We are ready to prove that θ : PSh V ; M S Ñ Cat V S is a cartesian fibration,and a presentable fibration if p V ; M q is a presentable pair. We will begin byproving that θ is a cartesian fibration. Proposition 4.16. If V is a monoidal -category, M is a left V -module,and S is a set, then the forgetful functor θ : PSh V ; M S Ñ Cat V S is a cartesianfibration, and a morphism f in PSh V ; M S is θ -cartesian if and only if its imageunder ev X : PSh V ; M S Ñ M is an equivalence for each X P S . Remark 4.17.
Since θ is a cartesian fibration, a map F : C Ñ D in Cat V S induces a functor F ˚ : PSh V p D ; M q Ñ PSh V p C ; M q . By the identificationof θ -cartesian morphisms, F ˚ p F q can be evaluated at objects by the formula F ˚ p F qp X q “ F p X q . In other words, each triangle commutes:PSh V p D ; M q F ˚ / / ev X & & ▲▲▲▲▲▲▲▲▲▲ PSh V p C ; M q ev X y y rrrrrrrrrr M . Lemma 4.18.
Suppose the pair p V ; M q is modeled by a cocartesian fibrationof quasicategories ş p V ; M q Ñ LM, and K is a simplicial set such that M admits K -indexed limits. Then:1. For every commutative square K / / _(cid:127) (cid:15) (cid:15) A PSh V ; M Sθ (cid:15) (cid:15) K Ÿ / / : : ttttt A Cat V S , there exists a dotted arrow as indicated, which is a θ -limit diagram.2. An arbitrary functor K Ÿ Ñ A PSh V ; M S is a θ -limit diagram if andonly if the composite K Ÿ Ñ A PSh V ; M S ev X ÝÝÑ M is a limit diagramfor each X P S .Proof. We follow HA 4.2.3.1, which is the | S | “ ş BV ˙ M q / / p % % ❑❑❑❑❑❑❑❑❑ ş BV p | | ①①①①①①①① ∆ op r n s P ∆ op are ş BV ˙ M n “ V ˆ n ˆ M and ş BV n “ V ˆ n .Lurie concludes in HA 4.2.3.1 p n ´ n q :1’. For every commutative square K / / _(cid:127) (cid:15) (cid:15) ş BV ˙ M nq n (cid:15) (cid:15) K Ÿ / / ttttt ş BV n , there exists a dotted arrow as indicated, which is a q -limit diagram.2’. An arbitrary map K Ÿ Ñ ş BV ˙ M n is a q -limit diagram if and only ifthe projection K Ÿ Ñ ş BV ˙ M n “ V ˆ n ˆ M Ñ M is a limit diagram.Applying HA 3.2.2.9,1”. For every commutative square K / / _(cid:127) (cid:15) (cid:15) Fun p ∆ op { S , ş BV ˙ M q θ (cid:15) (cid:15) K Ÿ / / ♦♦♦♦♦♦ Fun p ∆ op { S , ş BV q , there exists a dotted arrow as indicated, which is a θ -limit diagram.2”. An arbitrary map K Ÿ Ñ Fun p ∆ op { S , ş BV ˙ M q is a θ -limit diagram ifand only if for each t X ă ¨ ¨ ¨ ă X n u P ∆ { S , the projection K Ÿ ev X ÝÝÑ ş BV ˙ M n Ñ M is a limit diagram.In 1”-2”, we can just as well restrict to A Cat V S , which is a full subcategoryof Fun p ∆ op { S , ş BV q by Remark 4.13 (that is, the full subcategory of functorswith send inert morphisms to inert morphisms and are compatible with thefunctors down to ∆ op ).Let pre A PSh V ; M S be the full subcategory of Fun p ∆ op { S , ş BV ˙ M q spannedby those functors F for which θ p F q is an A -algebra. Then the restriction θ : pre A PSh V ; M S Ñ A Cat V S enjoys the same properties 1”-2”.By Proposition 4.15, A PSh V ; M is the full subcategory of pre A PSh V ; M spanned by those functors ∆ op { S Ñ ş BV ˙ M which send totally inert mor-phisms to p -cocartesian morphisms.Now we will prove part (1) of the lemma. It suffices to show that if ¯ g isa θ -limit diagram K Ÿ Ñ pre A PSh V ; M S and the restriction g to K factors43hrough A PSh V ; M S , then ¯ g factors through A PSh V ; M S . Any totally inertmorphism f : t X ă ¨ ¨ ¨ ă X n u Ñ t Y ă ¨ ¨ ¨ ă Y m u in ∆ op { S induces a naturaltransformation ¯ g X Ñ ¯ g Y of functors K Ÿ Ñ ş BV ˙ M , and we want to showthat each object of K Ÿ is sent to a p -cocartesian morphism in BV ˙ M .Since f is totally inert, f p m q “ n , and we have a commutative triangle,where the downward maps are just projection ş BV ˙ M n / / α % % ❑❑❑❑❑❑❑❑❑❑ ş BV ˙ M mβ y y ssssssssss M , and it suffices to show that ¯ t : α ¯ g X Ñ β ¯ g Y is an equivalence. By hypothesis,the restriction t : αg X Ñ βg Y is an equivalence, and then by (2”), ¯ t is aninduced natural transformation between limit diagrams. Therefore, ¯ t is alsoan equivalence, completing the proof of (1).As for (2), we need to prove that the following are equivalent for a functor¯ g : K Ÿ Ñ A PSh V ; M S :2”. Each K Ÿ ¯ g ÝÑ A PSh V ; M S ev X ÝÝÑ M is a limit diagram, where ev X isevaluation of ∆ op { S Ñ ş BV ˙ M at any X “ t X ă ¨ ¨ ¨ ă X n u P ∆ op { S ;2. Each ev X ¯ g is a limit diagram if n “
0; that is, X “ t X u P ∆ op { S .Obviously 2” implies 2. Conversely, assume 2. If X “ t X ă ¨ ¨ ¨ ă X n u ,then there is a totally inert morphism X Ñ t X n u which induces a commu-tative square A PSh V ; M p C q ev X / / ev t Xn u (cid:15) (cid:15) ş BV ˙ M n (cid:15) (cid:15) ş BV ˙ M / / M . Then ev X ¯ g “ ev t X n u ¯ g is a limit diagram by 2, completing the proof. Proof of Proposition 4.16.
Suppose the pair p V ; M q is modeled by a co-cartesian fibration of quasicategories ş p V ; M q Ñ LM. We will prove that θ : A PSh V ; M S Ñ A Cat V S is a cocartesian fibration of quasicategories. Wealready know θ is a categorical fibration (Remark 4.13), hence also an innerfibration.Recall that a functor ∆ Ñ A PSh V ; M S is a θ -limit diagram if and onlyif it is a θ -cartesian edge (HTT 4.3.1.4). Hence when K “ ˚ , part (1) of the44emma asserts θ is a cartesian fibration, and (2) asserts that f is θ -cartesianif and only if ev X p f q is an equivalence for each X P S .For the rest of this section, we will assume that p V ; M q is a presentable pair. Theorem 4.19. If p V ; M q is a presentable pair and C is V -enriched:1. PSh V p C ; M q is presentable;2. for each X P C , ev X : PSh V p C ; M q Ñ M preserves limits and colimits;3. If F : C Ñ D is a map in Cat V S , then F ˚ : PSh V S p D ; M q Ñ PSh V p C ; M q preserves limits and colimits.4. if N is presentable, a functor F : N Ñ PSh V p C ; M q preserves colimits(respectively limits) if and only if the composite ev X F : N Ñ M preserves colimits (limits) for each X P C . In particular, (1) is Theorem 1.2 of the introduction.
Proof.
We follow the proof of HA 4.2.3.4. Pick a cocartesian fibration ofquasicategories ş p V ; M q Ñ LM which realizes M as a left V -module. Define X to be the pullback X / / q (cid:15) (cid:15) ş BV ˙ M q (cid:15) (cid:15) ∆ op { S C / / ş BV , and say that a morphism of X is totally inert if its images in ∆ op { S and ş BV ˙ M are each totally inert. Recall (Remark 2.26) that q is a categoricalfibration and a locally cocartesian fibration. Therefore, q is as well.By Proposition 4.15, PSh V p C ; M q is equivalent to the quasicategory ofsections of q which send totally inert morphisms to totally inert morphisms.Applying HTT 5.4.7.11 using the subcategory Pr L Ď y Cat, we conclude: • PSh V p C ; M q is presentable; • F : N Ñ PSh V p C ; M q preserves colimits if and only if the compositeev X F : N Ñ ş BV ˙ M n does for each X “ t X ă ¨ ¨ ¨ ă X n u P S .At the end of the proof of Lemma 4.18 (where we proved conditions 2 and2” are equivalent), we proved that this second condition is equivalent to thecolimit formulation of (4). 45he limit formulation of (4) is Lemma 4.18 when K Ÿ Ñ A Cat V S is theconstant functor with value C .So we have (1) and (4). Applying (4) to the identity functor establishes(2). Applying (4) to the functor PSh V S p D ; M q Ñ PSh V p C ; M q along withRemark 4.17 establishes (3).We deduce two important corollaries. The first is a formulation of Corollary1.4 from the introduction. Corollary 4.20. If p V ; M q is a presentable pair and C is V -enriched:1. A functor p : K Ÿ Ñ PSh V p C ; M q is a limit diagram if and only ifev X p : K Ÿ Ñ M is a limit diagram for all X P C ;2. A functor p : K Ź Ñ PSh V p C ; M q is a colimit diagram if and only ifev X p : K Ź Ñ M is a colimit diagram for all X P C ; Corollary 4.21. If p V , M q is a presentable pair, θ : PSh V ; M S Ñ Cat V S is apresentable fibration.Proof. By Proposition 4.16, θ is a cartesian fibration. Now assume p V ; M q is a presentable pair. Let F : p Cat V S q op Ñ y Cat be the associated functor, sothat F p C q “ PSh V p C ; M q . By Theorem 4.19, F p C q is presentable for each C , and F p D q Ñ F p C q is a right adjoint functor (as it preserves limits andcolimits) for each enriched functor C Ñ D . Therefore, F factors throughPr R Ď y Cat, so θ is a presentable fibration. Remark 4.22. If F : C Ñ D is a map in Cat V S , then there is always afunctor F ˚ : PSh V p D ; M q Ñ PSh V p C ; M q . Corollary 4.21 asserts that F ˚ has a left adjoint F ˚ if p V ; M q is a presentable pair. Recall that F ˚ fits in acommutative triangle for each X P S (Remark 4.17),PSh V p D ; M q F ˚ / / ev X & & ▲▲▲▲▲▲▲▲▲▲ PSh V p C ; M q ev X y y rrrrrrrrrr M . Taking left adjoints, we have a commutative triangle M rep X b´ y y rrrrrrrrrr rep X b´ & & ▲▲▲▲▲▲▲▲▲▲ PSh V p C ; M q F ˚ / / PSh V p D ; M q , so F ˚ p rep X p C q b M q – rep X p D q b M . F : X Ñ Y is called monadic if either of the followingequivalent conditions hold (HA 4.7.3.5): • F has a left adjoint, is conservative, and preserves certain colimits ; • There is a monoidal -category E , an algebra E in E , and a left E -module structure on Y , such that F is equivalent to the forgetfulfunctor LMod E p Y q Ñ Y . E can be taken to be the endomorphism -category Fun p Y , Y q , and E themonad associated to F (the composite F ˝ L , where L is the left adjoint). Corollary 4.23. If p V ; M q is a presentable pair and C is V -enriched with set S of objects, let ev : PSh V p C ; M q Ñ M ˆ S be the product of all the functorsev X , as X P S varies. The functor ev is monadic.Proof. By Theorem 4.19(2), ev is a functor of presentable -categorieswhich preserves small limits and colimits. Therefore, it has a left adjoint.By Corollary 4.20 in the case K “ ˚ , ev is conservative, hence monadic. Corollary 4.24. If p V ; M q is a presentable pair and C is V -enriched, thenPSh V p C ; M q is generated under colimits by the free presheaves rep X b M ,where X P C and M P M .Proof. Since ev : PSh V p C ; M q Ñ M S is monadic, we can without loss ofgenerality replace ev with a functor LMod E p M S q Ñ M S . Every left E -module M is a colimit of free left E -modules by the bar construction for E b E M (HA 4.4.2). Therefore, PSh V p C ; M q is generated under colimitsby the image of the free functor M S Ñ PSh V p C ; M q . By Corollary 3.29,this free functor sends t M X u X P S to the coproduct š X P S rep X b M X , soPSh V p C ; M q is generated under colimits by free presheaves. Remark 4.25.
It should be possible to relax the presentability conditions on V and M as follows: • By Lemma 4.18, PSh V p C ; M q admits limits indexed by K as long as M admits limits indexed by K (with no condition on V ). • PSh V p C ; M q admits colimits indexed by K if M admits colimits in-dexed by K , which are compatible with the left V -module structure ina certain sense. If details are needed, consult HA 4.2.3.4-5.In each case, limits and colimits are detected by ev X : PSh V p C ; M q Ñ M . We won’t care which colimits are preserved, because all monadic functors we studywill preserve all small colimits. Tensor products of presheaves
Suppose that F P PSh V p C q is a presheaf, M is a left V -module, and M P M .Then there is a tensor presheaf F b M P PSh V p C ; M q given informally bythe formula p F b M qp X q “ F p X q b M .If M “ V , then this tensor product describes a right action of V onPSh V p C q . We begin in Section 5.1 by reviewing right modules. In the sameway that left modules were generalized to enriched presheaves, right modulesdescribe enriched copresheaves . We introduce enriched copresheaves, whichplay an important role in Section 6.In Section 5.2, we construct the right V -action on presheaves (Definition5.16, Theorem 5.17). In Section 5.4, we prove that it really is given by theformula above (Proposition 5.18).In the event that V is presentable and closed monoidal, PSh V p C q is evena presentable right V -module (Theorem 5.21) and assembles into a functorPSh V p´q : Cat V S Ñ RMod V p Pr L q (Corollary 5.24). In other words, if F : C Ñ D is an enriched functor, then F ˚ : PSh V p C q Ñ PSh V p D q is compatible with the right V -module structure,or F ˚ p F b A q – F ˚ p F q b A . Remark 5.1.
Of course, we expect that PSh V p´q extends to a functorCat V Ñ RMod V p Pr L q , where Cat V is the -category of all V -enriched cat-egories (that is, without a fixed set of objects). As always, we are delayingsuch results until a future paper in this series. Our discussion so far all assumed M “ V . In general, the construction F b M describes a functor PSh V p C q b V M Ñ PSh V p C ; M q . In Section 5.4,we prove the main theorem of this section: Theorem 1.6. If V is presentable and closed monoidal, C is a V -enrichedcategory, and M is a presentable left V -module, then the functor Ψ :
PSh V p C q b V M Ñ PSh V p C ; M q is an equivalence, where b V is the relative tensor product in Pr L . The construction of the V -module structure and this functor Ψ are quiteinvolved. However, most of our proofs rely only on the interaction of tensorpresheaves with evaluation and representable presheaves. That is: • p F b M qp X q – F p X q b M rep X b M is the free presheaf generated by M at X .In Section 3.3, we used the notation rep X b M for the free presheaf generatedby M at X . Now we see that this notation is justified: the free presheafrep X b M really is equivalent to the tensor presheaf rep X b M .We prove these facts in Proposition 5.18 in the case M “ V , and Lemma5.25 in general. First we introduce -operads RM S for each set S , whose algebras are co-presheaves . In the case | S | “
1, RM is the right module -operad, and analgebra over RM is a pair p A, M q , where A is an algebra and M a rightmodule.The -operad RM S is the reverse of LM S , in the sense of HA 4.1.1.7;that is, the underlying -category is equivalent, but the operad structure isreversed. We begin by reviewing reverse -operads.If Γ P Assoc S is a graph, let rev p Γ q denote Γ with the directions ofall arrows reversed. For example, rev p X, Y q “ p
Y, X q . This constructiondescribes a functor of categories rev : Assoc S Ñ Assoc S , which is an isomor-phism. Recall: Definition 5.2 (HA 4.1.1.7) . If O is an -operad, then the composite O Ñ Assoc rev
ÝÝÑ
Assoc endows the -category O with a distinct -operadstructure. We write O rev for this -operad, and call it the reverse -operad of O .If V : Assoc Ñ Cat is a monoidal -category, then the composite V ˝ revis another monoidal -category, which we denote V rev . Note that V and V rev have the same underlying -categories, but themonoidal operations are reversed; that is, X b rev Y “ Y b X .If O is an -operad and V is a monoidal -category, then the reversalisomorphism induces an equivalence Alg O p V q – Alg O rev p V rev q . Example 5.3.
The isomorphism rev : Assoc S Ñ Assoc S exhibits Assoc S as equivalent to its own reversal. Therefore, Cat V S – Cat V rev S . If C is a V -enriched category, then the corresponding V rev -enriched category is C op ,which has mapping spaces C op p X, Y q “ C p Y, X q . Definition 5.4.
For any set S , RM S “ LM rev S .When | S | “ , we also write RM “ RM S , which is the right moduleoperad of HA 4.2.1.36. emark 5.5. We say that a graph Γ P Assoc S ` is right modular (compare left modular , Definition 3.11) if for all edges e P Γ , t p e q ‰ ˚ . Then RM S is equivalent to the full subcategory of Assoc S ` spanned by right modulargraphs.This is because rev : Assoc S ` Ñ Assoc S ` restricts to an isomorphismbetween LM S and the full subcategory of right modular graphs. In the case | S | “
1, we think of an RM-algebra as a pair p A, M q , where A is an algebra and M is a right A -module.In general, we think of an RM S -algebra as a pair p C , F q , where C is a V -enriched category and F is an enriched copresheaf on C : Definition 5.6.
Suppose that V is a monoidal -category and N a right V -module category, so that the pair p V ; N q is an RM-monoidal -category.If C is a V -enriched category with set S of objects, an enriched copresheafon C with values in M is a lift of the Assoc S -algebra C to an RM S -algebrain p V ; N q . We write coPSh V ; N S “ Alg RM S { RM p V ; N q , coPSh V p C , N q “ coPSh V ; N S ˆ Cat V S t C u . We think of a copresheaf F P coPSh V p C , N q informally as a V -enrichedfunctor C Ñ N . Remark 5.7.
Because LM S and RM S are reverse -operads, • A right V -action on N canonically induces a left V rev -action on N ,and vice versa; • coPSh V ; N S – PSh V rev ; N S ; • coPSh V p C ; N q – PSh V rev p C op ; N q .Therefore, copresheaves may be regarded as examples of presheaves, so every-thing we have already proven about presheaves is also true of copresheaves. Now we are ready to describe the right V -module structure on PSh V p C q , forany monoidal -category V and V -enriched category C .Given two graphs Γ P LM S and Γ P RM T , define x Γ , Γ y P Assoc S > T as follows: 50 an edge of x Γ , Γ y is a pair p e , e q , where e i is an edge of Γ i and atleast one of the edges e , e touches the basepoint vertex ˚ ; • if one of the edges e i does not touch the vertex ˚ , then p e , e q has thesame source and target as e i ; • if e is an edge from some X to ˚ , and e is an edge from ˚ to Y , then p e , e q is an edge from X to Y . Example 5.8.
Let p X , . . . , X n q “ p X , X q b p X , X q b ¨ ¨ ¨ b p X n ´ , X n q denote the graph with a single path from X to X n . Then we have xp X , . . . , X m q , p Y , . . . , Y n qy “ Hxp X , . . . , X m , ˚q , p Y , . . . , Y n qy “ p Y , . . . , Y n qxp X , . . . , X m , ˚q , p˚ , Y , . . . , Y n qy “ p X , . . . , X m , Y , . . . , Y n q . Remark 5.9.
The pairing x´ , ´y is a functor LM S ˆ RM T Ñ Assoc S > T .Moreover, if Γ Ñ Γ is inert in LM S and Γ Ñ Γ is inert in RM S , then x Γ , Γ y Ñ x Γ , Γ y is inert.So it is even a marked functor x´ , ´y : LM § S ˆ RM § S Ñ Assoc § S > T . Remark 5.10.
The bimodule operad BM of HA 4.3.1.5 can be identifiedwith the full subcategory of Assoc t , u spanned by graphs that have no edgesfrom to . Then our pairing factors LM t u ˆ RM t u Ñ BM Ď Assoc t , u .This recovers the functor of HA 4.3.2.1, Pr : LM ˆ RM Ñ BM.
For now, we will just be interested in the case | S | “ | T | “ ˆ RM Ñ Assoc t , u Ñ Assoc.
Definition 5.11. If V is a monoidal -category, define ş V via the pullback ş V p (cid:15) (cid:15) / / ş V (cid:15) (cid:15) LM ˆ RM x´ , ´y / / Assoc , and say that a morphism in ş V is left inert if it is p -cocartesian, its pro-jection to LM is inert, and its projection to RM is an equivalence. We willwrite ş V ! for the left inert marking. emark 5.12. Identify RM “ RM t u , so that there are two kinds of edgesin RM: p˚ , q and p , q .Unpacking this construction, the fiber of ş V Ñ LM ˆ RM over any Γ P RM is an LM-monoidal -category, which describes:1. V acting on itself on the left if Γ “ p˚ , q ;2. acting on V on the left if Γ “ p , q (where is the trivial monoidal -category). Definition 5.13. If V is a monoidal -category, Ě Cat V S “ Fun :{ LM p Assoc § S , ş V ! q , Ě PSh V S “ Fun :{ LM p LM § S , ş V ! q . Consider the forgetful functors Ě Cat V S Ñ Fun :{ LM p Assoc § S , LM § ˆ RM q – Fun : p Assoc § S , RM q – RM , and similarly Ě PSh V S Ñ RM. (The rightmost equivalence is by Lemma 2.12.)These fit into a commutative triangle Ě PSh V S ¯ θ / / p ●●●●●●●●● Ě Cat V Sp { { ①①①①①①①①① RM . Lemma 5.14.
In this triangle:1. p and p are cocartesian fibrations of -operads;2. a morphism in Ě PSh V S (respectively Ě Cat V S ) is p -cocartesian ( p -cocartesian)if and only if the evaluation at each Γ P LM S (or Assoc S ) is a p -cocartesian morphism of ş V , where p is the map ş V Ñ LM ˆ RM;3. ¯ θ sends p -cocartesian morphisms to p -cocartesian morphisms.Proof. (3) is a direct corollary of (2). The proof of (1) and (2) is completelyidentical to the proof of HA 4.3.2.5, which uses categorical patterns. Ratherthan introduce the categorical pattern terminology here (which would ap-pear nowhere else in this paper), see the proof of HA 4.3.2.5, and replacethe construction ¯ X ÞÑ LM ˆ ¯ X with LM S for Ě PSh V S or Assoc S ˆ ´ for Ě Cat V S .The proof carries over without any modification, except that we observeLM S and Assoc S P p
Set ` ∆ q { P LM are cofibrant as well as LM .52y the lemma, ¯ θ is a functor of RM-monoidal -categories (HA 2.1.3.7). Lemma 5.15.
The RM-monoidal -category Ě PSh V S p ÝÑ RM describes aright V -action on PSh V S .The RM-monoidal -category Ě Cat V S p ÝÑ RM describes the (essentiallyunique) right action of the trivial monoidal -category on Cat V S .Proof. If Γ P RM, let ş V Γ Ñ LM denote the fiber of ş V Ñ LM ˆ RM overΓ, and note that the fiber of p over Γ is Fun :{ LM p LM § S , ş V § Γ q .By Remark 5.12, ş V p˚ , q “ ş p V ; V q , so the fiber of p over p˚ , q is PSh V S .On the other hand, ş V p , q “ ş p V q , so the fiber over p , q is PSh V S – V ;this equivalence is by Corollary 3.31. Therefore, p exhibits a right V -actionon PSh V S . Similarly, p exhibits a right action of Cat S – V .Summarizing our discussion so far, the forgetful functor θ : PSh V S Ñ Cat V S extends to an RM-monoidal functor, which is to say that it is compatiblewith the right V -action on PSh V S and the trivial action on Cat V S . It followsthat each fiber PSh V p C q “ PSh V S ˆ Cat V S t C u inherits the right V -action. Tomake this more clear, here is an explicit construction of the right V -actionon PSh V p C q : Definition 5.16.
By Lemma 5.15, Ě Cat V S – ş p Cat V S q . By Corollary 3.31,Cat V S – RMod p Cat V S q “ Fun :{ RM p RM § , Ě Cat V§ S q . If C P Cat V S , let C ˚ : RM Ñ Ě Cat V S be the associated functor, and define Ě PSh V p C q to be the pullback Ě PSh V p C q / / (cid:15) (cid:15) Ě PSh V S (cid:15) (cid:15) RM C ˚ / / Ě Cat V S . Theorem 5.17. If V is a monoidal -category and C P Cat V S , then thefunctor Ě PSh V p C q Ñ RM is a cocartesian fibration of -operads, and theassociated RM-monoidal -category describes a right V -action on PSh V p C q . In other words, we have constructed PSh V p C q as a right V -module.53 roof. The identity functor RM Ñ RM is a cocartesian fibration of -categories corresponding to the (unique) action of the monoidal -category0 on itself. By Lemma 5.14, C ˚ sends all morphisms to p -cocartesian mor-phisms, and the composite p C ˚ : RM Ñ RM is the identity functor.Therefore, C ˚ is a functor of RM-monoidal -categories, so the pullback Ě PSh V p C q Ñ RM is also an RM-monoidal -category, exhibiting an actionof V ˆ – V on PSh V S ˆ Cat V S t C u “ PSh V p C q . Proposition 5.18.
Suppose C P Cat V S and X P S . Then1. Evaluation ev X : PSh V S Ñ V promotes to a right V -module functor;2. If a morphism A Ñ F p X q exhibits F P PSh V p C q as freely generated by A at X (Definition 3.23), then A b B Ñ F p X q b B – p F b B qp X q exhibits F b B as freely generated by A b B at X . (1) asserts that F b A can be ‘computed’, in that p F b A qp X q – F p X q b A .(2) asserts that tensor products of free presheaves are free. That is, the freepresheaf rep X b A (notation introduced in Section 3.3) is equivalent to thetensor presheaf rep X b A (in the sense of the right V -action on presheaves).In light of Proposition 5.18, Corollary 4.24 can be restated: Corollary 1.5. If V is presentable and closed monoidal and C is V -enriched,then PSh V p C q is generated under colimits and the right V -action by the rep-resentable presheaves rep X .Proof of Proposition 5.18. Consider the functor s ev X : Ě PSh V S “ Fun :{ LM p LM § S , ş V ! q Ñ ş V given by evaluation at p X, ˚q P LM S composed with ş V Ñ ş V . By Lemma5.14, s ev X sends p -cocartesian morphisms to p -cocartesian morphisms, where p : ş V Ñ RM, so the induced functor Ě PSh V S Ñ ş p V ; V q “ ş V ˆ Assoc
RM is anRM-monoidal functor ev X : p V ; PSh V S q Ñ p V ; V q . Moreover, the restrictionto Assoc Ď RM is the identity on V , so this describes a right V -module func-tor ev X : PSh V S Ñ V . It agrees with the functor we called ev X in Section3.3 because they are both given by evaluation at p X, ˚q P LM S .The second claim follows directly from Definition 3.23 and (1).54 orollary 5.19. Let s ev X : Ě PSh V p C q Ñ ş p V ; V q be the RM-monoidal functorassociated to ev X , guaranteed by Proposition 5.18(1). Then s ev X has a rightadjoint Ď rep X b ´ , which is also an RM-monoidal functor.Proof. By HA 7.3.2.7, s ev X has a right adjoint which is a map of -operadsover RM. (Intuitively, this is the statement that the right adjoint to anRM-monoidal functor is canonically a lax RM-monoidal functor.)To prove that Ď rep X b ´ is not just lax but fully RM-monoidal, it sufficesto prove that for each A, B P V , the map p rep X b A q b B Ñ rep X b p A b B q is an equivalence of presheaves. But this is true by Proposition 5.18(2). Remark 5.20.
That is, in the adjunction ev X : PSh V p C q Ô V : rep X b ´ ,both adjoints promote to right V -module functors, and in a compatible way. For an arbitrary monoidal -category V , we can’t do any better than con-struct the right V -action on PSh V p C q . However, if V is presentable, we willprove that PSh V p C q varies functorially in C . Theorem 5.21. If V is presentable and closed monoidal and C P Cat V S , thenPSh V p C q is a presentable right V -module. Together with Proposition 5.18(1), this establishes Theorem 1.3 from theintroduction.
Proof.
Since PSh V p C q is presentable (Theorem 4.19), we just need to provethat PSh V p C q ˆ V Ñ PSh V p C q preserves colimits independently in each vari-able. By Proposition 5.18(1), the following square commutes:PSh V p C q ˆ V b / / ev X (cid:15) (cid:15) PSh V p C q ev X (cid:15) (cid:15) V ˆ V b / / V . By Theorem 4.19 (and taking the upper composite around the square), itsuffices to show that for each X P S , the composite PSh V p C q ˆ V Ñ V preserves colimits independently in each variable. However, we know ev X preserves colimits (Theorem 4.19) and V ˆ V b ÝÑ V preserves colimits inde-pendently in each variable (since V is closed monoidal), so it follows thatthe lower composite preserves colimits in each variable separately. Since thesquare commutes, this completes the proof. Proposition 5.22. If V is presentable and closed monoidal, Ě PSh V S Ñ Ě Cat V S is a cocartesian fibration. emma 5.23. Suppose we have a commutative triangle of -categories X F / / G (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ Y H (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ Z such that G and H are cocartesian fibrations, and F sends G -cocartesianmorphisms to H -cocartesian morphisms. Further assume that for any a P Z , F a : X a Ñ Y a is a cocartesian fibration, and for any a Ñ b in Z , theinduced functor X a Ñ X b sends F a -cocartesian morphisms to F b -cocartesianmorphisms. Then F is a cocartesian fibration.Proof. This lemma is proven in a special case in HA 4.8.3.15, but the proofis general. We repeat it here. Let f : A Ñ B be a morphism in Y lyingover α : s Ñ t in Z , and let M P X be a lift of A . We are looking for an F -cocartesian lift M Ñ N of f .By assumption, there is a G -cocartesian morphism f : M Ñ M lifting α , which is therefore also F -cocartesian. Write f : A Ñ A for F p f q . Alsoby assumption, f is a cocartesian lift of α , so f factors A f ❆❆❆❆❆❆❆❆ A f / / f ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ B, for some f which projects to the identity morphism id t in Z . By assump-tion, there is an F t -cocartesian lift f : M Ñ N of f . Moreover, any t Ñ t in Z induces a functor X t Ñ X t which sends f to an F t -cocartesianmorphism; therefore, f is F -cocartesian.So f f : M Ñ N is a composite of F -cocartesian morphisms, andtherefore itself an F -cocartesian morphism (HTT 2.4.1.7) lifting f . Hence F is a cocartesian fibration. Proof of Proposition 5.22.
It suffices to prove that the commutative triangle Ě PSh V S ¯ θ / / p " " ❋❋❋❋❋❋❋❋ Ě Cat V Sp | | ②②②②②②②② RM56atisfies the hypotheses of the lemma. By Lemma 5.14, p and p are co-cartesian fibrations, and ¯ θ sends p -cocartesian morphisms to p -cocartesianmorphisms.Next, we show that ¯ θ Γ : p Ě PSh V S q Γ Ñ p Ě Cat V S q Γ is a cocartesian fibrationfor any Γ P RM. If Γ “ p˚ , q , then ¯ θ Γ is just θ : PSh V S Ñ Cat V S , which is acocartesian fibration by Corollary 4.21. If Γ “ p , q , then ¯ θ Γ is just V Ñ ˚ ,which is a cocartesian fibration (as is any functor to ˚ ). For general Γ, ¯ θ Γ isa product of copies of ¯ θ p˚ , q and ¯ θ p , q , which are all cocartesian fibrations.Now consider f : Γ Ñ Γ in RM. We are reduced to showing that f ˚ : p Ě PSh V S q Γ Ñ p Ě PSh V S q Γ sends ¯ θ Γ -cocartesian morphisms to ¯ θ Γ -cocartesianmorphisms. Every morphism in RM is a product of morphisms of thesetypes:1. inert morphisms p , q Ñ H and p˚ , q Ñ H ;2. H Ñ p , q ;3. p˚ , q b p , q Ñ p˚ , q .Therefore it suffices to prove just in these three cases that f ˚ sends co-cartesian morphisms to cocartesian morphisms. For (1) and (2) this is clearbecause p PSh V S q H “ ˚ . For (3), the claim reduces to the following:If F : C Ñ D is a map in Cat V S and A P V , then for every F P PSh V p C q ,the map η F : F ˚ p F b A q Ñ F ˚ p F q b A is an equivalence of presheaves on D .(In other words, we are claiming F ˚ : PSh V p C q Ñ PSh V p D q is compatiblewith the right V -module structures; this is the main content of the proof.)If F “ rep X b B for some B P V , then the claim is true because freepresheaves are sent to free presheaves by both the constructions F ˚ (Remark4.22) and ´ b A (Proposition 5.18(2)). Hence we have a natural transfor-mation η : F ˚ p´ b A q Ñ F ˚ p´q b A of functors PSh V p C q Ñ PSh V p D q , bothfunctors preserve colimits, and η is an equivalence at representables. SincePSh V p C q is generated under colimits by representables (Corollary 1.5), η isan equivalence at all presheaves. Therefore ¯ θ satisfies the hypotheses of thelemma, so it is a cocartesian fibration.Now we are ready to show that the right V -modules PSh V p C q are functorialin C . Recall from Definition 5.16 that Cat V S – Fun :{ RM p RM § , Ě Cat V§ S q . By57djunction, we have a functor RM ˆ Cat V S Ñ Ě Cat V S . Form the pullback Ě PSh V p Cat V S q / / p (cid:15) (cid:15) Ě PSh V S ¯ θ (cid:15) (cid:15) RM ˆ Cat V S / / Ě Cat V S . By Proposition 5.22, ¯ θ is a cocartesian fibration, and therefore p is alsoa cocartesian fibration. Moreover, each fiber p C : Ě PSh V p C q Ñ RM is acocartesian fibration of -operads by Theorem 5.17. In other words, p isa cocartesian Cat V S -family of RM-monoidal -categories in the sense of HA4.8.3.1. By Theorem 5.21, it is also compatible with small colimits in thesense of HA 4.8.3.4.Therefore p is classified by a functor Cat V S Ñ Mod RM p Pr L q , which sends C to PSh V p C q as a right V -module (Ha 4.8.3.20). The composite with theforgetful functor Mod RM p Pr L q Ñ Mod
Assoc p Pr L q is constant with value V ,so PSh V p´q factors through the fiber RMod V p Pr L q Ď Mod RM p Pr L q over V P Mod
Alg p Pr L q . To summarize: Corollary 5.24. If V is presentable and closed monoidal, then the cocarte-sian fibration p : Ě PSh V p Cat V S q Ñ RM ˆ Cat V S classifies a functorPSh V p´q : Cat V S Ñ RMod V p Pr L q . We will end this section by proving Theorem 1.6: that for any presentablepair p V ; M q , there is an equivalence of -categoriesΨ : PSh V p C q b V M Ñ PSh V p C ; M q . Informally, F b M corresponds to the presheaf which assigns X ÞÑ F p X qb M .The construction of Ψ is quite technical. However, the only properties of Ψthat we will use are: Lemma 5.25.
Suppose p V ; M q is a presentable pair and C is V -enriched.1. For any X P C , the following diagram commutes:PSh V p C q b V M Ψ / / ev X (cid:15) (cid:15) PSh V p C ; M q ev X (cid:15) (cid:15) V b V M M . . For any X P C and M P M , M Ñ C p X, X q b M – Ψ p rep X b M qp X q exhibits Ψ p rep X b M q as freely generated by M at X , in the sense ofDefinition 3.23. In other words, (1) asserts that Ψ p F b M qp X q – F p X q b M , and (2) assertsthat Ψ p rep X b M q is the free presheaf rep X b M .We will proceed with the proof of Theorem 1.6, conditional on Lemma5.25. Then, at the end of the section, we will construct the functor Ψ andprove Lemma 5.25.First, we need two lemmas concerning monadic functors which respecta module structure. Lemma 5.26.
Suppose F : X Ñ Y is a map in RMod V p Pr L q : that is, a V -module functor with a right adjoint. Also suppose F is monadic and itsleft adjoint is compatible with the V -module structure (as in Corollary 5.19).Then there is a monoidal -category E such that Y is an p E , V q -bimodule,and an algebra A P E such that there is an equivalence f of right V -module -categories making the triangle commute: X f / / F (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ LMod A p Y q y y tttttttttt Y . Proof.
Let E “ End V Mod p Y q , the monoidal -category of V -module functorsfrom Y to itself. Then Y P RMod V p Cat q is a left E -module, so therefore Y is an p E , V q -bimodule by HA 4.3.3.8.Moreover, F “ Fun
RMod V p X , Y q is a left E -module. If G denotes theleft adjoint to F , then A “ F G is an algebra in E , and F is a left A -module. Therefore, F factors X f ÝÑ LMod A p Y q Ñ Y , where f is a right V -module functor. By Barr-Beck (HA 4.7.3.16), f is an equivalence, becausethe induced morphism of monads A Ñ A is an equivalence by construction. Lemma 5.27. If F : X Ñ Y is a monadic right V -module functor as in thelast lemma, and M is a left V -module, X b V M Ñ Y b V M is also monadic.Proof. By the last lemma, we can assume without loss of generality that X “ LMod A p Y q and F is the forgetful functor, where A is an algebra insome E , such that Y is an p E , V q -bimodule. By HA 4.8.4.6, F factorsLMod A p Y q – LMod A p E q b E Y F ÝÑ E b E Y – Y . A p E q b E Y b V M Ñ Y b V M . This is equivalent (by HA 4.8.4.6 again) toLMod A p Y b V M q Ñ Y b V M , which is indeed monadic. Theorem 1.6. If M is a presentable left V -module, then Ψ :
PSh V p C ; M q Ñ PSh V p C q b V M is an equivalence.Proof (conditional on Lemma 5.25). Suppose C has set S of objects. Theproof is by Barr-Beck, following HA 4.8.4.6, which is the case | S | “
1. ByLemma 5.25(1), the following triangle commutes:PSh V p C q b V M Φ / / G & & ◆◆◆◆◆◆◆◆◆◆◆ PSh V p C ; M q G y y rrrrrrrrrr M S , where G and G are given by evaluation at each object in S (and G factorsthrough the equivalence V S b V M – M S ). We know G preserves smalllimits and colimits by Theorem 4.19(2), so it has a left adjoint F . It is alsoconservative by Corollary 4.20. Therefore, it is monadic. By Lemma 5.27, G is also monadic; call its left adjoint F .According to the Barr-Beck Theorem (HA 4.7.3.16-17), Φ is an equiv-alence if G, G are both monadic and the induced natural transformation G F Ñ GF of functors M S Ñ M S is an equivalence.We will prove this is true. Since G F and GF preserve colimits, it sufficesto check for each X, Y P S that G X F Y Ñ G X F Y is a natural equivalence offunctors M Ñ M , which is to say Φ p rep Y b M qp X q Ñ rep Y p X q b M is anequivalence for all M P M . This is true by Lemma 5.25(2).Finally, we will construct the functor Ψ and prove that it satisfies Lemma5.25. The construction follows HA 4.8.4.4, which is the case | S | “ emark 5.28. Let X “ p ş PSh V p C q ¸ BV q ˆ ş BV p ş BV ˙ M q as in Propo-sition 2.31, which is equipped with a cocartesian fibration r : X Ñ ∆ op . Weshould think of an object of X as a tuple p F , A , . . . , A n , M q P X n , where F P PSh V p C q , A i P V , and M P M .By Proposition 2.31, constructing Ψ :
PSh V p C q b V M Ñ PSh V p C ; M q is equivalent to constructing a functor X Ñ PSh V p C ; M q which sends r -cocartesian morphisms to equivalences and (Remark 2.32) such that the re-striction PSh V p C q ˆ V ˆ n ˆ M Ñ PSh V p C ; M q preserves colimits for each n .By Proposition 4.15, we have PSh V p C ; M q – Fun :{ ş BV § p ∆ op ;{ S , ş BV ˙ M ; q .Hence we need to construct a functor ψ : X ˆ ∆ op { S Ñ ş BV ˙ M such that:1. ψ sends r -cocartesian morphisms in the first coordinate to equiva-lences;2. ψ sends totally inert morphisms in the second coordinate to totallyinert morphisms;3. the following square commutes X ˆ ∆ op { S ψ / / (cid:15) (cid:15) ş BV ˙ M (cid:15) (cid:15) ∆ op { S C / / ş BV ;
4. for each n , PSh V p C q ˆ V ˆ n ˆ M Ñ PSh V p C ; M q preserves colimits. Let ‹ : p ş BV ˙ V q ˆ p ş BV ˙ M q Ñ ş BV ˙ M denote concatenation, definedby p A , . . . , A m q ‹ p A m ` , . . . , A n , M q “ p A , . . . , A n , M q . Define φ , φ tobe the functors (respectively) X ˆ ∆ op { S Ñ PSh V p C q ˆ ∆ op { S Ñ Fun p ∆ op { S , ş BV ˙ V q ˆ ∆ op { S Ñ ş BV ˙ V , X ˆ ∆ op { S Ñ X Ñ ş BV ˙ M , and φ “ φ ‹ φ , which is also a functor X ˆ ∆ op { S Ñ ş BV ˙ M . Explicitly,if T “ pp F , A , . . . , A n , M q , p X ă ¨ ¨ ¨ ă X m qq P X ˆ ∆ op { S , then φ p T q “ p C p X , X q , . . . , C p X m ´ , X m q , F p X m qq P ş BV ˙ V ,φ p T q “ p A , . . . , A n , M q P ş BV ˙ M , p T q “ p C p X , X q , . . . , C p X m ´ , X m q , F p X m q , A , . . . , A n , M q P ş BV ˙ M . By construction, the following diagram commutes: X ˆ ∆ op { S φ / / (cid:15) (cid:15) ş BV ˙ M q (cid:15) (cid:15) ∆ op { S ˆ ∆ op { S ‹ / / ∆ op { S , where p X ă ¨ ¨ ¨ ă X m q ‹ p X m ` ă ¨ ¨ ¨ ă X n q “ p X ă ¨ ¨ ¨ ă X n q . Call thecomposite ¯ φ : X ˆ ∆ op { S Ñ ∆ op { S . On the other hand, call ¯ ψ : X ˆ ∆ op { S Ñ ∆ op { S the projection onto the second coordinate. There is a canonical naturaltransformation ¯ φ Ñ ¯ ψ of inert morphisms, given essentially by inclusion X Ď X ‹ Y in ∆ op { S .We know q : Fun p X ˆ ∆ op { S , ş BV ˙ M q Ñ Fun p X ˆ ∆ op { S , ∆ op { S q is a cocarte-sian fibration since ş BV ˙ M Ñ ∆ op { S is a cocartesian fibration. Therefore,there is a q -cocartesian lift φ Ñ ψ of ¯ φ Ñ ¯ ψ , essentially defined by ψ p T q “ p C p X , X q , . . . , C p X m ´ , X m q , F p X m q b A b ¨ ¨ ¨ b A n b M q . Unpacking, ψ satisfies conditions (1)-(3) of Remark 5.28, and it also satisfiescondition (4) by Theorem 4.19(4), so there is an induced functorΨ : PSh V p C q b V M Ñ PSh V p C ; M q . We are finally ready to prove Lemma 5.25, which we restate here for refer-ence.
Lemma 5.25.
Suppose p V ; M q is a presentable pair and C is V -enriched.1. For any X P C , the following diagram commutes:PSh V p C q b V M Ψ / / ev X (cid:15) (cid:15) PSh V p C ; M q ev X (cid:15) (cid:15) V b V M M .
2. For any X P C and M P M , M Ñ C p X, X q b M – Ψ p rep X b M qp X q exhibits Ψ p rep X b M q as freely generated by M at X , in the sense ofDefinition 3.23. roof of Lemma 5.25. By construction, ev X ˝ Ψ corresponds to the functor X ˆ tp X qu Ď X ˆ ∆ op { S ψ ÝÑ ş BV ˙ M Ñ M ;this sends p F , A , . . . , A n , M q to F p X qb A b¨ ¨ ¨b A n b M , which is the tensorproduct of X Ñ PSh V p C q ev X ÝÝÑ V with X Ñ ş p V ; M q b ÝÑ M . Unpackingdefinitions, this implies (1).To prove (2), we just need to check Definition 3.23; that is, we need toprove that C p Y, X q b M Ñ Ψ p rep X b M qp Y q is an equivalence for all Y P C .But this is true by (1). In ordinary category theory, we have a Yoneda embedding Y : C Ñ PSh p C q ,which exhibits PSh p C q as freely generated by C under colimits.In other words, if D is a presentable category, then restriction along Y induces an equivalence of categoriesFun L p PSh p C q , D q Ñ Fun p C , D q . In this section, we will prove the analogous statement for enriched -categories. We will always assume the enriched -category V is presentableand closed monoidal.Because PSh V p C q is a right V -module, and not a priori a V -enriched cat-egory, we will not want to speak of V -enriched functors to D , but rather V -enriched copresheaves with values in D . (We introduced enriched co-presheaves in Section 5.1.)We will construct the Yoneda embedding in the guise of a copresheaf Y P coPSh V p C ; PSh V p C qq , and then we will prove: Theorem 6.1. If N is a presentable right V -module, and F : PSh V p C q Ñ N is a colimit-preserving right V -module functor, let Y ˚ p F q denote the push-forward of Y along F ˚ : coPSh V p V ; PSh V p C qq Ñ coPSh V p V ; N q . Then Y ˚ : Fun L RMod V p PSh V p C q , N q Ñ coPSh V p V ; N q is an equivalence of -categories. This theorem is essentially equivalent to Theorem 1.7 from the introduction,that PSh V p C q and coPSh V p C q are dual V -modules. We will conclude Theo-rem 1.7 at the end of this section. 63e begin by constructing the Yoneda copresheaf.Identify the V -enriched category C with a marked functor Assoc § S Ñ ş V § ,and consider the compositeLM § S ˆ RM § S x´ , ´y ÝÝÝÝÑ
Assoc § S > S ∇ ÝÑ Assoc § S C ÝÑ ş V § , where ∇ : S > S Ñ S is the identity on each component (the codiagonal),and x´ , ´y is the pairing of Section 5.2. Because LM S ˆ RM S Ñ Assoc S isnatural in S , the following diagram commutes:LM S ˆ RM S ∇ x´ , ´y / / (cid:15) (cid:15) Assoc S C / / % % ❏❏❏❏❏❏❏❏❏❏ ş V (cid:15) (cid:15) LM ˆ RM ∇ x´ , ´y / / Assoc . Therefore, there is an induced marked functorLM § S ˆ RM § S Ñ ş V § “ ş V ˆ Assoc p LM ˆ RM q . By adjunction, we have Y : RM S Ñ Ě PSh V S “ Fun :{ LM p LM § S , ş V ! q , which iscompatible with the functors down to RM:RM S Y / / ●●●●●●●●● Ě PSh V Sp (cid:15) (cid:15) RM . Moreover, Y sends inert morphisms to p -cocartesian morphisms by Lemma5.14, so therefore it describes a copresheaf. Since Ě PSh V S – ş p V ; PSh V S q byLemma 5.15, Y P coPSh V ;PSh V S S . We will see shortly: • that the underlying enriched category is C , so that Y P coPSh V p C ; PSh V S q ; • that the copresheaf evaluated at any object of C is an enriched presheafon C , so that Y P coPSh V p C ; PSh V p C qq .We will regard Y as the enriched Yoneda embedding. Now we prove the twopoints in the next two lemmas; they amount to the observations that tp X, ˚qu ˆ Assoc S Ď LM S ˆ RM S ∇ x´ , ´y ÝÝÝÝÝÑ
Assoc S , Assoc S ˆ tp˚ , X qu Ď LM S ˆ RM S ∇ x´ , ´y ÝÝÝÝÝÑ
Assoc S are each the identity functor by construction of x´ , ´y .64 emma 6.2. Y is a copresheaf on C ; that is, Y P coPSh V p C ; PSh V S q .Proof. The underlying enriched category of Y is described by the restrictionAssoc S Ď RM S Y ÝÑ Ě PSh V S . Since Ě PSh V S – ş p V ; PSh V S q , this composite is justa V -enriched category.For any object X P S , recall that ev X : PSh V S Ñ V is a right V -modulefunctor, described explicitly (as in the proof of Proposition 5.18) as s ev X : Ě PSh V S “ Fun :{ LM p LM § S , ş V ! q Ñ ş V , which is evaluation at p X, ˚q P LM S . As a right V -module functor, s ev X restricts to an equivalence on Ě PSh V S ˆ RM Assoc – ş V .In particular, this means that the compositeAssoc S Ď RM S Y ÝÑ Ě PSh V S Ď ev X ÝÝÑ ş V , which is a priori a V -enriched category, actually recovers the underlying V -enriched category of the copresheaf Y (regardless of which X P S waschosen).By construction of Y , this composite is also tp X, ˚qu ˆ Assoc S Ď LM S ˆ RM S x´ , ´y ÝÝÝÝÑ
Assoc S C ÝÑ ş V , but tp X, ˚qu ˆ Assoc S Ñ Assoc S is the identity by construction of x´ , ´y , sothis composite recovers the enriched category C , completing the proof. Lemma 6.3.
The following square commutes:RM S Y / / (cid:15) (cid:15) Ě PSh V S ¯ θ (cid:15) (cid:15) RM C ˚ / / Ě Cat V S . Proof.
Since Ě Cat V S – ş p
0; Cat V S q , each composite RM S Ñ Ě Cat V S is a co-presheaf in coPSh V S S . Each evaluation map ev X : coPSh V S S Ñ Cat V S isan equivalence by Corollary 3.31.Therefore, it suffices to show (for a single chosen X P S ) that the twocomposites RM S Ñ Ě Cat V S are equivalent when evaluated at p˚ , X q P RM S .65he lower composite evaluated at p˚ , X q is C , by construction, while theupper composite evaluated at p˚ , X q is the enriched categoryAssoc S ˆ tp˚ , X qu Ď LM S ˆ RM S x´ , ´y ÝÝÝÝÑ
Assoc S C ÝÑ ş V . As in the proof of the last lemma, the composite Assoc S ˆtp˚ , X qu Ñ Assoc S is the identity, so this enriched category is also C , completing the proof. Remark 6.4.
By Lemma 6.3, Y factors through a marked functor of theform RM S Ñ Ě PSh V p C q “ Ě PSh V S ˆ Ě Cat V S RM, which is to say a copresheaf withvalues in PSh V p C q . By Lemma 6.2, the underlying enriched category of thiscopresheaf is C , so that we have a Yoneda copresheaf Y P coPSh V p C ; PSh V p C qq . Now we will prove Theorem 6.1, that the functor Y ˚ : Fun L RMod V p PSh V p C q , N q Ñ coPSh V p V ; N q is an equivalence of -categories. Proof of Theorem 6.1.
The proof is by Barr-Beck, following HA 4.8.4.1 andvery similar to Theorem 1.6. As S P X varies, the right V -module functorsrep X b´ : V Ñ PSh V p C q assemble into a V -module functor V ˆ S Ñ PSh V p C q .Precomposition with this functor inducesFun L RMod V p PSh V p C q , N q Ñ Fun L RMod V p V ˆ S , N q – N S , and the following triangle commutes (where T denotes evaluation at each X P S ) Fun L RMod V p PSh V p C q , N q Y ˚ / / T ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗ coPSh V p C ; N q T x x rrrrrrrrrrr N S , because Y ˚ p F qp X q – F p rep X q . We proved T is monadic in the proof ofTheorem 1.6 (end of Section 5). We claim T is also monadic.Indeed, T preserves colimits by construction (as it is a map of Pr L ), and ithas a left adjoint U by Corollary 5.19, which is given by precomposition withev : PSh V p C q Ñ V ˆ S . To finish the proof that T is monadic, we need onlyshow T is conservative. Let F, G : PSh V p C q Ñ N be two right V -module66unctors and η : F Ñ G a natural transformation such that T p η q is anequivalence. In other words, η is an equivalence at rep X for all X P S . Butsince PSh V p C q is generated by rep X as a right V -module in Pr L (Corollary1.5), it follows that η is an equivalence everywhere.Therefore, T and T are monadic. By HA 4.7.3.16-17, to complete theproof that Y ˚ is an equivalence, it suffices to show that the induced naturaltransformation T U Ñ T U of functors N S Ñ N S is an equivalence, where U, U are the left adjoints to T , respectively T .Since T U and T U preserve colimits, it suffices to check for each
X, Y P S that T X U Y Ñ T X U Y is a natural equivalence of functors N Ñ N . But boththese functors are given by ´ b C p Y, X q , and unpacking, the map is anequivalence. This completes the proof.Finally, we will conclude Theorem 1.7, that PSh V p C q P RMod V p Pr L q is leftdual to coPSh V p C q P LMod V p Pr L q (in the sense of HA 4.6.2.3). Proof of Theorem 1.7.
We have coPSh V p C ; PSh V p C qq – PSh V p C qb V coPSh V p C q by Theorem 1.6, so we may regard Y as an object of this tensor product, ora colimit-preserving functor ´ b Y : Top Ñ PSh V p C q b V coPSh V p C q . Here the -category Top of spaces is the unit of the monoidal structure onPr L . By HA 4.6.2.18, we just need to prove that for each D P Pr L and N P RMod V p Pr L q , the compositeiFun L RMod V p D b PSh V p C q , N q ´b V coPSh V p C q (cid:15) (cid:15) iFun L p D b PSh V p C q b V coPSh V p C q , N b V coPSh V p C qq Y (cid:15) (cid:15) iFun L p D , N b V coPSh V p C qq is an equivalence of spaces ( -groupoids), where iFun denotes the maximalsubgroupoid of the functor -category. However, we have equivalencesFun L RMod V p D b PSh V p C q , N q Ñ Fun L p D , coPSh V p C ; N qq , Fun L p D , N b V coPSh V p C qq Ñ Fun L p D , coPSh V p C ; N qq , the first by Theorem 6.1 and the second by Theorem 1.6, and (unpacking)they are compatible with the functor above. This establishes duality.67 eferences [1] C. Barwick. On the algebraic K-theory of higher categories. Journal ofTopology -categories. arXiv: 2006.10851 (2020).Preprint.[3] J. Berman. THH and traces of enriched categories. Int. Math. Res. Not.IMRN , to appear.[4] T. Elmendorf, I. Kriz, M.A. Mandell, and J.P. May. Rings, modules,and algebras in stable homotopy theory.
Mathematical surveys andmonographs
47, AMS, Providence, RI (1997).[5] D. Gepner, M. Groth, and T. Nikolaus. Universality of multiplicative in-finite loop space machines. Algebr. Geom. Topol. 15: 3107-3153 (2015).[6] D. Gepner and R. Haugseng. Enriched -categories via non-symmetric -operads. Adv. Math. , 279:575-716 (2015).[7] V. Hinich. Yoneda lemma for enriched infinity categories. arXiv:1805.07635 (2018). Preprint.[8] M. Hoyois, S. Scherotzke, and N. Sibilla. Higher traces, noncommu-tative motives, and the categorified Chern character.
Adv. Math.
Ann. of Math. Stud. „„