aa r X i v : . [ m a t h . C T ] M a r YONEDA LEMMA FOR ENRICHED ∞ -CATEGORIES VLADIMIR HINICH
Abstract.
We continue the study of enriched ∞ -categories, using a definitionequivalent to that of Gepner and Haugseng. In our approach enriched ∞ -categories are associative monoids in an especially designed monoidal categoryof enriched quivers. We prove that, in the case where the monoidal structure inthe basic category M comes from the direct product, our definition is essentiallyequivalent to the approach via Segal objects. Furthermore, we compare ournotion with the notion of category left-tensored over M , and prove a versionof Yoneda lemma in this context. We apply the Yoneda lemma to the studyof correspondences of enriched (for instance, higher) ∞ -categories. Introduction
Overview.
The objective of this paper is the study of enriched ∞ -categories,examples of which include A ∞ -categories, DG categories and higher ∞ -categories.Enriched ∞ -categories abound in derived algebraic geometry, mirror symme-try, and so on. However, category theory is unthinkable without the Yonedalemma and this is what is lacking in the existing approaches .Our approach to the notion of enriched category is based on the followingobservation. It is clear how to define an M -enriched ∞ -category with one objectfor a monoidal ∞ -category M ; this is just an associative algebra in M . Thismeans that one can expect to define enriched ∞ -categories as associative algebraobjects in a certain category of enriched quivers.Our definition of enriched category is very close to that of Gepner-Haugseng [GH] . The definitions are equivalent, but we work with enriched categories slightlydifferently. The details are below.1.1.1. Given a space X , Gepner and Haugseng construct a planar operad whichwe denote as Ass X such that Ass X -algebras in a monoidal category M are pre-cisely enriched M -(pre)categories with the space of objects X .Our definition is based on the observation that direct product with a flatplanar operad (see Definition 2.8.1) admits a right adjoint which we will denoteas Funop Ass in this paper. The planar operad
Ass X is flat, so this allows us to See, however, [GR], Appendix, for the Yoneda lemma for ( ∞ , The paper [Hau2] of R. Haugseng contains a basically equivalent approach to enrichedcategories via a monoidal structure on enriched quivers, see 4.1 op. cit. define a planar operad Funop
Ass ( Ass X , M ) which is the operad of M - quivers withthe space of objects X . Enriched precategories are just associative algebras in it.1.1.2. The above construction makes sense for any planar operad M and forany category X . It gives a monoidal category if M is a monoidal category havingenough colimits. The definition is equivalent to that of [GH] when X is a space.1.1.3. Simultaneously with defining a planar operad of quivers, we define its(weak) action on the category Fun( X, M ) compatible with the right M -action.In the most general form, our construction gives, for any ( ∞ -) category X andany BM -operad M , a BM -operad Quiv BM X ( M ).This is important for several reasons. First, it turns out that, under mild re-strictions on M and X , the monoidal category of quivers can be defined as anendomorphism object for the right M -module Fun( X, M ). We use this character-ization to compare the notion of enriched category with the Segal-type definition,in the case where the monoidal structure in M is given by direct product.Second, we use this extended construction to define M -functors from an en-riched M -precategory A to a left M -module B . This latter is the basis for ourapproach to the Yoneda lemma, described for the convential enriched categoriesin the short note [H.Y]. It is also a basis of the construction assigning, whenpossible, an M -enriched category to a left M -module.The detailed content of the sections is presented below.1.2. ∞ -categories. In this paper we use a model-independent language of ∞ -categories. The idea of this approach is that, once we understand the ∞ -categorical Yoneda lemma, we can reformulate all the theory without mentioninga specific model.Section 2 of the present paper is devoted to developing this language, and so,most of it consists of well-known results and constructions.In it we reformulate in the model-independent language the standard notionsof theory of ∞ -categories (which can be found in [L.T]) as well as the languageof operads from [L.HA].In this section we sketch the basic notions of the language of infinity categories(left fibrations, cocartesian fibrations, the Yoneda lemma, localizations) and thelanguage of operads (operads, operads over a given operad, approximation, flatoperads, internal mapping object, tensor product). Some of the notions differslightly from their original version in [L.HA].We also present a number of operads and their approximations important forthe present work.On the technical level, we reevaluate the role of the conventional category ∆of finite totally ordered sets in the theory of infinity categories. This category This means, a pair of planar operads weakly acting on the left and on the right on a categoryin a compatible way. usually appears in a model category (of simplicial or bisimplicial sets) describinginfinity categories. We use it in the identification of the ∞ -category Cat as a(Bousfield) localization of the ∞ -category of simplicial spaces.In this way most of the constructions with infinity categories (for instance, theopposite category C op or the twisted arrows category Tw( C )) can be describedpurely in terms of conventional categories (by functors ∆ → ∆ or similar).Note a few places of Section 2 containing a less standard material. In Sec-tion 2.10 we develop a notion of tensor product of operads slightly more generalthat the notion described in [L.HA], 3.2.4. In Section 2.12 we study operadicsieves which describe a nice class of morphisms of operads leading to a cartesianfibration of the corresponding categories of operad algebras. In 2.8.9 we presentan explicit description of the internal Hom in operads, Funop C ( P , Q ), in the casewhen P and Q are C -monoidal categories.1.3. Quivers.
In Section 3 we present the construction of the categories of en-riched quivers. There are different versions of the construction, the most generalamong them assigns to an ∞ -category X (of objects) and to a BM -operad M , a BM -operad Quiv BM X ( M ). The construction is represented by a (strict) functor as-signing to any simplex σ : [ n ] → BM a poset F BM ( σ ) describing the combinatoricsof compositions of arrows in a category. This allows us to define, for any X ∈ Cat ,a BM -operad denoted BM X , as a functor (∆ / BM ) op → S , carrying σ : [ n ] → BM tothe space Map( F BM ( σ ) , X ). We also verify that for any X the BM -operad BM X isflat.In Section 4 we study conditions on M and X that ensure that the enrichedquivers form a monoidal category. Roughly speaking, M is required to be monoidal,with the tensor product commuting with enough colimits (with respect to thesize of X ), see Theorem 4.4.8. We also provide the description of Quiv X ( M ) asthe category of endomorphisms of Fun( X, M ), see Proposition 4.5.3.1.4. The case where M has a cartesian monoidal structure. In Section 5we compare our notion of enriched precategory over M with the existing Segal-type definition, in the case where the monoidal structure on M comes from thedirect product.Here we proceed as follows. To any category M with fiber products we assigna family of BM -monoidal categories, cartesian over M × M . Its fiber at ( X, Y ) ∈ M × M consists of a pair of monoidal categories, M /X × X and M /Y × Y , acting onthe left and on the right on the category M /X × Y .This construction yields a monoidal structure on M /X × X ; associative algebrasin M /X × X identify with the simplicial objects A : ∆ op → M satisfying Segalcondition and having A = X . VLADIMIR HINICH
In Section 5.4 we define prototopoi, categories with fiber products satisfyingsome weakened topos properties. In particular, topoi, as well as the categories of( ∞ , n ) categories, are prototopoi.Finally, in 5.5, in the case when M is a prototopos and X is in the im-age of the standard colimit preserving functor S → M , we identify the triple( M /X × X , M /X , M ) with Quiv BM X ( M ). This implies that enriched precategories inthis case are equivalent to simplicial objects A : ∆ op → M , satisfying the Segalcondition and having A in the image of S , see 5.6.1. In proving this result weuse the description of Quiv X ( M ) as the endomorphism object of Fun( X, M ) andthe full BM -category structure on the quivers.1.5. The Yoneda Lemma.
In Section 6 we develop the notion of enrichedpresheaves and prove a version of the Yoneda lemma. Let us try to imaginewhat a Yoneda lemma could mean for enriched categories. Let M be a monoidalcategory and A be an M -enriched precategory. Enriched preshaves should beenriched functors F : A op → M . Note that A op is an enriched precategory over M rev , the monoidal category with reverse multiplication. On the other hand, itis not at all obvious that M is enriched over M rev (or over M ).These problems can already be seen in the conventional setting. As we showedin [H.Y], they disappear if one carefully distingushes between two different typesof enrichment: M is not necessarily enriched over itself, but it is definitely a left(and a right) M -module.The basis of our approach to the Yoneda lemma is a notion of an M -functor F : A → B from an M -enriched precategory A to a left M -module B . Let A ∈ Quiv X ( M ). An M -enriched functor F : A → B is given by a functor f : X → B , together with some extra data. The formalism of quiver categoriesprovides an action of the monoidal category Quiv X ( M ) on the category of functorsFun( X, B ). The extra data on f : X → B is precisely the A -module structureon f , see 6.1.3.The notion of M -functor described above is exactly what is needed for theYoneda lemma. Any associative algebra gives rise to a bimodule in an appropriatesense. Applying this general principle to an associative algebra A in Quiv X ( M ),we get a bimodule which can be interpreted as an M × M rev -functor from A ⊠ A op to M , or, even better, as an M -functor from A to the category of M -presheaves P M ( A ) = Fun M rev ( A op , M ). The Yoneda lemma 6.2.7 claims that this M -functoris fully faithful in an appropriate sense.The passage from A - B -bimodules to modules over the tensor product A ⊗ B op that we used in the above explanation seems very plain; but it is not competelyobvious in Higher Algebra. The corresponding general construction is presentedin 3.6. The construction is presented as a “folding functor” carrying a BM -operadwith components ( P a , P m , P b ) to an LM -operad with components ( P a × P rev b , P m ). M -functor allows one, when possible, to convert aleft M -module B into an M -enriched category. In general, for any pair of objects x, y ∈ B , the left M -module structure on B gives rise to a presheaf hom B ( x, y )(in the usual, non-enriched sense) on M . Given a functor F : X → B such thatfor any x, y ∈ X hom B ( F ( x ) , F ( y )) is representable, we can construct a universal M -morphism A → B where A is an M -enriched precategory with the categoryof objects X .1.5.2. Completeness.
According to 1.4, the notion of S -enriched precategory witha space of objects X , is equivalent to the notion of Segal space with the space ofobjects X . This means that properly defined enriched categories should take intoaccount a version of the completeness condition. This issue is already addressedin [GH]. We use the Yoneda lemma to present an alternative construction of thecompletion functor [GH], 5.6.Choosing B = P M ( A ) and X the subspace of representable M -presheaves on A ,we get a universal arrow A → E from an M -enriched precategory to a complete M - enriched precategory, see 7.2.1.6. Correspondences.
In Section 8 we apply the developed technique to study-ing correspondences between enriched ∞ -categories. Let M be a monoidal cate-gory with colimits, C and D two M -enriched categories. A correspondence from C to D can be defined as an M -functor D → P M ( C ). We prove that, in thecase where M is a prototopos endowed with a cartesian monoidal structure (forinstance, if M is the category of ( ∞ , n )-categories), then the category of M -correspondences is equivalent to Cat ( M ) / [1] . This result, for ( ∞ , ∞ , Acknowledgment.
We are grateful to Nick Rozenblyum who informed usabout the work [GH] at very early stages of the work. Our work does not relyon [GH], and can be read completely independently. This led to a certain overlap,which the author has not found a way to avoid.A part of this work was done during the author’s stay at Radboud Universityat Nijmegen, Utrecht University, MPIM and UC Berkeley. The author is verygrateful to these institutions for the excellent working conditions. Numerousdiscussions with Ieke Moerdijk were very helpful. John Francis’ advice for aproof of Proposition 3.6.7 is appreciated. We are very grateful to R. Haugsengwho pointed out an error in the first version of the manuscript. We are also verygrateful to the anonymous referee who has made an incredible job trying to makethe manuscript better. The work was supported by ISF 446/15 grant.
VLADIMIR HINICH The language of ∞ -categories Introduction.
Throughout this paper a language of ( ∞ , ∞ , ∞ , Generalities.
We will try to make our usage of ( ∞ , ∞ -category of ∞ -categories which we will simply denote by Cat . This means that we will notuse the usual categorical notions of fiber product, coproduct, or more generallimits and colimits — but replace them with corresponding ∞ -categorical no-tions. Since a Quillen equivalence of model categories induces an equivalence ofthe underlying ∞ -categories, one can use any existing model for ∞ -categories toprove claims formulated in this model-independent language.In particular, we will use the notions of left, right, cartesian or cocartesianfibration in a sense slightly different from the one defined in [L.T]. For instance,our left fibrations are arrows in Cat representable by a left fibration in sSet inthe sense of [L.T], Ch. 2. The notion of cocartesian fibration in
Cat has a similarmeaning: this is an arrow in
Cat which is equivalent to a cocartesian fibration inthe sense of [L.T], Ch. 2.We will use the following standard notation throughout the paper. S is the ∞ -category of spaces, Cat is the ∞ -category of ∞ -categories.In what follows we will use the word “category” instead of “ ∞ -category”, and“conventional category” instead of “category”.2.2.1. Basic vocabulary.
In what follows we will use the following notation.The category
Cat has products, and is closed, that is, it has internal mappingobject denoted Fun(
C, D ) or D C . The spaces form a full subcategory S of Cat .The embedding S → Cat has a right adjoint functor (of maximal subspace) and aleft adjoint functor (of total localization). For a category C and x, y ∈ C a spaceMap C ( x, y ) of maps from x to y is defined, canonically “up to a contractible spaceof choices”.Any category C defines a conventional category π ( C ) and a canonical functor C → π ( C ). The conventional category π ( C ) has the same objects as C and theset Hom π ( C ) ( x, y ) defined as π (Map C ( x, y ). An arrow f ∈ Map C ( x, y ) is calledequivalence if its image in π ( C ) is an isomorphism.We will write sometimes x = y for a natural equivalence between two objectsof a category. For a category C the maximal subspace of C is denoted C eq . It is obtained from C by “discarding all arrows which are not equivalences”.2.2.2. Subcategory.
Note from the very beginning that our notion of subcategorydoes not generalize the conventional notion — the latter is not invariant underequivalences.For X ∈ S a subspace of X is a morphism Y → X which is an equivalence toa union of connected components of X .A subcategory D of C ∈ Cat is a morphism f : D → C in Cat defining, forany A ∈ Cat , the space Map( A , D ) as a subspace of Map( A , C ). In particular,the maximal subspace D eq of D is a subspace of C eq and for each pair of objects x, y ∈ D the space Map D ( x, y ) is a subspace of Map C ( f ( x ) , f ( y )). This impliesthat a subcategory D ⊂ C is uniquely defined by a (conventional) subcategory D in the category π ( C ) as the fiber product(1) D −−−→ C y y D −−−→ π ( C ) , where the embedding D → π ( C ) satisfies the additional property saying that itinduces an injective map on the sets of isomorphism classes of objects. In thiscase one has D = π ( D ).Vice versa, any subcategory D as above of π ( C ) defines a subcategory D suchthat the diagram (1) is cartesian.2.2.3. Left fibrations.
A map f : C → B in Cat is called a left fibration if themap C [1] → B [1] × B C induced by the embedding [0] = { } → [1] is an equivalence. Note that the abovedefinition is invariant under equivalences in Cat .Grothendieck construction for left fibrations yields a canonical equivalence
Left ( B ) → Fun( B, S ), where Left ( B ) denotes the full subcategory of Cat /B spanned by the left fibrations over B .2.2.4. The Yoneda lemma.
Let C ∈ Cat . The assignment ( x, y ) Map C ( x, y ) isfunctorial. This means one has a canonical functor Y : C op × C → S . It can beotherwise presented by a left fibration(2) Tw( C ) → C op × C VLADIMIR HINICH corresponding, via Grothendieck construction, to Y , see [L.HA], 5.2.1 . Later inthis paper we will use the opposite right fibration(3) Tw( C ) op → C × C op classified by the same functor Y .Yoneda embedding Y : Cat → Fun(
Cat op , S ) restricted to the subcategory∆ ⊂ Cat , yields a fully faithful embedding(4)
Cat → Fun(∆ op , S ) . In terms of this embedding the category Tw( C ) is defined as the composition C ◦ τ where C in this formula is interpreted as a simplicial object in S , and τ : ∆ → ∆is the functor carrying a finite totally ordered set I to the join I op ⋆ I .In general, for C ∈ Cat , we define P ( C ) = Fun( C op , S ). This is the category ofpresheaves (of spaces) on C . The functor Y : C op × C → S can be rewritten asa funtor Y : C → P ( C ) which is fully faithful. This is the ∞ -categorical versionof the classical Yoneda lemma. The category P ( C ) can be otherwise interpretedas the category of right fibrations X → C . The right fibration correspondingto Y ( x ), x ∈ C , is the forgetful functor C /x → C , where the overcategory C /x is defined as the fiber product C [1] × C { x } , with the map C [1] → C defined by { } → [1].2.2.5. Cocartesian fibrations.
The left fibration (2) is functorial in X . This im-plies that, given f : X → B in Cat , any arrow a : [1] → X with a (0) = x, a (1) = y gives rise to a commutative diagram(5) X y/ / / (cid:15) (cid:15) X x/ (cid:15) (cid:15) B f ( y ) / / / B f ( x ) / An arrow a in X is called f -cocartesian if the diagram (5) is cartesian. Themap f : X → B is called a cocartesian fibration if for any x ∈ X and for any¯ a : f ( x ) → b there exists an f -cocartesian arrow a : x → y lifting ¯ a .The category Coc ( B ) of cocartesian fibrations over B is defined as follows. Thisis a subcategory of Cat /B . It is spanned by the cocartesian fibrations over B . Amorphism of cocartesian fibrations over B is in Coc ( B ) iff it carries cocartesianarrows to cocartesian arrows.Grothendieck construction for left fibrations extends to cocartesian fibrations.It yields a canonical equivalence(6) Coc ( B ) → Fun( B, Cat ) . Note that we use the different convention, as in [L.HA], 5.2.1, the functor Y is encoded bya right fibration. The adjoint pair i : S −→←− Cat : K , with i the obvious embedding and K given bythe formula K ( C ) = C eq , extends to the adjoint pair i : Left ( B ) −→←− Coc ( B ) : K ,with i the embedding and K ( C ) defined as the subcategory of C spanned by thecocartesian arrows.2.2.6. Cartesian fibrations and bifibrations.
An arrow f : X → B in Cat is calleda cartesian fibration if the arrow f op : X op → B op is a cocartesian fibration. ViaGrothendieck construction cartesian fibrations over B correspond to functors B op → Cat . Sometimes a “mixed” Grothendieck construction is more appropri-ate: a functor B op × C → Cat can be converted into f : X → B × C which iscartesian over B and cocartesian over C . We call such maps bifibrations .Here are the details. A functor f : B op × C → Cat is the same as a func-tor B op → Fun( C, Cat ) =
Coc ( C ). Composing this with the forgetful func-tor Coc ( C ) → Cat , we get a contravariant functor from B to Cat which, byGrothendieck construction, converts to a cartesian fibration p : X → B . Theconstant functor B op → Cat with value C converts by the Grothendieck con-struction to the projection B × C → C . This yields a decomposition of p as X → B × C → B .In the opposite direction, given a map f : X → B × C such that its compositionwith the projection to B is a cartesian fibration, we get a map B op → Cat /C . Ifthe image of this map belongs to the subcategory Coc ( C ) of Cat /C , this definesa functor B op × C → Cat .2.2.7.
Correspondences. Adjoint functors.
Given a pair of categories C , D , a cor-respondence from C to D is a left fibration p : E → C op × D .A correspondence is called left-representable if for each x ∈ C the base changeof p with respect to D → C op × D determined by x , defines a representablepresheaf on D op .A correspondence p : E → C op × D is called right-representable if for each y ∈ D the base change of p with respect to the morphism C op → C op × D , determinedby y , corresponds to a representable presheaf on C .A left-representable correspondence comes from a unique (up to usual ambigu-ity) functor C → D . A right-representable correspondence comes from a uniquefunctor D → C .A left fibration π : E → C op × D that is both left and right representable,determines a pair of functors, F : C → D and G : D → C . In this case F is calledleft adjoint to G and G is called right adjoint to F . Since F and G are defined,uniquely up to equivalence, by φ , and, vice versa, φ is defined by any one of F , G , an adjoint functor, if it exists, is unique. Lurie [L.T] defines bifibration as the maps f : X → B × C corresponding to functors B op × C → S . f : C → D defines an adjoint pair f ! : P ( C ) −→←− P ( D ) : f ∗ , with f ! being the colimit-preserving functor defined by the composition of f withthe Yoneda embedding Y : D → P ( D ), and f ∗ is defined by restriction of presheafto C . The functor f ∗ is also colimit-preserving.2.3. Flat morphisms in
Cat . A morphism f : C → D in Cat is called flat ifthe pullback f ∗ : Cat / D → Cat / C admits a right adjoint.If f : C → D is flat, f ∗ : Cat / C → Cat / D denotes the functor right adjoint to f ∗ . For X ∈ Cat / C , the fiber of f ∗ ( X ) at d ∈ D is given by the formula(7) f ∗ ( X ) d = Fun C ( C d , X ) . In this form the definition belongs to [AFR], A.3, where the term exponentiable is used instead of flat .Note that, if f : C → D is flat, an internal mapping object Fun Cat / D ( C , X ) isdefined for any X ∈ Cat / D by the formula Fun Cat / D ( C , X ) = f ∗ ( f ∗ ( X )).Flat morphisms were introduced by J. Lurie in [L.HA], B.3, and were calledthere flat categorical fibrations . The following characterizations of flat morphismscan be found in [L.HA], B.3.2 and [AFR], A.16.2.3.1. Proposition. A map f : C → D is flat iff for any s : [2] → D the base change C × D [2] → [2] is flat. A functor p : C → [2] is flat iff the natural map C { , } ⊔ C C { , } → C where C i , resp., C { i,j } , are defined by base change of C with respect to { i } → [2] , resp., { i, j } → [2] , is an equivalence. A functor p : C → [2] is flat iff for any arrow f : A → C in C , with p ( A ) = { } , p ( C ) = { } , the full subcategory of Fun [2] ([2] , C ) spanned by s : [2] → C such that d ( s ) = f , is weakly contractible. (cid:3) In this paper we will use an interpretation of flatness in terms of correspon-dences.Let f : C → [1] be a functor with fibers C , C at 0 and 1. We denote by φ i : C i → C the embeddings of the fibers.The composition P ( C ) φ → P ( C ) φ ∗ → P ( C ) is a colimit-preserving functor defin-ing a correspondence from C to C .Given f : C → D , the base change along any arrow α : d → d ′ in D givesrise, therefore, to a colimit preserving functor f α : P ( C d ′ ) → P ( C d ) between the presheaves on the fibers. This assignment, however, is not necessarily functorial.We will show that it is functorial precisely when f is flat.Let f : C → [2]. The maps φ i : C i → C and φ i,j : C { i,j } → C are fully faithful,so that the unit of adjunction id → φ ∗ ◦ φ ! is an equivalence for φ = φ i , φ i,j . Thefunctor f defines correspondences f : P ( C ) → P ( C ) and f : P ( C ) → P ( C )whose composition is given as P ( C ) φ → P ( C ) φ ∗ → P ( C ) φ → P ( C ) φ ∗ → P ( C ) . Applying the counit of adjunction φ ◦ φ ∗ → id, we get a morphism of functors(8) f ◦ f → f . We have2.3.2.
Proposition.
A map f : C → [2] is flat iff the morphism of the functors(8) from P ( C ) to P ( C ) is an equivalence.Proof. We choose A ∈ C , C ∈ C , and evaluate both functors P ( C ) → P ( C )at C (getting a morphism of presheaves on C ) and at A (getting a morphismof spaces). We have to verify when this morphism of spaces is an equivalence.The target is just Map C ( A, C ). Let us calculate the source. Denote by Y C thepresheaf in P ( C ) represented by C ∈ C ⊂ C . It is defined by the right fibration p C : C /C → C , so φ ∗ ( Y C ) is defined by the right fibration ( C ) /C → C defined asthe base change of p C with respect to the embedding C → C . Thus, φ ∗ ( Y C ) =colim(( C ) /C → C → P ( C )). Applying φ , we get φ φ ∗ ( Y C ) = colim(( C ) /C → C → C → P ( C )) . The morphism (8) evaluated at C is the canonical map φ φ ∗ ( Y C ) → Y C , and itis given by the canonical map of colimitscolim(( C ) /C → C → P ( C )) → colim( C /C → C → P ( C )) . To find out when is it an equivalence, we will evaluate the colimits at an arbitrary A ∈ C ⊂ C . The results are the colimits of the compositions ( C ) /C → C e A → S and C /C → C e A → S respectively, where e A ( B ) = Map C ( A, B ). Note that thefunctor e A : C → S is the left Kan extension of the terminal functor t : C A/ → S ,with respect to the left fibration π : C A/ → C .Therefore, evaluating (8) at A and C , we get a morphism of spaces(9) | ( C ) A/ × C ( C ) /C | → | C A/ × C C /C | = Map C ( A, C ) . It is an equivalence iff all its fibers are equivalences. This, by Proposition 2.3.1(3),is equivalent to f being flat. (cid:3) Localization.
The embedding S → Cat admits a left adjoint L : Cat → S called the total Dwyer-Kan localization .More generally, for f : C ◦ → C the Dwyer-Kan localization L ( f ) = L ( C , C ◦ ) isdefined as the colimit L ( C , C ◦ ) = L ( C ◦ ) ⊔ C ◦ C .This is the most general notion of localization available in the ( ∞ , Marked categories.
One usually localizes along C ◦ which is a subcategoryof C having the same objects as C . It is also assumed that C ◦ ⊃ C eq . Such a pair( C , C ◦ ) is called a marked category .The subcategory C ◦ determines for each pair of objects x, y a set of marked connected components in Map( x, y ). The trivially marked category C ♭ is definedas the pair ( C , C eq ); the maximally marked category C is ( C , C ). Sometimes acategory C has a “standard” subcategory C ◦ . Then we denote C ♮ = ( C , C ◦ ).The category of marked categories is denoted by Cat + . It is defined as a fullsubcategory of Fun([1] , Cat ) spanned by the embeddings C ◦ → C .2.4.2. A typical example of Dwyer-Kan localization is the functor from a modelcategory to its underlying ∞ -category.Let M be a model category with the collection of weak equivalences W ⊂ M .Dwyer-Kan localization L ( M, W ) was the main object of study in the originalseries of papers [DK1]–[DK3] by Dwyer and Kan. We call this localization the ∞ -category underlying M .2.4.3. Bousfield localization.
A functor L : C → L is called a (Dwyer-Kan) lo-calization if it induces an equivalence L ( C , C ◦ ) → L where C ◦ = f − ( L eq ). Alocalization L is called a Bouslfield localization if it admits a fully faithful rightadjoint.A typical example is the total DK localization L : Cat → S whose right adjointis the standard embedding of the category of spaces S to Cat .Another example is the functor Fun([1] , Cat ) → Cat + carrying an arrow W → C to the marked category ( C , C ◦ ) where C ◦ is the subcategory of C generated by C eq and the image of W .Finally, the DK localization functor L : Cat + → Cat is itself a Bousfieldlocalization: its right adjoint C C ♭ is fully faithful.2.4.4. Complete Segal spaces.
The embedding
Cat → Fun(∆ op , S ) described in(4) has a left adjoint which is another example of Bousfield localization. Thisadjoint pair of ∞ -categories is usually deduced from the Rezk (complete Segal)model structure on bisimplicial sets obtained from the Reedy model structure by(what is classically called) Bousfield localization of a model category.2.5. Marked categories. For instance, the category of finite pointed sets
Fin ∗ has the subcategory Fin ◦∗ spanned bythe inert arrows. f : C ♮ → D ♮ between marked categories defines an adjoint pairof functors(10) f ! : Cat + / C ♮ −→←− Cat + / D ♮ : f ∗ , with f ∗ defined by pullback along f : C ♮ → D ♮ and f ! given by composition with f .2.5.2. In the case where f is flat, the functor f ∗ : Cat + / D ♮ → Cat + / C ♮ has a rightadjoint ¯ f ∗ whose explicit description is presented below.For X ♮ = ( X, X ◦ ) ∈ Cat + / C ♮ the marked category ¯ f ∗ ( X ♮ ) over D ♮ is definedas ( Y, Y ◦ ) where Y is the full subcategory of f ∗ ( X ) spanned by the objects y : C d → X over d ∈ D carrying C d ∩ C ◦ to X ◦ . An arrow in Y over a : d → d ′ in D is given by a functor α : C × D { a } → X over C ; it belongs to Y ◦ iff a is markedin D and α preserves markings.Let Z ♮ ∈ Cat + /Y ♮ . The space Map Cat + / D ♮ ( Z ♮ , ¯ f ∗ ( X ♮ )) is, by definition, thesubspace of Map Cat / D ( Z, f ∗ ( X )) = Map Cat / C ( Z × D C , X ) spanned by the maps φ : Z × D C → X such that for any α : [1] → Z ◦ over a : [1] → D thecomposition [1] ♯ × D C → X preserves markings. This subspace is preciselyMap Cat + / C ♮ ( f ∗ ( Z ♮ ) , X ♮ ), so ¯ f ∗ is right adjoint to f ∗ .2.6. Categories with decomposition.
The following definition is an adap-tation of (a special case of) Lurie’s notion of categorical pattern, see [L.HA],App. B.Categories with decomposition are an important technical tool in working withoperads and operad-like objects.2.6.1.
Definition.
A category with decomposition is a triple ( C , C ◦ , D ) where( C , C ◦ ) is a marked category and D is a collection of “decomposition diagrams” { ρ d,i : C d → C di } d ∈ D in C ◦ .We will use the notation C (cid:11) = ( C , C ◦ , D ) for a category with decomposition.Some special cases: for a marked category C ♮ = ( C , C ◦ ) we denote by C ♮, ∅ thecategory with decomposition ( C , C ◦ , ∅ ). In particular, C ♭, ∅ and C ♯, ∅ are defined bythe minimal and the maximal marking.The first important example of a category with decomposition is the categoryof finite pointed sets Fin ∗ .2.6.2. Example.
The decomposition structure on
Fin ∗ is defined as follows. Wedefine Fin ◦∗ to be the category spanned by the inert arrows. For any I ∗ ∈ Fin ∗ any presentation I = ⊔ J i defines a decomposition diagram consisting of the inertarrows I ∗ → J i ∗ . Let C (cid:11) = ( C , C ◦ , D ) be a category with decomposition.We will now define a subcategory Fib ( C (cid:11) ) of the category Cat / C .2.6.3. Definition. (see [L.HA], 2.3.3.28) An object p : X → C of Cat / C is called fibrous if the following conditions are satisfied.(Fib1) For any x ∈ X any marked α : p ( x ) → C has a cocartesian lifting.In particular, any marked arrow α : C → C ′ in C defines a functor α ! : X C → X C ′ .(Fib2) For any d ∈ D the collection of maps ρ d,i ! : X C d → X C di defines anequivalence of categories X C d → Q X C di .(Fib3) For any d ∈ D and any x ∈ X C d the diagram of cocartesian liftings of ρ d,i , { x → x i } , is a p -product diagram, [L.T], 4.3.1.1.Given a category with decomposition C (cid:11) = ( C , C ◦ , D ), we define Fib ( C (cid:11) ) asthe subcategory of Cat / C spanned by the fibrous arrows X → C , with morphismspreserving the cocartesian liftings of arrows in C ◦ .For example, Fib ( C ♭, ∅ ) = Cat / C and Fib ( C ♯, ∅ ) = Coc ( C ).2.6.4. Let C (cid:11) = ( C ♮ , D ) be a category with decomposition. The functor i : Fib ( C (cid:11) ) → Cat + / C ♮ carrying p : X → C to the marked category X ♮ over C ♮ , withthe marking defined by the cocartesian lifting of the marked arrows in C , is fullyfaithful. Lemma.
The functor i has a left adjoint presenting Fib ( C (cid:11) ) as Bousfield local-ization of Cat + / C ♮ .Proof. This is an ∞ -categorical version of Lurie’s Theorem B.0.20 [L.HA] where Fib ( C (cid:11) ) is described as the ∞ -category underlying a model structure on markedsimplicial sets over C ♮ .An arrow f : X → Y in Cat + / C ♮ will be called C (cid:11) -equivalence if, for any Z ∈ Fib ( C (cid:11) ), it induces and equivalence Map( Y, Z ) → Map(
X, Z ). We have to verifythat, for any X ∈ Cat + / C ♮ , there exists a C (cid:11) -equivalence X → X ′ with X ′ ∈ Fib ( C (cid:11) ).We will deduce this fact from [L.HA], B.0.20. Let X ♮ be a marked categoryover C ♮ . We will represent X ♮ and C ♮ by pairs of quasicategories ( X, X ◦ ) and( C , C ◦ ).According to [L.HA], B.0.20, there is a trivial cofibration X ♮ → X ′ ♮ to a fibrantobject X ′ ♮ . This means that for any fibrant Z a homotopy equivalenceMap C ♮ ( X ′ ♮ , Z ♮ ) → Map C ♮ ( X ♮ , Z ♮ )is induced, where Map defines the simplicial structure on the category sSet + / C ♮ It remains to verify that Map in our cases represents correctly the mapping space in Cat + . This is actually true for any Z ∈ Fib ( C ♮, ∅ ). In fact, it is easy tosee that Map Cat + / C ♮ ( X ♮ , Z ♮ ) is represented by the maximal Kan simplicial subsetof Map C ♮ ( X ♮ , Z ♮ ). In the case where Z ♮ is fibrant, Map C ♮ ( X ♮ , Z ♮ ) is Kan. (cid:3) Arrows of
Cat + / C ♮ whose image is an equivalence in Fib ( C (cid:11) ), will be called C (cid:11) -equivalences .2.6.5. Let C (cid:11) be a decomposition category. A collection of arrows ρ i : C → C i in C ◦ is called a weak decomposition diagram if any X ∈ Fib ( C (cid:11) ) satisfies theproperties (Fib2) and (Fib3) with respect to the collection { ρ i : C → C i } .2.6.6. A functor between decomposition categories f : C (cid:11) → D (cid:11) is defined as afunctor f : C → D preserving the markings and carrying decomposition diagramsto weak decomposition diagrams.A functor f preserving the markings defines an adjoint pair ( f ! , f ∗ ) of functors,see (2.5.1).If f is a functor between decomposition categoires, f ∗ preserves the fibrous ob-jects. This implies that the functor f ! : Cat + / C ♮ → Cat + / D ♮ carries C (cid:11) -equivalencesto D (cid:11) -equivalences, see [L.HA], B.2.9. In particular, an adjoint pair of functors(11) f ! : Fib ( C (cid:11) ) −→←− Fib ( D (cid:11) ) : f ∗ is defined, where f ∗ is just the restriction of the base change functor (2.5.1), and f ! commutes with the localizations Cat + / C ♮ → Fib ( C (cid:11) ) and Cat + / D ♮ → Fib ( D (cid:11) ).In particular, applying the above adjunction to the functor id : C ♭, ∅ → C (cid:11) , wededuce the following.2.6.7. Corollary.
The forgetful functor
Fib ( C (cid:11) ) → Cat / C preserves limits. (cid:3) C (cid:11) , D (cid:11) be categories with decomposition, and let f : C ♮ → D ♮ be amap of the corresponding marked categories. Assume that f is flat.We call f a cofunctor between the decomposition categories if f ∗ : Cat + / D ♮ → Cat + / C ♮ carries D (cid:11) -equivalences to C (cid:11) -equivalences. A cofunctor between decom-position categories gives rise therefore to an adjoint pair(12) f ∗ : Fib ( D (cid:11) ) −→←− Fib ( C (cid:11) ) : ¯ f ∗ where ¯ f ∗ is just the restriction of the functor defined in 2.5.2 to the subcategory Fib ( C (cid:11) ), and f ∗ commutes with the localizations. Note that a cofunctor between decomposition categories does not necessarily carry decom-position diagrams in C (cid:11) to decomposition dieagrams in D (cid:11) (even though in the cases we have Lurie [L.HA], B.4.1, provides sufficient conditions for a marked map f to bea cofunctor. We will use them in 2.8 in the study of flat operads. We presentbelow his result specialized to the context of decomposition structures.2.6.9. Proposition. (see [L.HA] , B.4.1). Let C (cid:11) , D (cid:11) be categories with decom-position, and let f : C ♮ → D ♮ be a flat marked functor. Then f is a cofunctorbetween the decomposition categories, provided the following conditions are ful-filled. For any marked arrow α : [1] → D ◦ the fiber products f α : C α = [1] × D C → [1] and f ◦ α : C ◦ α = [1] × D ◦ C ◦ → [1] are cartesian fibrations, and theembedding C ◦ α → C α is a map of cartesian fibrations. For any marked arrow α : [1] → D ◦ appearing in a decomposition diagramin D the maps f α , f ◦ α are also cocartesian fibrations, and the embedding C ◦ α → C α is a map of cocartesian fibrations. A cocartesian lifting of a decomposition diagram in D is a weak decom-position diagram in C .Proof. Lurie [L.HA], B.4.1, lists nine conditions. His conditions (ii), (iii), (iv) arepart of our requirements. Condition (vi) is void in the context of decompositioncategories. Conditions (i) and (v) are equivalent to our condition (1), conditions(vii) and (viii) are equivalent to our (2), and (ix) is just our (3). (cid:3) C (cid:11) = ( C , C ◦ , D ), D (cid:11) = ( D , D ◦ , D ′ ),we define their product as the category E = C × D , with E ◦ = C ◦ × D ◦ , anddecompositions ( C, D ) → ( C i , D j )) where ( C → C i ) and ( D → D j )are weakdecomposition diagrams .The product of categories over C and D defines a functor Cat / C × Cat / D → Cat / C × D . This functor carries pairs of fibrous categories to a fibrous category, so defininga functor(13)
Fib ( C (cid:11) ) × Fib ( D (cid:11) ) → Fib ( C (cid:11) × D (cid:11) ) . One has2.6.11.
Proposition. The product map (13) preserves colimits in each of the two arguments. in mind this does happen). So the functor f ∗ in the formula (12) does not necessarily coincidewith the one in (11). Note that we have not defined the category of decomposition categories. It is functorial: a pair of maps f : C (cid:11) → C ′ (cid:11) and g : D (cid:11) → D ′ (cid:11) givesrise to a commutative diagram (14) Fib ( C (cid:11) ) × Fib ( D (cid:11) ) / / f ! × g ! (cid:15) (cid:15) Fib ( C (cid:11) × D (cid:11) ) ( f × g ) ! (cid:15) (cid:15) Fib ( C ′ (cid:11) ) × Fib ( D ′ (cid:11) ) / / Fib ( C ′ (cid:11) × D ′ (cid:11) ) . .Proof. This follows from [L.HA], B.2.5 and B.2.9. The product map (13) canbe presented by a left Quillen bifunctor, and f ! is also presented by left Quillenfunctor. The corresponding functors commute on the level of model categories,so this proves the claim. (cid:3) Operads.
The category of operads Op is defined as Fib ( Fin (cid:11) ∗ ).Marked arrows in Fin (cid:11) ∗ are inert arrows of Fin ∗ . Cocartesian liftings in O ofinert arrows in Fin ∗ are called inert arrows in O .Thus, Op is a subcategory of Cat /F in ∗ spanned by the operads, with the arrowspreserving the inerts.Our definition is essentially equivalent to the one given in [L.HA], Section 2. Infact, p : O → Fin ∗ is an operad if an only if it satisfies the conditions of definition[L.HA], 2.1.1.10 when presented by a categorical fibration.An arrow f : I ∗ → J ∗ is called active if the preimage of ∗ ∈ J ∗ consists of ∗ only. An arrow in O over an active arrow in Fin ∗ is also called an active arrow.The fiber of p : O → Fin ∗ at h i , denoted in what follows by O , is the categoryof colors of O .2.7.1. Strong approximation of operads.
This is a version of Lurie’s approxima-tion of operads described in [L.HA], 2.3.3.Let O ∈ Cat / Fin ∗ be an operad. Definition.
A map f : C → O p → Fin ∗ in Cat / Fin ∗ is called a strong approxima-tion if it satisfies the following conditions.(1) Let C ∈ C have image h n i in Fin ∗ ; then there exist p ◦ f -locally cocartesianliftings a i : C → C i of the standard inerts ρ i : h n i → h i , and f ( a i ) areinert in O .(2) Let a : X → f ( C ) be active in O (that is, its image in Fin ∗ is active).Then there exists an f -cartesian lifting e X → C of a .(3) The map f induces an equivalence C := C × Fin ∗ {h i} → O of the corresponding categories of colors. Strong approximation of operads allows one to have an alternative descriptionof Op / O .Let f : C → O be a strong approximation. We endow C with an induceddecomposition structure as follows. C ◦ is spanned by the arrows whose imagein O is inert. Decompositions are given by cocartesian liftings of the standardinerts ρ i : h n i → h i . As with the operads, the arrows of C ◦ are called inert.Active arrows in C are defined as the cartesian liftings of the active arrows in O .Also as with operads, strong approximation inherits a factorization system fromits operad: any arrow in C decomposes as an inert followed by an active arrow,see[L.HA], 2.3.3.8 and 2.1.2.5.Fibrous objects over C (cid:11) will also be called C -operads. We denote Op C = Fib ( C (cid:11) ).One has the following2.7.2. Proposition.
Let f : C → O be a strong approximation of an operad O .Then the base change with respect to f induces an equivalence Op / O → Op C .Proof. See [L.HA], 2.3.3.26. (cid:3)
The above equivalence is functorial: given a commutative diagram(15) P g ′ / / f ′ (cid:15) (cid:15) C f (cid:15) (cid:15) O ′ g / / O where g is a morphism of operads and f, f ′ are strong approximations, the basechange with respect to g ′ gives rise to a commutative diagram Op / O / / ∼ (cid:15) (cid:15) Op / O ′ ∼ (cid:15) (cid:15) Op C / / Op P . Strong approximations of operads are preserved by the base change.2.7.3.
Lemma. (see [L.HA] , 2.3.3.9.) Let f : C → O be a strong approximationand let p : O ′ → O be a map of operads. The the map f ′ : C ′ → O ′ obtained bybase change from f , is also a strong approximation. (cid:3) O -monoidal categories. Let f : C → O be a strong approximation. Wedenote by Mon O (resp., Mon C ) the subcategory of Op / O (resp., Op C ) whose objectsare cocartesian fibrous objects and whose morphisms preserve the cocartesianarrows. The equivalence 2.7.2 induces an equivalence Mon O → Mon C . One candescribe Mon C as Fib ( C (cid:11) ) for the following decomposition structure on C : (Mon1) C ◦ = C .(Mon2) Decomposition diagrams are the same as in C (cid:11) .Since cocartesian fibrations X → O can be described by functors C → Cat , Mon C identifies with the full subcategory Fun lax ( C , Cat ) ⊂ Fun( C , Cat ) spannedby the functors F satisfying the following version of Segal condition: each de-composition diagram { C → C i } gives rise to an equivalence F ( C ) → Q i F ( C i ),see [L.HA], 2.4.7.1, where such functors are called lax cartesian structures.2.7.5. Cat -enrichment.
Let O ′ , Q ∈ Op / O . The category Alg O ′ / O ( Q ) is defined asthe full subcategory of Fun O ( O ′ , Q ) spanned by the O -operad maps.Given a pair of strong approximations as in (15), and Q ∈ Op C , one defines Alg P / C ( Q ) as the full subcategory of Fun C ( P , Q ) spanned by the maps P → Q over C preserving the inerts . Note that for P , Q ∈ Op C one has Alg P / C ( Q ) eq =Map Op C ( P , Q ), so that the assignment P , Q Alg P / C ( Q ) provides (a sort of) Cat -enrichment on Op C . This enrichment is independent of the approximation, as thefollowing lemma shows. Lemma.
Let Q = C × O Q ′ for Q ′ ∈ Op / O . Then the base change along C → O provides a natural equivalence Alg O ′ / O ( Q ′ ) → Alg P / C ( Q ) .Proof. One can describe
Alg O ′ / O ( Q ′ ) as the fiber of the map Alg O ′ ( Q ′ ) → Alg O ′ ( O )at g ∈ Alg O ′ ( O ). Similarly, Alg P / O ( Q ′ ) is the fiber of the map Alg P ( Q ′ ) → Alg P ( O )at f ◦ g ′ . These two categories identify by [L.HA], 2.3.3.23. Finally, the fiber at f ◦ g ′ can be calculated in two steps, starting with the base change with respectto Alg P ( C ) → Alg P ( P ). This identifies Alg P / O ( Q ′ ) with Alg P / C ( Q ). (cid:3) Similarly, for P , Q ∈ Mon C we define Fun ⊗ C ( P , Q ) as the full subcategory ofFun C ( P , Q ) spanned by the maps preserving cocartesian arrows.2.8. Internal mapping object in operads.
The category Op is not cartesianclosed as direct product of operads does not commute, in general, with colimits.We impose an extra condition on operads to correct the problem.2.8.1. Definition.
Let C be a strong approximation of an operad. A C -operad p : P → C is called flat if any of the equivalent conditions of Lemma 2.8.2 belowis satisfied.2.8.2. Lemma.
Let C be a strong approximation of an operad and let p : P → C be a C -operad. The following conditions are equivalent. (1) For any pair of composable arrows in C given by σ : [2] → C , with d σ active, the base change P × C [2] → [2] is flat. (2) For any pair of composable active arrows in C given by σ : [2] → C , thebase change P × C [2] → [2] is flat. Note that P is not necessarily a C -operad. (3) For any pair of composable active arrows in C given by σ : [2] → C , with σ (2) ∈ C , the base change P × C [2] → [2] is flat.Proof. The implications (1) ⇒ (2) ⇒ (3) are clear.Let σ : [2] → C have γ = d σ active. We decompose d σ = β ◦ α with β activeand α inert. If now τ : [3] → C is composed of α, β and γ , one has a commutativediagram of functors between the categories of presheaves(16) ( p α ◦ p β ) ◦ p γ θ α,β / / ◗◗◗◗◗◗◗◗◗◗◗◗ ◗◗◗◗◗◗◗◗◗◗◗◗ p β ◦ α ◦ p γ / / p γ ◦ β ◦ α p α ◦ ( p β ◦ p γ ) / / p α ◦ p γ ◦ βθ α,γ ◦ β O O Since α is inert, θ α,β and θ α,γ ◦ β are equivalences. This proves that (2) implies (1).Any base change with respect to a pair of active arrows decomposes as aproduct of base changes as in (3). Thus, (3) ⇒ (2). (cid:3) Example. C -monoidal categories are flat C -operads as cocartesian fibrations areflat. Symmetric promonoidal categories in the sense of [BGS], 1.4, are flat op-erads. The operads BM X , LM X and Ass X constructed in Section 3, are flat, see3.3.6.The flatness of an operad P ∈ Op / O is independent of the choice of strongapproximation. In fact, let p : C → O be a strong approximation. Any σ : [2] → O defined by a pair of active arrows and satisfying the condition σ (2) ∈ O , as incondition (3) above, factors through p : C → O . This means that P is flat as O -operad iff its base change to C is flat as C -operad.Proposition 2.8.3 below is a version of Lurie’s [L.HA], 2.2.6, where a Dayconvolution monoidal structure is presented as a special case of internal mappingobject in operads. This result also generalizes a part of [BGS], 1.6.2.8.3. Proposition.
Let C be a strong approximation of an operad and let P ∈ Op C be flat. Then the product functor × P : Op C → Op C has a right adjoint. The functor right adjoint to × P will be denoted as Funop C ( P , ).Lurie in loc. cit. requires P to be O -monoidal. We were able to weaken therequirement, using an observation proven in Lemma 2.8.2. The rest of the proofis just a variation of the proof of [L.HA], 2.2.6.20. Proof.
The claim is independent of the choice of a strong approximation, so wewill assume C = O .We will present a category with decomposition P ′ endowed with a markedfunctor π : P ′ → P , such that the composition p ′ = p ◦ π : P ′ → O is a cofunctorin the sense of 2.6.8. Moreover, the composition π ! ◦ π ∗ will be equivalent to identity on Op P , so the functor × P : Op O → Op O , being the composition p ! ◦ p ∗ , isequivalent to p ! ◦ π ! ◦ π ∗ ◦ p ∗ = p ′∗ ◦ p ′ ! .Since the factors p ′∗ and p ′ ! admit right adjoints, this will prove the claim. Definition of P ′ . We define P ′ ∈ Cat + / P ♮ as follows. Denote by Fun in ([1] , O ) the category of inertarrows in O , with two maps, s, t : Fun in ([1] , O ) → O assigning to an arrow itssource and target.As a category over P , P ′ = Fun in ([1] , O ) × O P , where we use the target map t in the fiber product. Two maps, s, t : P ′ → O aredefined as compositions with the projection P ′ → Fun in ([1] , O ).An arrow in P ′ is marked if and only if its images in P and in O under s areinert.For any inert arrow u : [1] → O the base change P ′ u = [1] × O P ′ → [1] is acocartesian fibration, see [L.HA], proof of Proposition 2.2.6.20, property (5). Thisdefines a decomposition structure on P ′ : for x ∈ P ′ a decomposition diagram for x is obtained by a cocartesian lifting of the decomposition diagram for s ( x ) ∈ O . An equivalence id → π ! ◦ π ∗ . The diagonal embedding δ : O → Fun in ([1] , O ) induces δ : P → P ′ which gives,for any P -operad R , a map(17) R → R × P P ′ in Cat + / P ♮ . Thus, we have a morphism of endofunctors id → π ! ◦ π ∗ on Cat + / P ♮ . Themap (17) is P (cid:11) -equivalence; that is, it induces an equivalence of endofunctors on Op P . In fact, for Q ∈ Op P the marked maps R × O P ′ → Q are precisely the rightKan extensions of the marked maps R → Q . This proves the map (17) inducesan equivalence id → π ! ◦ π ∗ of endofunctors on Op P .It remains to verify that the composition p ′ = p ◦ π : P ′ → P → O is a cofunc-tor of categories with decomposition. We will verify that the the conditions ofProposition 2.6.9 hold. Flatness of p ′ is proven in Lemma 2.8.4 below. Condition(3) holds by definition of decomposition structure on P ′ . Conditions (1) and (2)of 2.6.9 are verified in [L.HA], Proposition 2.2.6.20, where they correspond toconditions (4), (7) and (5), (8) respectively. (cid:3) Lemma.
Let p : P → O be an O -operad and let P ′ = Fun in ([1] , O ) × O P bedefined as above. If P is a flat O -operad, then s : P ′ → O is flat.Proof. We will use the flatness criterion 2.3.1(3). Fix σ : [2] → O with the edges α, β and γ = β ◦ α . Let f ′ : [1] → P ′ be an arrow in P ′ over γ . Let f : [1] → P be the image of f ′ in P . Its image in O is γ ′ = t ( f ′ ). These data define a commutative diagram in O described below.(18) u ′ i ′′ " " γ ′ ❀❀❀❀❀❀❀❀❀❀❀❀❀❀ (cid:29) (cid:29) ❀❀❀❀❀❀❀❀❀❀❀❀❀ α ′ ❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋ u γ ' ' i ′ α (cid:30) (cid:30) ❂❂❂❂❂❂❂❂ i / / u ′′ (cid:15) (cid:15) v β (cid:31) (cid:31) ❃❃❃❃❃❃❃❃ j / / v ′ β ′ (cid:15) (cid:15) w k / / w ′ . The commutative square formed by the inerts i ′ , k and the arrows γ, γ ′ is definedby the image of f ′ in Fun in ([1] , O ).We decompose k ◦ β into an inert map j followed by an active map β ′ . Thenwe decompose j ◦ α into an inert map i followed by an active map u ′′ → v ′ .We decompose γ ′ into an inert i ′′ followed by an active map. Essential unique-ness of decomposition of the map u → w ′ ensures that u ′′ can be choosen to bea target of i ′′ . We finally denote by α ′ the composition u ′ → v ′ . We denote by σ ′ : [2] → O the 2-simplex given by the commutative triangle with the edges α ′ , β ′ γ ′ .We have to prove that the full subcategory D ⊂ Fun O ([2] σ , P ′ ) spanned by thesections τ : [2] → P ′ satisfying d τ = f ′ , is weakly contractible .We will compare D to the full subcategory D ′ ⊂ Fun O ([2] σ ′ , P ) spanned by thesections τ ′ : [2] → P satisfying d τ ′ = f . The category D ′ is weakly contractibleas P is a flat operad (and d σ ′ = β ′ is active).The commutative diagram (18) yields a map Σ : [2] → Fun in ([1] , O ) such that s ◦ Σ = σ, t ◦ Σ = σ ′ . This defines a functor ǫ : D ′ → D carrying τ ′ to the pair(Σ , τ ′ ). We will now show that ǫ has left adjoint; this will imply that D is alsoweakly contractible.Let τ : [2] → P ′ be in D and τ ′ : [2] → P be in D ′ . One has τ = (Σ ′′ , τ ′′ )where Σ ′′ : [2] → Fun in ([1] , O ) satisfies s ◦ Σ ′′ = σ, t ◦ Σ ′′ = σ ′′ , d σ ′′ = γ ′ , and τ ′′ : [2] → P with p ◦ τ ′′ = σ ′′ . The first and the last vertex of σ ′′ are u ′ and w ′ ;the remaining vertex will be denoted by v ′′ .Let F be the full subcategory of Fun([2] , Fun in ([1] , O )) spanned by the functorsΦ satisfying the conditions s ◦ Φ = σ, d ( t ◦ Φ) = γ ′ . It is easy to see that Σ ∈ F is a terminal object, so that we have a unique morphism Σ ′′ → Σ in F ; moreover,the corresponding map v ′′ → v ′ is inert.A map τ → ǫ ( τ ′ ) consists, by definition, of a map of triangles Σ ′′ → Σ in F anda compatible morphism τ ′′ → τ ′ of triangles in P over the unique map σ ′′ → σ ′ . We denote by [2] σ the category over O defined by σ : [2] → O . In Lemma 2.8.5 we verify that the map θ : σ ′′ → σ ′ in Fun([2] , O ) admits alocally cocartesian lifting to Fun([2] , P ). This means that a map τ ′′ → τ ′ over θ can be rewritten as a map θ ! ( τ ′′ ) → τ ′ over σ ′ .Therefore, the assignment τ = (Σ ′′ , τ ′′ ) θ ! ( τ ′ ) defines a functor left adjointto ǫ . (cid:3) Lemma.
Let p : P → O be a functor, σ : [ n ] → O be a simplex x f → x f → . . . f n → x n . Assume that for a certain k f k : [1] → O admits a cocartesian lifting to P . Thesimplex σ determines an arrow ˜ σ : d k σ → d k − σ in Fun([ n − , O ) , having thecomponent f k : x k − → x k at place k − and identity elsewhere. Then ˜ σ admitsa locally cocartesian lifting to Fun([ n − , P ) .Proof. We have to verify that the base change map Q := [1] × Fun([ n − , O ) Fun([ n − , P ) → [1]is a cocartesian fibration. The category Q over [1] defines a presheaf on ∆ / [1] carrying τ : [ k ] → [1] to the space of sections Q τ = Map [1] ([ k ] , Q ) which is thefiber of the mapFun([ k ] , Fun([ n − , P )) → Fun([ k ] , Fun([ n − , O ))at ¯ σ ◦ τ .This space is identified with the spaceMap O ([ k ] × [ n − , P ) , where the map [ k ] × [ n − → O is defined by ˜ σ ◦ τ . In the proof below wedenote k -simplices τ in [1] by monotone sequences of 0 and 1 of length k + 1. Let x ∈ Q , y ∈ Q be the source and the target of f ∈ Q . The arrow f in Q iscocartesian iff the composition { y } × Q Q f → { x } × Q Q × Q Q ∼ ← { x } × Q Q d → { x } × Q Q is an equivalence (see Proposition 3.3.2 for a more general statement). Applyingthis criterion to the description of Q τ , we deduce that an arrow f ∈ Q definedby the diagram in P (19) y g (cid:15) (cid:15) / / . . . / / y k − g k − (cid:15) (cid:15) / / y k +1 g k (cid:15) (cid:15) / / . . . / / y ng n − (cid:15) (cid:15) y ′ / / . . . / / y ′ k − / / y ′ k +1 / / . . . / / y ′ n is cocartesian if and only if the arrow g k − is a cocartesian lifting of f k : x k − → x k and the rest of g i are equivalences. (cid:3) We will now deduce some easy consequences of the existence of Funop C ( P , Q ).2.8.6. Corollary.
For P , Q , R ∈ Op C with P flat, one has a natural equivalence (20) Alg R / C (Funop C ( P , Q )) = Alg R × C P / C ( Q ) . Proof.
Immediately follows from the definition of
Alg P / Q , see 2.7.5. (cid:3) For O ∈ Op we denote by O the fiber of the projection O → F in ∗ at h i .This notation extends to the objects of Op ( C ): P is the fiber of the composition P → C → O → Fin ∗ at h i . In the case where P , Q are C -operads and P → C factors through C ◦ , one has Alg P / C ( Q ) = Fun C ( P , Q ).Applying formula (20) to R = C ◦ , we get2.8.7. Corollary.
Assume C is a space. For P , Q ∈ Op C with P flat, one has Alg C ◦ / C (Funop C ( P , Q )) = Alg C ◦ × C P / C ( Q ) = Fun C ( P , Q ) . Base change.
Let C → O be a strong approximation, q : D → C be in Op ( C ). The category D inherits from C a decomposition structure ( D , D ◦ , D ),with D ◦ being determined by inerts in D and D by the locally cocartesian liftingof the standard inerts ρ i : h n i → h i .The base change functor Cat / C → Cat / D restricts to a functor q ∗ : Op C → Op D having a left adjoint q ! : Op D → Op C . Thisis a special case of the functor described in 2.6.6.One has Lemma.
For P , Q ∈ Op C with P flat, one has an equivalence (21) q ∗ Funop C ( P , Q ) → Funop D ( q ∗ P , q ∗ Q ) . Proof.
The adjoint pair(22) q ! : Cat + / D ♮ −→←− : Cat + / C ♮ : q ∗ satisfies the projection formula: for X ∈ Cat + / D ♮ and for P ∈ Op C ⊂ Cat + / C ♮ , onehas a natural equivalence(23) q ! ( X × q ∗ ( P )) → q ! ( X ) × P , with the first projection induced by X × q ∗ ( P ) → X and the second projectionby X × q ∗ ( P ) → q ∗ ( P ) and by the counit of the adjunction q ! ◦ q ∗ → id.Since P is flat, the functor × P preserves C (cid:11) -equivalences, so the equivalence(23) induces the projection formula for the localized adjunction(24) q ! : Fib ( D (cid:11) ) −→←− : Fib ( C (cid:11) ) : q ∗ . Now our claim immediately follows from this projection formula.Map(
X, q ∗ Funop C ( P , Q )) = Map( q ! X, Funop C ( P , Q )) = Map( q ! X × P , Q )= Map( q ! ( X × q ∗ P ) , Q ) = Map( X × q ∗ P , q ∗ Q )(25) = Map( X, Funop D ( q ∗ P , q ∗ Q ) . (cid:3) Alternative description of
Funop( P , Q ) . Let C → O be a strong approxi-mation and let P , Q be C -monoidal categories.We present below a very explicit description of the C -operad Funop C ( P , Q ). Let Ar be the full subcategory of Cat / [1] spanned by the cocartesian fibrations over[1]. The objects of Ar are defined by the arrows A → B in Cat ; the arrows in Ar are given by 2-diagrams A (cid:15) (cid:15) / / B (cid:15) (cid:15) A ′ : B ⑤⑤⑤⑤⑤⑤⑤ ⑤⑤⑤⑤⑤⑤⑤ / / B ′ . The category Ar is closed under products in Cat / [1] , so the embedding Ar → Cat / [1] preserves the products. The restriction to the ends of [1] defines a functor p = ( p , p ) : Ar → Cat × Cat . We denote by Ar × the respective SM category(presented as a cocartesian fibration over Com = Fin ∗ ). The functor p preservesproducts, so it induces a functor p × : Ar × → Cat × × Cat × . The functor p × isobviously fibrous. We define an operad F P , QC as the fibre product(26) F P , QC (cid:15) (cid:15) / / Ar × p × (cid:15) (cid:15) C ( P , Q ) / / Cat × × Cat × , where the lower horizontal arrow is defined by the pair of C -monoidal categories P and Q .We claim Proposition.
The operad F P , QC defined by (26) is naturally equivalent to Funop C ( P , Q ) . We delay the proof of this result till the end of Section 6 when we developmeans to compare F P , QC to Funop C ( P , Q ). The proof is given in 6.3.9. C = Com . We have an even nicer description of Funop( P , Q )if Q has a cartesian SM structure. Define an operad F − , Q by the cartesian diagram(27) F P , Q Com (cid:15) (cid:15) / / F − , Q (cid:15) (cid:15) / / Ar × (cid:15) (cid:15) Com { P } / / Cat × id ×{ Q } / / Cat × × Cat × , We have the following.
Proposition. The operad F − , Q is a SM category. If Q is cartesian, F − , Q is also cartesian.Proof.
1. The underlying category of F − , Q is the cartesian fibration over Cat classified by the functor
Cat op → Cat carrying C to Fun( C , Q ). Thus, the spaceof the colors of our operad is the space ( Cat / Q ) eq of the categories over Q . Let uspresent a locally cocartesian lifting of an active arrow α : I ∗ → J ∗ in Com . Given f = { f i : C i → D } , i ∈ I , α ! ( f ) is given by the collection g = { g j } defined as Y i : α ( i )= j C i → Y i : α ( i )= j Q → Q , where the second arrow is the cocartesian lifting of α in the SM category Q . Thelocally cocartesian arrows defined as above are closed under composition, so themap F − , Q → Com is a cocartesian fibration.2. Let us now assume that Q is cartesian. The unit of F − , Q is obviously theterminal object t : C → Q , the one that factors through the terminal object[0] → Q . It remains to verify that for any pair of objects f i : C i → Q , i = 0 , f ⊗ f → f ⊗ t = f and f ⊗ f → t ⊗ f = f form a product diagram. This is an easy straightforward check. (cid:3) Examples.
We will now present some operads and their approximationsappearing in this paper. Two of them, the operads governing associative algebrasand left modules, are described in [L.HA]. We will also need an approximationfor the operad governing bimodules. The approximations for associative algebrasand left (right) modules can be realized as its full subcategories.2.9.1. For C ∈ Cat we denote as ∆ / C the fiber product(28) ∆ × Cat
Cat [1] × Cat { C } , that is, the category, whose objects are functors a : [ n ] → C , with a morphismfrom a to b : [ m ] → C given by a commutative triangle [ n ] → [ m ] → C . BM = (∆ / [1] ) op . The objects of BM are functors σ : [ n ] → [1],that is length n + 1 monotone sequences of 0 and 1.The operad for bimodules (which we denote as BM ⊗ to distinguish from BM )governs triples ( A, M, B ) where A and B are associative algebras and M hascompatible left A -module and a right B -module structures. An object of BM ⊗ over I ∗ ∈ Fin ∗ is a map f : I → { a, m, b } . An arrow from f : I → { a, m, b } to g : J → { a, m, b } over α : I ∗ → J ∗ is a collection of total orders at each α − ( j ), j ∈ J , such that • if g ( j ) = a then f ( i ) = a for all i ∈ α − ( j ). • if g ( j ) = b then f ( i ) = b for all i ∈ α − ( j ). • if g ( j ) = m then there exists a unique i ∈ α − ( j ) with f ( i ) = m ; moreover, f ( i ′ ) = a for i ′ < i and b for i ′ > i .The map ι : BM → BM ⊗ converts a sequence σ of 0 and 1 of length n + 1 intoa sequence of a, m, b of length n , each encoding a pair of consecutive numbers, a for 00, m for 01 and b for 11. This sequence of letters defines a map I := { , . . . , n } → { a, m, b } , that is an object of BM ⊗ . Given an arrow σ → τ in BM ,with ι ( σ ) = ( I, f ) and ι ( τ ) = ( J, g ), the arrow (
I, f ) → ( J, g ) in BM ⊗ is definedby the arrow I → J and the orders on the preimages of j ∈ J , induced from thetotal order on I .Degeneracies in BM correspond to inserting a letter a or b , inner faces correspondto “multiplications” aa → a , am → m , mb → m , bb → b , and outer faces erasethe leftmost or the rightmost letter. Lemma.
The map ι : BM → BM ⊗ is a strong approximation.Proof. We will verify the properties of Definition 2.7.1. The property (3) isobvious: one has BM = { a, m, b } = BM ⊗ . To verify (1), let σ : [ n ] → [1] representan object C ∈ BM and let ρ i : h n i → h i be the standard inert. The locallycocartesian lifting of ρ i is given by the inert map [1] → [ n ] carrying { } to { i − } and { } to { i } . Finally, to verify (2), let a : ( J, g ) → ι ( C ) be an active arrowin BM ⊗ . If ι ( C ) = ( I, f ), I acquires a total ordering. The active arrow a definestotal orderings on J . Overall, this allows one to uniquely define a total orderingon J compatible with the ordering on the fibers of a , such that a is monotone.This is equivalent to lifting of a to BM . This lifting is obviously cartesian. (cid:3) Outer faces are inerts in BM , and inner faces (as well as the degeneracies) areactive.2.9.3. Strong approximations to the operads Ass ⊗ , LM ⊗ , RM ⊗ can be found amongfull subcategories of BM . Here they are. • The subcategory
Ass is spanned by constant maps [ n ] → [1] with value0. It is isomorphic to ∆ op ; the object of Ass corresponding to [ n ] op ∈ ∆ op will be denoted as h n i . The map from Ass to the operad for associativealgebras
Ass ⊗ is compatible with the “left” embedding Ass ⊗ → BM ⊗ . Itcoincides with the map Cut : ∆ op → Ass ⊗ defined by Lurie in [L.HA],4.1.2.9. • The category LM is the full subcategory of BM spanned by σ : [ n ] → [1]satisfying σ (0) = 0 and having at most one value equal to 1. This categoryis isomorphic to ∆ op × [1] and the map LM → LM ⊗ coincides with the map γ : ∆ op × [1] → LM ⊗ defined by Lurie in [L.HA], 4.2.2. The isomorphism LM → ∆ op × [1] carries a n to ([ n ] ,
1) and a n m to ([ n ] , n ] , → ([ n ] ,
1) corresponds to the inert a n m → a n forgetting m . • Similarly, RM is the full subcategory spanned by σ : [ n ] → [1] satisfying σ ( n ) = 1 and having at most one value 0.2.9.4. One has embeddings(29) Ass → LM → BM ← RM ← Ass . We denote the two copies of
Ass inside BM as Ass − (the left copy) and Ass + (theright copy).2.9.5. The category Op Ass is called the category of planar operads. We do nothave special names for the categories Op BM , Op LM , etc. Definition.
For a BM -operad O the planar operads Ass ± × BM O are called the Ass ± -components of O , denoted as O a and O b , and the category { m } × BM O iscalled the m -component of O .One has a projection π : BM → Ass induced by [1] → [0]. This defines a basechange functor π ∗ : Op Ass → Op BM .2.9.6. Here is one more approximation. Let C n be the free planar operad gen-erated by one n -ary operation. It has n + 1 colors 0 , , . . . , n , and one n -aryoperation with the inputs of colors 1 , . . . , n , and the output of color 0. It has thestrong approximation Q n → C n defined as follows. Q n has one object { , . . . , n } over h n i , and the objects 0 , . . . , n over h i . One has inert arrows { , . . . , n } → i for i = 1 , . . . , n , and an active arrow { , . . . , n } → Q n is also a strong approximation of the free (non-planar) operad C n ∈ Op generated by one n -ary operation.2.9.7. Remark.
The projection π , as well as the maps (2.9.4), are not fibrous,so BM is not a planar operad and, for instance, Ass is not a BM -operad. Bilinear maps of operads. Tensor product.
In this subsection wepresent a slightly generalized version of Lurie’s [L.HA], 3.2.4. In this form it alsoincludes a presentation of BM as the tensor product of LM and RM , as well as apresentation of the operad C ⊔ , defined by a category C as in [L.HA], 2.4.3, as atensor product of Com with C .Let P (cid:11) , Q (cid:11) , R (cid:11) be categories with decomposition (in practice they are usuallyapproximations of operads). A bilinear map µ : P (cid:11) × Q (cid:11) → R (cid:11) is a map ofcategories with decomposition , where the decomposition structure of the productis defined as in 2.6.10.Given a bilinear map µ : P (cid:11) × Q (cid:11) → R (cid:11) and X ∈ Fib ( R (cid:11) ), one defines acategory p : Alg µ Q / R ( X ) → P over P by the formula(30) Map Cat / P ( K, Alg µ Q / R ( X )) = Map Cat + / R ♮ ( K ♭ × Q ♮ , X ♮ ) , where Q ♮ , R ♮ and X ♮ denote the corresponding marked categories. One has2.10.1. Proposition. (Compare to [L.HA] , 3.2.4.3). The formula (30) defines a category
Alg µ Q / R ( X ) over P . Alg µ Q / R ( X ) is P -fibrous. The equivalence (30) extends to an equivalence (31) Map
Cat + / P ( K ♮ , Alg µ Q / R ( X )) = Map Cat + / R ♮ ( K ♮ × Q ♮ , X ♮ ) . If π : X → R is a cocartesian fibration, p is also a cocartesian fibration.Proof.
1. The right-hand side of (30) preserves limits as a functor of K ∈ Cat op / P .Since Cat / P is presentable, Alg µ is correctly defined, see [L.T], 5.5.2.2.2. We apply 2.6.11 to f : P ♭, ∅ → P (cid:11) and g = id Q (cid:11) .The commutative dia-gram (14) restricted to Q ∈ Fib ( Q (cid:11) ) yields a commutative diagram of colimitpreserving functors(32) Fib ( P ♭, ∅ ) m ♭ / / f ! (cid:15) (cid:15) Fib ( P ♭, ∅ × Q (cid:11) ) ( f × id) ! (cid:15) (cid:15) Fib ( P (cid:11) ) m / / Fib ( P (cid:11) × Q (cid:11) ) µ ! / / Fib ( R (cid:11) ) ,m and m ♭ being evaluations of the product maps (13) at Q ∈ Fib ( Q (cid:11) ). All cate-gories are presentable, so this defines a commutative diagram of the correspondingright adjoints, which we define µ ∗ , m ∗ and m ∗ ♭ respectively. By definition Alg µ Q / R ( X ) = m ∗ ♭ ◦ ( f × id) ∗ ◦ µ ∗ ( X ) . Therefore,
Alg µ Q / R ( X ) = f ∗ ◦ m ∗ ◦ µ ∗ ( X ) , which means that this category over P is P -fibrous.3. Note that we simultaneously proved that the functor X Alg µ Q / R ( X ) isright adjoint to the composition µ ! ◦ m . This yields the equivalence (31).4. We can replace the decomposition structures on P and on R , retaining thedecomposition diagrams and assuming P ◦ = P , R ◦ = R . The functor Alg µ Q / R doesnot change after this replacement. This implies the required claim. (cid:3) µ : P (cid:11) × Q (cid:11) → R (cid:11) as above and X ∈ Fib ( R (cid:11) ), we define BiFun R ( P , Q ; X ) as the full subcategory of Fun R ( P × Q , X ) spanned by the mapspreserving the marked arrows. According to 2.10.1 (3), BiFun R ( P , Q ; X ) = Alg P ( Alg µ Q / R ( X )).2.10.3. Examples.
1. If P , Q , R are operads and µ is a bilinear map of operads, our definitionof Alg Q / R is just Lurie’s [L.HA], 3.2.4.2. For P = [0] a bilinear map µ is just a map Q → R of decomposition cate-gories. In the case when Q and R are strong approximations of operads,we recover the definition of Alg Q / R ( X ) given in 2.7.5.2.10.4. Tensor product.
Let(33) µ : C (cid:11) × C ′ (cid:11) → C ′′ (cid:11) be a bilinear map of categories with decomposition. We define a µ -tensor product Fib ( C (cid:11) ) × Fib ( C ′ (cid:11) ) → Fib ( C ′′ (cid:11) )as follows. Given P ∈ Fib ( C (cid:11) ), Q ∈ Fib ( C ′ (cid:11) ), define P ⊗ µ Q ∈ Cat + / C ′′ as theobject representing the functor(34) R BiFun C ′′ ( P , Q ; R ) , where the decomposition structure on P and Q is induced from C and C ′ and thebilinear map P × Q → C ′′ is defined as the composition P × Q → C × C ′ µ → C ′′ .The tensor product can be calculated as the image of P × Q under the local-ization functor Cat + / C ′′ → Fib ( C ′′ (cid:11) ).2.10.5. We will mention a few instances of the tensor product defined above.0. C = C ′ = C ′′ = [0]. In this case our definition gives a product in Cat .1. C = C ′ = C ′′ = Fin ∗ . We endow Fin ∗ with the smash product µ ( I ∗ , J ∗ ) =( I × J ) ∗ . The corresponding tensor product is the standard tensor productof operads defined in [L.HA], 3.2.4.2. C = C ′ = C ′′ = CM where M is the operad governing pairs ( A, M ) where A is a commutative monoid and M is an A -module. Its objects over I ∗ ∈ Fin ∗ are subsets I ⊂ I and a morphism from ( I, I ) to ( J, J ) isa map φ : I ∗ → J ∗ such that φ − ( j ) ∩ I is empty if j J and is a singleton otherwise. The functor µ : CM × CM → CM is defined by theformula µ (( I, I ) , ( J, J )) = ( K, K ) where(35) K = ( I × J ) I × J a ( I × J ) , K = I × J .
3. We choose as the bilinear map µ the identity map [0] × C → C where C isa strong approximation of an operad.One has Fib ([0]) =
Cat , so this type of tensor product assigns to a pair( K, P ) ∈ Cat × Op C a C -operad which we will denote by P K . The C -operad P K is independent of C in the following sense. If f : C → C ′ is a morphismof strong approximations as described by the diagram (15), the (derived)functor f ! , see 2.6.6, carries the C -operad C K to the C ′ -operad with thesame name.Here is the main property of the tensor product defined above.2.10.6. Proposition.
Let µ : C (cid:11) × C ′ (cid:11) → C ′′ (cid:11) be a bilinear map, and let P ∈ Fib ( C (cid:11) ) , Q ∈ Fib ( C ′ (cid:11) ) with R = P ⊗ µ Q ∈ Fib ( C ′′ (cid:11) ) . Then for any X ∈ Fib ( R (cid:11) ) one has a natural equivalence (36) Alg P ( Alg µ Q / R ( X )) = Alg R ( X ) . Conversely, a bilinear map P × Q → R lifting µ and inducing an equivalence 36,induces an equivalence P ⊗ µ Q → R .Proof. Immediately follows from 2.10.2. (cid:3) LM ⊗ and RM ⊗ are operads over CM . The bilinear structureon CM described in 2.10.5 (2) lifts to a map Pr : LM ⊗ × RM ⊗ → BM ⊗ , see [L.HA],4.3.2.1.Theorem 4.3.2.7 of [L.HA] asserts that the map Pr induces the equivalence(37) Alg RM ( Alg µ LM / BM ( X )) = Alg BM ( X ) . In other words, this means that Pr induces an equivalence(38) LM ⊗ ⊗ µ RM ⊗ = BM ⊗ . Another connection between bimodules and left modules is described in 3.6.2.10.8. We are back to Example 3 from 2.10.5. If X is a P K -operad, Alg P / P K ( X )is a category over K . In case X = π ∗ ( Y ) for a P -operad Y and π : P K → P the natural projection, Alg P / P K ( X ) = K × Alg P ( Y ), so Proposition 2.10.6 yields Alg P K ( Y ) = Fun( K, Alg P ( Y )).Let now X be a P K -monoidal category. Then, by Proposition 2.10.1, (4), themap p : Alg P / P K ( X ) → K is a cocartesian fibration.Note that a P K -monoidal category is given by a P K -algebra in Cat , that is,by a functor χ : K → Mon P to P -monoidal categories. The composition K χ → Mon P Alg P → Cat classifies the cocartesian fibration
Alg P / P K ( X ) → K , see 2.11.4and 2.11.5 for a more general claim.2.11. Operad families.
Definition.
Let C → O be a strong approximation. Let K be a category.We define on K × C the structure of a category with decomposition 2.6.1 asfollows. • The inerts of K × C are defined as ( K × C ) ◦ = K eq × C ◦ . • Decomposition diagrams are defined by pairs ( x, d ) ∈ K × D , D beingthe set of decomposition diagrams d = { ρ d,i : C d → C di } in C , as thecollection of maps (id x , ρ d,i ) : ( x, C d ) → ( x, C di ) . Definition. A C -operad family indexed by K is an object of Fib ( K × C ) (cid:11) . Remarks.
1. In the case C = Fin ∗ our notion is equivalent to Lurie’s notionof a family of operads, [L.HA], 2.3.2.10.2. Lurie’s notion of a family of operads is equivalent to the notion of gen-eralized operad, see [L.HA], 2.3.2.11. This is not so for general C .For instance, a family of planar operads is not the same as a generalizedplanar operad introduced in the work of Gepner-Haugseng [GH], 2.4.1.A C -operad family O → K × C is cartesian if the projection O → K is a cartesianfibration. The Grothendieck construction converts a cartesian C -operad familyover K into a functor K op → Cat / C whose essential image belongs to Op C . Thus,the notion of cartesian C -operad family is equivalent to a (contravariant) functorto Op C . Cocartesian families of operads are defined in the same way.Similarly, a C -operad family O → K × L × C is bifibered if the projection O → K × L is a bifibration. Bifibered C -operad family over K × L is equivalentto a functor K op × L → Op C .2.11.2. Cartesian families of monoidal categories.
Let O be an operad and C → O a strong approximation. A cartesian C -operad family M → K × C is called acartesian family of C -monoidal categories if for each x ∈ K the fiber M x → C is a C -monoidal category . Cartesian families of C -monoidal categories over K forma category FamMon cart C ( K ), with arrows M → M ′ inducing C -monoidal functors M x → M ′ x for each x ∈ K . This category has a very simple description. Let Cat cart /K denote the full subcategory of Cat /K spanned by the cartesian fibrations.2.11.3. Proposition.
There is a natural equivalence
FamMon cart C ( K ) = Alg C ( Cat cart /K ) . basically since C may not have an initial object. Note that the cartesian liftings M x ′ → M x of arrows x → x ′ in B are lax monoidal. Proof.
The category
FamMon cart C ( K ) is, by definition, a subcategory of Cat /K × C .Its objects are bifibered families with the Segal condition. Morphisms are mor-phisms of families preserving cocartesian liftings of arrows in C .The right-hand side can be described as the category of lax functors, see 2.7.4,Fun lax ( C , Cat cart /K ). This is also a subcategory of Cat /K × C with the same descrip-tion of objects and arrows. (cid:3) Families of operads and of monoidal categories.
There is a category
FamOp C of C -operad families together with a cartesian fibration FamOp C → Cat , whosefiber at X is Fib ( X × C ) ♮ . Any C -operad O is a trivial C -operad family ( X isa point) and a morphism from it to a K -indexed family P consists in a choiceof x ∈ K and a morphism O → P x of C -operads. In this context the category Alg O ( P ) of O -algebras in the K -indexed family P of C -operads is defined as theobject of Cat /K representing the functor(39) X Map
Cat + /K♭ × C ♮ ( X ♭ × O ♮ , P ♮ ) . Let P be a cocartesian family of C -operads classified by a functor f : K → Op C .In this case the functor (39) is represented by the cocartesian fibration p : A → K classified by the composition K f → Op C Alg O → Cat . In fact, for a C -operad Q one has Alg O ( Q ) is a full subcategory of Fun C ( O , Q ), so A is a full subcategory ofFun K C ( O , P ) := Fun C ( O , P ) × Fun( O ,K ) K, so Map Cat /K ( X, Fun K C ( O , P )) is a full subcategory of Map Cat /K × C ( X × O , P ). Oneeasily verifies that A represents (39).Thus, in the case when is a cocartesian family of C -operad classified by f : K → Op C , the category Alg O ( P ) is classified by the composition Alg O ◦ f .The category FamMon C is defined as a subcategory of FamOp C . Its objects arethe families O → X × C whose fibers at any x ∈ X are C -monoidal categories.The morphism are those inducing C -monoidal functors on the fibers.Similarly to the above, we extend the notation Fun ⊗ C ( P , Q ) to families ofmonoidal categories.2.11.5. Remark.
Let P be a C K -operad. Then P ′ = P × C K ( K × C ) is a familyof C -operads. The category Alg C ( P ′ ) of C -algebras in the family P ′ identifies bydefinition with Alg C / C K ( P ). This is the special case of the formula (30) applied to the identity functor µ = id K × C . Operadic sieves.
Recall [L.T], 6.2.2.1, that a sieve C → C is a full sub-category such that, if y ∈ C and f : x → y in C , then x ∈ C . Equivalently, C is a sieve in C if there is a functor p : C → [1] such that C = p − (0).We define operadic sieves in a similar way. Recall that for an operad P and acategory K , one assigns a new operad P K governing K -diagrams of P -algebras.Then the operad Com [1] governing maps of commutative algebras will play, foroperads, the role of [1] in the definition of a sieve in a category.2.12.1.
Definition.
An operadic sieve on P is a suboperad Q presentable as thefiber of a map P → Com [1] with respect to the embedding
Com → Com [1] definedby { } ∈ [1].An operadic sieve j : Q → P is uniquely determined by a sieve Q in P .In this subsection we will prove the following.2.12.2. Proposition.
Let Q be an operadic sieve in P . Then For any P -operad C the restriction functor j ∗ : Alg P ( C ) → Alg Q ( C ) is a cartesian fibration. For any map f : C → C ′ of P -operads the induced map f ! : Alg P ( C ) → Alg P ( C ′ ) preserves j ∗ -cartesian arrows. Example.
The suboperad
Ass ⊂ LM is an operadic sieve. Similarly, Ass − ⊔ Ass + ⊂ BM is an operadic sieve.The claim of Proposition 2.12.2 can be slightly strengthened.2.12.4. Corollary.
Let Q be an operadic sieve in P and let C be a cartesian familyof P -operads. Then the restriction functor j ∗ : Alg P ( C ) → Alg Q ( C ) is a cartesian fibration.Proof. If C is a cartesian family of P -operads with base K ∈ Cat , Alg P ( C ) and Alg Q ( C ) are cartesian fibrations over K , so that j ∗ is a map of cartesian fibrations.According to 2.12.2(1), the map j ∗ x : Alg P ( C x ) → Alg Q ( C x ) is a cartesian fibrationfor any x ∈ K . Then [L.T], 2.4.2.11 implies that j ∗ is a locally cartesian fibration.Moreover, an arrow e : A → A ′ in Alg P ( C ) is j ∗ -locally cartesian iff it is equivalentto a composition e = e ′′ ◦ e ′ where e ′ id j x -cartesian and e ′′ is a cartesian liftingof an arrow in K . Now the second part of Proposition 2.12.2 implies that thecollection of locally cartesian arrows is closed under composition. This provesthe claim. (cid:3) P K . Note that, by definition, P K is a P -operad such that, for any P -operad C , the category of P K -algebras in C is canonically equivalent to Fun( K, Alg P ( C )).We present below the digest of loc. cit. Proposition. P K = P × Com K . Com K is the operad K ⊔ defined by J. Lurie in [L.HA] , 2.4.3.1. As acategory over Com = Fin ∗ , it represents the functor B Map( B × Fin ∗ Γ ∗ , K ) , where Γ ∗ is the conventional category of pairs ( I ∗ , i ) with I ∗ ∈ Fin ∗ and i ∈ I , with the arrows ( I ∗ , i ) → ( J ∗ , j ) given by arrows I ∗ → J ∗ carrying i to j , and the functor Γ ∗ → Fin ∗ carries ( I ∗ , i ) to I ∗ . Com K is a flat operad. In particular, the assignment P P K preservescolimits.Proof. The first two claims are in [L.HA], Theorem 2.4.3.18. The third claimfollows from the explicit description of
Com B and the criterion 2.3.1, (3). (cid:3) Proposition 2.12.2 will be proven in 2.12.9, after a certain preparation.2.12.6.
Lemma. The functor B Com B preserves limits. Let q : A → B be a cartesian fibration. The induced map of operads Com A → Com B defines Com A as a flat Com B -operad.Proof. The first claim is a direct consequence of Proposition 2.12.5(2). To verifythe second claim, we will show that any active arrow in
Com B admits a cartesianlifting. In fact, an active map in Com B over an active arrow α : I ∗ → J ∗ in Com isgiven by a collection of arrows b i → b ′ α ( i ) with b i , b ′ j ∈ B . Its cartesian lifting isjust a collection of cartesian liftings of the separate maps b i → b ′ α ( i ) . (cid:3) p : P → Com [1] be an operadic sieve and let q : A → [1] be a cartesianfibration. The fiber product P q := P × Com [1]
Com A has a beautiful presentation asa colimit. Let j : Q ⊂ P be defined by p . Let, furthermore, A and A be thefibers of q at 0, 1, and let φ : A → A be the functor classifying the cartesianfibration q . We define a map(40) η : Q A ⊔ Q A P A → P q as the one induced by the presentation A = A ⊔ A ( A × [1]). Proposition 2.12.8below claims that (40) is an equivalence of operads.2.12.8. Proposition. Let q : A → [1] , A = colim A α in Cat / [1] , so that q, as well as q α : A α → A → [1] , are cartesian fibrations. Then the maps P q α → P q form a colimitdiagram of operads over Com [1] . In particular, the map η (40) is an equivalence of operads. The proof of the proposition is given in 2.12.12.2.12.9.
Proof of 2.12.2.
We denote by j : Q → P the embedding. The idea is toexpress the the cartesian lifting for the functor Alg P ( C ) → Alg Q ( C ) via algebrasover operads of type P q for certain cartesian fibrations q : A → [1]. We will havea plethora of operads of such form. • The first operad of this form, P , is defined by σ : A = [2] → [1]. Herewe have A = [1] , A = [0] , φ (0) = 1. Thus, by 2.12.8(2), P = Q [1] ⊔ Q P ,so that P -algebras are triples ( C, B, f ) with C ∈ Alg Q , B ∈ Alg P , f : C → j ∗ ( B ). • The second operad, P , is defined by σ : A = [2] → [1]. Here we have A = [0] , A = [1]. By 2.12.8(2), P = Q ⊔ Q [1] P [1] , so that P -algebras aremorphisms g : B → B ′ of P -algebras such that j ∗ ( g ) is an equivalence. • The operad P is defined by A = [1] × [1] → [1] given by the projectionto the first factor. Obviously, P = P [1] .The map σ : [2] → [1] has two sections ∂ , ∂ : [1] → [2]. The map ∂ inducesan obvious map P → P determined by decomposition of P ; the map ∂ definesa more interesting map u : P → P . The map u ∗ : Alg P → Alg P assigns a P -algebra C ′ to a map C → j ∗ ( B ) such that C = j ∗ ( C ′ ). Example.
Let j be the embedding Ass ⊗ → LM ⊗ . Then u ∗ : Alg P → Alg P assigns to a pair ( C → B, B M ) the C -module M .The map ∂ : [1] → [2] defines as well a map v : P → P . In the above example, v ∗ : Alg P → Alg P assigns to a map of B -modules M → N the B -module N .Presenting the square [1] × [1] as glued from two triangles along the commonhypotenuse, we get, by 2.12.8(1), an equivalence P = P ⊔ P P . In our examplethis means that a morphism from an A -module M to a B -module N is given bya pair ( f, g ) where f : A → B is a morphism of algebras and g : M → f ∗ ( N ) is amorphism of A -modules. The projection P → P induced by σ : [2] → [1] yields,together with the decomposition P = P ⊔ P P , a projection π : P → P . Theinduced map of algebras defines a (contravariant) lifting of arrows in Alg Q withrespect to the functor j ∗ : Alg P → Alg Q . The same decomposition of P = P [1] can be now interpreted as a proof that this lifting is locally cartesian.Note that at this point we have already verified that j ∗ are locally cocartesianfibrations and that for any morphism of P -operads f : C → C ′ the functor f ! : Alg P ( C ) → Alg P ( C ′ ) preserves the locally cocartesian liftings.To prove that j ∗ is actually a cartesian fibration, we have to consider a slightlymore complicated diagram of the same type. We need two more operads. • P = Q [2] ⊔ Q [1] P [1] where the map Q [1] → Q [2] is induced by ∂ : [1] → [2].Algebras over P are given by a morphism f of P -algebras, together with apresentation of j ∗ ( f ) into a composition of two morphisms of Q -algebras. • P = P ⊔ P P [1] , with u : P → P and the map P → P [1] induced by { } → [1]. P -algebras are defined by a pair of P -algebras B ′′ , B , a mapof Q -algebras a : C → j ∗ ( B ), and a map of P -algebras B ′′ → B ′ where B ′ is obtained by the local cartesian lifting of a .Using Proposition 2.12.8, one easily deduces that both P and P are naturallyequivalent to P q where q : A → [1] is the cartesian fibration classified by the map ∂ : [1] → [2].This proves the assertion. (cid:3) C be a Com [1] -monoidal category classified by a SM functor C → C and let q : A → [1] be a cartesian fibration classified by the map φ : A → A . Thecategory Fun [1] ( A, C ) is a cocartesian fibration over [1] classified by the SM functor φ ∗ : Fun( A , C ) → Fun( A , C ). Therefore, it is a Com [1] -monoidal category.We will now show that this
Com [1] -monoidal category is canonically equivalentto Funop
Com [1] ( Com A , C ). This fact will easily lead to the proof of 2.12.8.Note that Fun [1] ( A, C ) is a cocartesian fibration over [1] and, therefore, it hasa standard presentation as a fiber product, see [H.L], 9.8.8, formula (72), as(41) Fun [1] ( A, C ) = Fun [1] ( A × [1] , C ) × Fun [1] ( A ⊳ , C ) Fun [1] ( A ⊳ , C ) , where K ⊳ = [0] ⊔ K ( K × [1]). Recall that Com A is flat over Com [1] , see 2.12.6(2).We can now define a canonical map(42) θ A : Funop Com [1] ( Com A , C ) → Fun [1] ( A, C )using the decomposition A = A ⊔ A ( A × [1]) and (41).We have2.12.11. Proposition.
The map θ A (42) is an equivalence of Com [1] -operads.Proof.
Let us first calculate the fibers of both operads at h i ∈ Com . For theright-hand side we have, obviously, the cocartesian fibration Fun [1] ( A, C ) over [1].The fiber at h i of the left-hand side is Alg ι ([1]) / Com [1] (Funop
Com [1] ( Com A , C )) = Alg ι ([1]) × Com [1]
Com A / Com [1] ( C ) = Alg ι ( A ) / Com [1] ( C ) = Fun [1] ( A, C ) , the same cocartesian fibration over [1]. The morphism is constructed to be iden-tity for the special cases A = A × [1], A = K ⊳ , so it is an equivalence in general.In order to prove that the map (42) is an equivalence, it is therefore sufficientto verify that it induces an equivalence of spaces of active maps. These canbe expressed via the category of algebras Alg C n (see 2.9.6) with values in theseoperads. Let s : [1] → Com [1] be an active arrow over the active arrow h n i → h i in Com . We denote the colors of
Com [1] as 0, 1, so s is either (0 , . . . , →
0, or(0 k n − k ) → k ∈ { . . . , n } . We will study in detail the second case as itis less obvious. Let f i : A → C , i = 1 , . . . , k , g j : A → C , j = 1 , . . . , n − k , g : A → C be the colors over 0 or 1 ∈ Com [1] respectively. We will calculate thespace(43) Map( f ⊕ . . . ⊕ g n − k , g )in Funop Com [1] ( Com A , C ). The map s extends (essentially) uniquely to a map ofoperads which we will denote by ˜ s : C n → Com [1] . The collection of colors f i , g j , g determine a map C ◦ n → Funop
Com [1] ( Com A , C × Com [1] ), and the space (43) identifieswith the fiber of the functor
Alg C n / Com [1] (Funop
Com [1] ( Com A , C )) → Alg C ◦ n / Com [1] (Funop
Com [1] ( Com A , C ))(44)at ( f i , g j , g ). The categories of algebras in question can be easily described usingthe strong approximation Q n → C n . The result is described as follows.Denote as φ s : A → A k × A n − k the map with components φ and id A re-spectively, depending on the target. Let A s be the cartesian fibration over [1]classifying the map φ s . Then the map (44) identifies with(45) Fun [1] ( A s , C ) → Fun( A , C ) k × Fun( A , C ) n − k . Description of the space of maps in the
Com [1] -monoidal category Fun [1] ( A, C )gives obviously the same result. (cid:3) Proof of Proposition 2.12.8.
A map φ : P → P ′ of operads is an equiva-lence iff for any SM category C the induced map φ ∗ : Alg P ′ ( C ) → Alg P ( C )is an equivalence. This easily follows from [L.HA], 2.2.4.10, where any operad P is proven to be equivalent to a full subcategory of a SM category.The operad Com A is flat over Com [1] , therefore, for any SM category C Alg P q ( C ) = Alg P / Com [1] (Funop
Com [1] ( Com A , C × Com [1] )) =
Alg P / Com [1] (Fun [1] ( A, C × [1]))by Proposition 2.12.11. The first part of Proposition 2.12.8 is a direct con-sequence of this formula. In fact, if A = colim A α , then Fun [1] ( A, C × [1]) =lim Fun [1] ( A α , C × [1]), and, therefore, Alg P q ( C ) = lim Alg P qα ( C ). This proves theassertion.To verify the second claim of Proposition 2.12.8, we apply the first claim tothe decomposition A = A ⊔ A ( A × [1]). This is a presentation of A as a colimitof cartesian fibrations over [1], so it gives a presentation of P q as a colimit. Itremains to add that P × Com [1]
Com ( A × [1]) = P A and P × Com [1]
Com A i = Q A i where A i → [1] factors through { } → [1]. (cid:3) π : P = P [1] → P σ = P be the map defined in 2.12.9, corre-sponding to a projection of a square to one of its halves. Let Y be a P -monoidalcategory and X = π ∗ ( Y ). The P [1] -monoidal category X is presented by a P -monoidal functor X → X . We have2.12.14. Proposition.
The diagram (46)
Alg P ( X ) (cid:15) (cid:15) / / Alg P ( X ) (cid:15) (cid:15) Alg Q ( X ) / / Alg Q ( X ) is cartesian. Let us show the meaning of this claim for P = LM . The LM [1] -monoidal category X is defined by an LM -monoidal functor f : ( C , M ) → ( C , M ) between LM -monoidal categories, such that the functor M → M is an equivalence. Thenour claim amounts to saying that, for any associative algebra A in C the categoryof A -modules in M is equivalent to the category of f ( A )-modules in M . Proof.
Look at the pair of bilinear maps µ : [1] × Q → Q [1] → P and ν : [1] × P → P [1] π → P . The P -monoidal category Y defines cocartesian fibrations p : Alg ν P / P ( Y ) → [1]and q : Alg ν Q / P ( Y ) → [1] by 2.10.1(4). These cocartesian fibrations are classifiedby the functors Alg P ( X ) → Alg P ( X ) and Alg Q ( X ) → Alg Q ( X )respectively.We have P = Q [1] ⊔ Q P , so Alg P ( Y ) = Alg Q [1] ( Y ) × Alg Q ( Y ) Alg P ( Y ) . Now,
Alg P ( Y ) = Fun [1] ([1] , Alg ν P / P ( Y )), and we denote by Alg coc P ( Y ) the subcate-gory of cocartesian section of Alg ν P / P ( Y ). We define in the similar way Alg coc Q [1] ( Y ). Alg coc P ( Y ) is the full subcategory of Alg P ( Y ) spanned by the P -sections car-rying the arrow [1] ∂ → [2] → P × Com [1]
Com [2] = P to a cocartesian arrow in Y .This implies that Alg coc P ( Y ) = Alg coc Q [1] ( Y ) × Alg Q ( Y ) Alg P ( Y ) . Thus, the fiber of the map
Alg coc P ( Y ) → Alg coc Q [1] ( Y ) at a point is equivalent to thefiber of Alg P ( X ) → Alg Q ( X ) at the image of this point. This proves the claim. (cid:3) Opposite monoids and opposite algebras.
All imaginable meanings ofthe notion “opposite” in category theory can be expressed in terms of the functor I I op assigning to a totally ordered finite set the same set with the oppositeorder. We denote this functor by op : ∆ → ∆. We denote by the same letter theendofunctor on Ass = ∆ op 14 C be a category with products. If A : Ass → C is an associativemonoid (that is a functor satisfying Segal condition and such that A ( h i ) isterminal), the composition A ◦ op is also a monoid called the opposite monoid and denoted by A op .If A : Fin ∗ → C is a commutative monoid, one has a canonical equivalence A = A op .This construction applies, in particular, to C = Cat , which gives the notion ofthe opposite monoidal category. In order to avoid confusion with the notion ofopposite category, we will denote the monoidal category opposite to M as M rev and will call it reversed monoidal category .2.13.2. The notion of opposite category can be also extracted from the functorop : Ass → Ass . Categories, according to our favorite description, are completeSegal spaces, that is functors C : Ass → S satisfying completeness and Segal conditions. Composing C with op, we getanother complete Segal functor Ass → S , that is a new category denoted by C op .Note that, in this description, a space X coincides “identically” with its opposite.2.13.3. The endofunctor op : Ass → Ass extends to op : BM → BM carrying LM to RM and vice versa. This identifies left modules over A with right modules over A op in a category with products.In particular, the categories left-tensored over M identify with the categoriesright-tensored over M rev .2.13.4. Let now A be an associative algebra in a monoidal category M . Bydefinition, A is a section of the canonical projection p : M → Ass . Composing A with op : Ass → Ass , we get a section of the base change of M with respect to op,which is precisely M rev . Thus, for A ∈ Alg
Ass ( M ) one has A op ∈ Alg
Ass ( M rev ).If M is symmetric monoidal, M rev = M , and we get A op as an algebra in M . Note that
Ass stands for a very concrete approximation of the operad for associativealgebras. Were we to mean the operad for associative algebras, we would denote it by
Ass ⊗ . In the case where M is cartesian both constructions of opposite algebra in M coincide.If M is a monoidal category and A is an associative algebra in M , one has acanonical equivalence LMod A ( M ) = RMod A op ( M rev ) .2.13.5. The notion of reversed monoidal category extends to planar operads. Iffact, if p : P → Ass is a planar operad, the composition op ◦ p : P → Ass is alsoa planar operad called the reversed operad . Note that the functor op : BM → BM provides an isomorphism of BM with BM rev .If P is flat, P rev is also flat. One has ( P × Q ) rev = P rev × Q rev , so Funop( P , Q ) rev =Funop( P rev , Q rev ). 3. Enriched quivers
Introduction.
In this section we present a construction that assigns, toeach X ∈ Cat and M ∈ Op Ass , a new planar operad Quiv X ( M ) called the planaroperad of M -enriched X -quivers.With the aim of describing a universal property of this construction later on,we present two more versions of this construction: Quiv LM X ( M ) ∈ Op LM of an LM -operad M and Quiv BM X ( M ) for any M ∈ Op BM .The constructions are functorial in X and in M , as explained in 3.1.3, 3.1.5and 3.5 below.The category of colors Quiv X ( M ) of the planar operad Quiv X ( M ) isFun( X op × X, M ). These are functors from X op × X to the category of colorsof M ; we interpret them as quivers, with the category of objects X , and withvalues in M .The most important for us is the case when M is a monoidal category havingcolimits (precise requirements are given below); in this case Quiv X ( M ) is also amonoidal category. We weaken the requirements on X and M in order to betterunderstand the functoriality of the construction.The ultimate goal of the paper are M -enriched ∞ -categories. Similarly to theconventional case where categories can be defined as associative algebra objectsin the appropriate monoidal category of quivers, we define M -enriched categoriesas associative algebras in Quiv X ( M ) satisfying some extra (completeness) prop-erties. The full meaning of the following definitions will become clear later.3.1.1. Definition.
Let X be a category and M be a planar operad. An M -enriched precategory with the category of objects X is an associative algebraobject in Quiv X ( M ). M and M rev have the same underlying category! presented here as an advertisement. Definition.
Let X be a space, M a monoidal category with colimits. An M -enriched category A with the space of objects X is an M -enriched precategorysatisfying a completeness condition, see Definition 7.1.1.3.1.3. Dependence on M . The BM -operad Quiv BM X ( M ) is defined, using the internalmapping object in Op BM , as(47) Quiv BM X ( M ) = Funop BM ( BM X , M ) , where M is a BM -operad and BM X is a flat BM -operad depending of X ∈ Cat ,defined below.If M is in Op Ass , we will sometimes write Quiv BM X ( M ) instead of Quiv BM X ( π ∗ M ),where π : BM → Ass is the natural projection, see 2.9.4,The construction of BM X is presented below, after a certain preparation in 3.1.4.We will only mention now that the Ass − -component of BM X is equivalent tothe planar operad O X defined in [GH], 4.2.4, when X is presented by a simplicialcategory.3.1.4. Relative categories.
Recall that the embedding ∆ → Cat , together withthe Yoneda embedding, identifies
Cat with the full subcategory of Fun(∆ op , S )spanned by the complete Segal objects. We will now present a similar descriptionfor categories over a fixed category C ∈ Cat .Recall from 2.9.1 that ∆ / C denotes the full subcategory of Cat / C spanned bythe objects σ : [ n ] → C .An object X ∈ Cat / C yields a Yoneda map Y X : ( Cat / C ) op → S , whose restriction to ∆ / C defines a presheaf F X : (∆ / C ) op → S . Note the following
Lemma.
The presheaf F X ∈ P (∆ / C ) is presented by the right fibration ∆ / X → ∆ / C induced by the map X → C .Proof. The Yoneda lemma implies that Y X ∈ P ( Cat / C ) is presented by the rightfibration ( Cat / C ) / X → Cat / C . Proposition 2.1.2.5 in [L.T] asserts that (
Cat / C ) / X is canonically equivalent to Cat / X . The required result follows from this by base change along ∆ → Cat . (cid:3) In practice we will mostly be interested in categories over a conventional cat-egory B having no nontrivial isomorphisms. In this case ∆ /B is a conventionalcategory whose objects are the functors a : [ n ] → B , with a morphism from a to b : [ m ] → B given by a commutative triangle [ n ] → [ m ] → B . Given X ∈ Cat /B , the corresponding functor F X : (∆ /B ) op → S carries σ :[ n ] → B to the space of sections of the base change(48) X × B [ n ] → [ n ] . Conversely, a functor F : (∆ /B ) op → S determines a simplicial space X over B which is a category over B iff it satisfies a version of completeness and Segalconditions. We will be only interested in the special case when B is a conventionalcategory with no nontrivial isomorphisms. In this case for each n the space B n is discrete and X n = ⊔ σ ∈ B n F ( σ ).Therefore, for B a conventional category with no nontrivial isomorphisms X isa category over B iff F satisfies the following properties.(CS1) Let σ : [ n ] → B be given by a sequence of arrows f i : x i − → x i , i =1 , . . . , n , in B . Then the canonical map F ( σ ) → F ( f ) × F ( x ) . . . × F ( x n − ) F ( f n )is an equivalence.(CS2) For any x ∈ B the Segal space obtained by the composition∆ op = (∆ / { x } ) op → (∆ /B ) op F → S , is complete.The property (CS1) implies that X is Segal; the property (CS2) ensures it iscomplete.3.1.5. Dependence on X . The BM -operad BM X will be defined by a functor F BM X : (∆ / BM ) op → S , whose dependence on X is seen from the formula(49) F BM X ( σ ) = Map( F BM ( σ ) , X ) , where F BM : ∆ / BM → Cat is independent of X ; it is defined below.3.2. Functor F BM . The category ∆ / BM has objects σ : [ n ] → BM , and a map from σ to τ : [ m ] → BM is given by the commutative diagram[ n ] / / σ ! ! ❇❇❇❇❇❇❇❇ [ m ] τ } } ④④④④④④④④ BM . A general description.
The functor F BM will have values in conventionalcategories. It will also satisfy the following property which will ensure that F BM X satisfies Segal condition.Let σ : [ n ] → BM be given by a collection of arrows f , . . . , f n , f i : v i − → v i in BM . Then the natural map(50) F BM ( f ) ⊔ F BM ( v ) F BM ( f ) ⊔ F BM ( v ) ⊔ . . . ⊔ F BM ( v n − ) F BM ( f n ) → F BM ( σ )is an equivalence.Having this in mind, we will describe F BM first on σ : [ n ] → BM with n = 0,then for σ with n = 1, and then use the formula (50) to define F BM ( σ ) for general σ as the left-hand side of the formula.Finally, we will have to define the functors between the F BM ( σ ) correspondingto the inner faces.In what follows we will write | σ | = n for σ : [ n ] → BM .The functor F BM will assign discrete categories to 0-dimensional simplices, andsome disjoint unions of [1] to one-dimensional simplices.3.2.2. | σ | = 0 . An object σ : [0] → BM is given by a length n + 1 monotonesequence w of 0’s and 1’s. Let us, following 2.9.2, translate w into a length n sequence of letters a, m, b — it should have the form a k m α b l , where k, l ≥ α ∈ { , } , and α = 1 if both k, l are nonzero, with n = k + l + α .The functor F BM assigns to σ the discrete category with 2 k + α objects, denotedby x r and y r with r = 1 , . . . , k , and y if α = 1. Note that there are two objectsof BM of length n = 0 which have an empty presentation as an amb -word. Thevalue of F BM at these two objects is the empty category. In the pictures below wedraw the objects in the following order:(51) x k ◦ y k • . . . x ◦ y • ( α = 0) y • x k ◦ y k • . . . x ◦ y • ( α = 1)Note that x -type objects are denoted by ◦ , whereas y -type objects are denotedby • .3.2.3. | σ | = 1 . An object σ : [1] → BM is defined by an arrow f : w → w ′ in BM .We assume w = a k m α b l and w ′ = a k ′ m α ′ b l ′ , using the standard notation.The arrow f : w → w ′ in BM = (∆ / [1] ) op defines (and is defined by) an arrow π ( f ) : h n i → h n ′ i in Ass , so by an arrow φ : [ n ′ ] → [ n ] in ∆. The object w is afunctor w : [ n ] → [1] and w ′ is the composition w ◦ φ .The objects of F BM ( f ) are the disjoint union of objects of F BM ( w ) and F BM ( w ′ ).The nontrivial arrows of F BM ( f ) are all disjoint, and they are specified below. We will denote the objects of F BM ( w ), as in 3.2.2, by x r , y r ( r = 1 , . . . , k ) or y ,and the objects of F BM ( w ′ ) as x ′ r , y ′ r ( r = 1 , . . . , k ′ ) or y ′ .Each segment { i − , i } of [ n ′ ], with i = 1 , . . . , k ′ defines φ ( i ) − φ ( i −
1) + 1nontrivial disjoint arrows in F BM ( f ) according to the following rule. • If φ ( i −
1) = φ ( i ), one has an arrow from x ′ i to y ′ i . • If φ ( i − < φ ( i ), one has arrows x ′ i → x φ ( i ) , y φ ( i − → y ′ i , as well as φ ( i ) − φ ( i − − y j to x j − , for j = φ ( i −
1) + 2 , . . . , φ ( i ).Furthermore, in the case α ′ = 1 (and also, since f : w → w ′ exists, α = 1),there are φ ( k ′ + 1) − φ ( k ′ ) more nontrivial arrows in F BM ( f ); these are the arrow y → x k , y j → x j − for j = k, . . . , φ ( k ′ ) + 2 ( j decreases by 1), as well as y φ ( k ′ )+1 → y ′ .As a result, the above description yields a category F BM ( f ) having 2( k + k ′ ) + α + α ′ objects (note that α = α ′ if f : w → w ′ exists) and k ′ + φ ( k ′ + α ′ ) − φ (0)disjoint nontrivial arrows.3.2.4. | σ | = 1 , in pictures. In the pictures below we denote by ◦ the x -objectsand • the y -objects. The upper row describes w , and the lower row describes w ′ .First of all, the picture (52) below presents F BM ( f ) for a typical inert arrowcorresponding to φ : [2] → [4] carrying i = 0 , , i + 1 (here w = a , w ′ = a ).(52) x ◦ y • x ◦ y • (cid:15) (cid:15) x ◦ y • (cid:15) (cid:15) x ◦ y • x ′ ◦ O O y ′ • x ′ ◦ O O y ′ • The next picture is of a typical active arrow [1] → [3] (here w = a , w ′ = a ).(53) x ◦ y • / / x ◦ y • / / x ◦ y • x x qqqqqqqqqqqqqqq x ′ ◦ f f ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ y ′ • The following picture describes F BM ( f ) in a case where the corresponding φ :[ n ′ ] → [ n ] is neither inert, nor active or injective. Here n = 4 , n ′ = 2 ( w = a , w ′ = a ), and the map φ carries 0 to 1, 1 to 3 and 2 to 3.(54) x ◦ y • x ◦ y • / / x ◦ y • (cid:15) (cid:15) x ◦ y • x ′ ◦ / / y ′ • x ′ ◦ f f ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ y ′ • The last example demonstrates F BM ( f ) for the active arrow f and α = 1. Here w = a m and w ′ = am . The corresponding map φ : [2] → [3] carries 0 to 0, 1 to1 and 2 to 3.(55) y • / / x ◦ y • (cid:1) (cid:1) ✄✄✄✄✄✄✄✄✄ x ◦ y • (cid:1) (cid:1) ✄✄✄✄✄✄✄✄✄ y ′ • x ′ ◦ A A ✄✄✄✄✄✄✄✄ y ′ • Description of F BM ( σ ) . Given an object σ in ∆ / BM presented by a sequenceof arrows w f → w → . . . f n → w n , one defines F BM ( σ ) as the colimit(56) F BM ( f ) ⊔ F BM ( w ) F BM ( f ) ⊔ . . . ⊔ F BM ( w n − ) F BM ( f n ) . This is the category freely generated by the union of n diagrams correspondingto the arrows f i as shown in the pictures above. It is convenient to present it asa multi-story graph, with the i -th story describing F BM ( f i ).3.2.6. Examples.
Let σ = ( w f → w ′ g → w ′′ ) be given by w : [ n ] → [1] and asimplex [ n ′′ ] ψ → [ n ′ ] φ → [ n ]. One has w ′ = w ◦ φ , w ′′ = w ◦ φ ◦ ψ .1. We assume ψ ( i ) > ψ ( i −
1) but φ ( ψ ( i )) = φ ( ψ ( i − F BM ( σ ) ( “a cap”) is connected.(57) x ′ ψ ( i ) ◦ / / y ′ ψ ( i ) • x ′ ψ ( i − ◦ / / y ψ ( i − • z z ✈✈✈✈✈✈✈✈✈✈✈✈✈ x ′′ i ◦ _ _ ❅❅❅❅❅❅❅❅❅❅ y ′′ i •
2. Let now assume φ ( i ) > φ ( i −
1) = . . . = φ ( i − k ) > φ ( i − k − r such that ψ ( r − ≤ i − k − , ψ ( r ) ≥ i .In this case we get another connected fragment of the graph of F BM ( σ ) ( “acup”).(58) y φ ( i − • } } ⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤ x φ ( i − k ) ◦ y ′ i • / / x ′ i − ◦ y ′ i − k +1 • / / x ′ i − k • a a ❈❈❈❈❈❈❈❈❈❈ Note that in this case the vertices y φ ( i − and x φ ( i − k ) are neighbors and φψ ( r −
1) + 2 ≤ φ ( i −
1) + 1 ≤ φψ ( r ) . In fact, φ ( i −
1) + 1 ≤ φ ( i ) ≤ φψ ( r ) as i ≤ φ ( r ) and φψ ( r − ≤ φ ( i − k − ≤ φ ( i − k ) − φ ( i − − ψ ( r − ≤ i − k − F BM ( σ ) is a free category defined by this multi-story graph.This graph has very pleasant properties. Let us start with a few remarks. • We consider the graph of F BM ( σ ) as embedded into the plane as describedabove. This embedding makes it a planar graph (its arrows have nointersection). • The horizontal arrows are directed rightward. • The ends of the upward arrows are x -objects, and the ends of the down-ward arrows are y -objects. All vertical arrows come in pairs ( α, β ) sothat α is upward, β is downward, and the source of α is a neighbor of thetarget of β . Such pairs of arrows will be called adjoint.3.2.8. Lemma. No vertex in the graph F BM ( σ ) has two incoming or two outgoing arrows. Let a connected path in the graph of F BM ( σ ) , having two ends at the samelevel, starts with an upward arrow α and ends with a downward arrow β .Then the ends of the path are neighbors and α and β are adjoint. bis . Let a connected path in the graph of F BM ( σ ) , having two ends at the samelevel, starts with an downward arrow β and ends with an upward arrow α . Then the ends of the path are neighbors. The graph F BM ( σ ) has no closed cycles.Proof.
1. An arrow in the graph F BM ( σ ) cannot go up or down for more than onefloor. Therefore, we can assume the simplex σ consists of two consecutive arrows, f and g , and the vertex in question belongs to F BM ( w ) where w is the target of f and the source of g . If there are two incoming arrows for a vertex, one of themshould originate from F BM ( f ), and the other from F BM ( g ). But the one originatedfrom F BM ( f ) should be y -vertex, whereas the one originated from F BM ( g ) shouldbe x -vertex. A similar reasoning shows that a vertex cannot have two outgoingarrows.2. A simplest example of a connected path in question is given in 3.2.6. Theclaim is proven by induction in the number of u upward arrows (which is thesame as the number of downward arrows). If u = 1, our path consists of α ,followed by a number of horizontal arrows, followed by β . This is only possiblewhen β and α are adjoint. In case u >
1, let us write down, moving along thepath, all ups and downs. Choose a pair of neighboring α ′ , β ′ , with the downwardarrow β ′ following the upward arrow α ′ . The corresponsing segment of the pathsatisfies the condition of 3.2.8(2), and so, its ends are a pair of neighboring x -and y -objects. Our segment looks just as in the Example 3.2.6. Replacing, inthe notation of loc. cit , the value of the map ψ : [ n ′′ ] → [ n ′ ] at i , declaring ψ ( i ) := ψ ( i − x to y . Weget a shorter path for which the claim is assumed to be already proven.2 bis . The proof is identical to the proof of claim 2.3. Assume there is a closed loop in the graph of F BM ( σ ). Look at the vertices atthe lowest row. There should be a upward arrow from (at least) one of them, anda downward arrow to one of them. According to part 2 of the lemma, these twoare adjoint. On the other hand, the source of the upward adjoint arrow cannotbe connected to thetarget of the downward adjoint arrow. Contradiction. (cid:3) The lemma above implies that F BM ( σ ), considered as a category, is a disjointunion of a finite number of finite totally ordered sets.3.2.9. Two examples.
For later reference, we present two examples of F BM ( σ ). Inboth cases σ : [ k ] → BM is the composition of a map h : [ k ] → [1] with an activemap α : [1] → BM given by an arrow in BM . The map h is given by a sequenceof k + 1 zeros and ones, say, h = 0 i +1 j with i + j = k . We will present twoexamples of α : the first one, is α : a n → a , and the second is α : a n − m → m .The pictures of F BM ( α ◦ h ) and F BM ( α ◦ h ) are presented below.(59) ◦ • (cid:15) (cid:15) ◦ • (cid:15) (cid:15) ◦ • (cid:15) (cid:15) ◦ O O • (cid:15) (cid:15) ◦ O O • (cid:15) (cid:15) ◦ O O • (cid:15) (cid:15) ◦ O O • / / ◦ O O • / / ◦ O O • w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ ◦ g g ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ • (cid:15) (cid:15) ◦ O O • — for F BM ( σ ), and (60) • (cid:15) (cid:15) ◦ • (cid:15) (cid:15) ◦ • (cid:15) (cid:15) • (cid:15) (cid:15) ◦ O O • (cid:15) (cid:15) ◦ O O • (cid:15) (cid:15) • / / ◦ O O • / / ◦ O O • s s ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ • (cid:15) (cid:15) • for F BM ( σ ) (compare to the pictures (53) and (55)).3.2.10. F BM is a functor. To define the functor F BM , we have to describe the opera-tion of “erasing a row of objects” corresponding to an inner face d i : [ n − → [ n ], i = 0 , n .Since F BM ( σ ) are posets, the inner face maps F BM ( d i σ ) → F BM ( σ ) are uniquelydefined once we verify the following. Lemma. The embedding
Ob( F BM ( d i σ )) → Ob( F BM ( σ )) is a map of posets. If i = 0 , | σ | , this is an embedding of posets (that is, two elements of Ob( F BM ( d i σ )) are comparable iff their images are).Proof. The first property is obvious for | σ | = 1; The decomposition formula (56)reduces both claims to the case | σ | = 2 and i = 1.From now on we assume | σ | = 2 and i = 1.We have to prove that for v, v ′ ∈ F BM ( gf ) v is less than v ′ in F BM ( gf ) if andonly if it is less as an element of F BM ( σ ).We will verify this in four possible cases separately (the cases A,B,C and Dbelow).We use the following notation. The simplex σ = ( w f → w ′ g → w ′′ ) is given by w : [ n ] → [1] and a simplex [ n ′′ ] ψ → [ n ′ ] φ → [ n ]. One has w ′ = w ◦ φ , w ′′ = w ◦ φ ◦ ψ .A. Assume v = x ′′ i and v ′ is of y -type. The condition v < v ′ in F BM ( gf ) meansthat φψ ( i ) = φψ ( i − F BM ( σ ) means that there is a pathconnecting v and v ′ . This is only possible if the path is as described by 57 andthis happens precisely when φψ ( i ) = φψ ( i − v = y j , and v ′ is of x -type. The condition v < v ′ in F BM ( gf ) meansthat there is r such that φψ ( r −
1) + 2 ≤ j ≤ φψ ( r ). The same condition in F BM ( σ ) means that there is a path connecting v and v ′ . This is only possible if the path is as described by 58 and we will now show that this happens preciselywhen there is r such that φψ ( r −
1) + 2 ≤ j ≤ φψ ( r ).Choose i such that φ ( i −
1) + 1 ≤ j ≤ φ ( i ). If j ≥ φ ( i −
1) + 2, this means thatour horizontal arrow belongs to F BM ( g ). Otherwise j = φ ( i −
1) + 1; we choose amaximal k such that j = φ ( i − k ) + 1. Then we have a downward arrow y j → y ′ i ,an upward arrow x ′ i − k → x j − , and all horizontal arrows connecting y ′ i with x ′ i − k .C. Assume F BM ( gf ) has an upward arrow, say, going from v = x ′′ i to v ′ = x φψ ( i ) .This means that φψ ( i − < φψ ( i ) which implies ψ ( i − < ψ ( i ). This givesa vertical arrow x ′′ i → x ′ ψ ( i ) in F BM ( σ ). Choose k ∈ [ ψ ( i − , ψ ( i ) − φ ( k ) < φψ ( i ). Then there is a vertical arrow x ′ k +1 → x φψ ( i ) . We also have arrows from x ′ j to y ′ j for j = k, . . . , ψ ( i ) as in thisrange φ ( j −
1) = φ ( j ), and also arrows from y ′ j to x ′ j − for j = k +2 , . . . , ψ ( i ). Thisgives a path from v to v ′ in F BM ( σ ). In the opposite direction, if there is a path in F BM ( σ ) connecting v = x ′′ i to v ′ = x j , it should be necessarily weakly monotone, asotherwise it would contain a subpath having both ends in w ′′ (downstairs) whichis impossible by the description of such paths in part A of the proof. This impliesthat the path starts with an upward arrow, continues horizontally, and ends withanother upward arrow. This is only possible if j = φψ ( i ) and φψ ( i ) > φψ ( i − F BM ( gf ) has adownward arrow, say, going from v = y φψ ( i − to v ′ = y ′′ i . This means that φψ ( i − < φψ ( i ) which implies ψ ( i − < ψ ( i ). This means one has a verticalarrow y ′ ψ ( i − → y ′′ i . Choose k minimal among those satisfying the condition φ ( k −
1) = φψ ( i − y φψ ( i − → y ′ k . We also havearrows from x ′ j to y ′ j for j = ψ ( i − , . . . , k − φ ( j −
1) = φ ( j ),and also arrows from y ′ j to x ′ j − for j = ψ ( i −
1) + 2 , . . . , k . This gives a pathfrom v to v ′ in F BM ( σ ). In the opposite direction, if there is a path in F BM ( σ ),connecting v = y j to v ′ = y ′′ i , it should be necessarily weakly monotone. Thisimplies that the path starts with a downward arrow, continues horizontally, andends with another downward arrow. This is only possible if j = φψ ( i −
1) + 1and φψ ( i ) > φψ ( i − (cid:3) BM X and variants. We now define, for X ∈ Cat , a functor F BM X : (∆ / BM ) op → S by the formula(61) F BM X ( σ ) = Map( F BM ( σ ) , X ) . We will prove in the next subsection that the functor F BM X represents a BM -operadwhich we will denote by BM X .The object that mostly interests us is its Ass − - component which we will denoteby Ass X . This is a planar operad represented by the functor(62) F X ( σ ) = Map( F ( σ ) , X ) , where F is the restriction of F BM : ∆ / BM → Cat to ∆ / Ass (embedded as the leftcopy). It also makes sense to consider the LM -component of BM X which we willdenote by LM X . This will be an LM -operad represented by the functor(63) F LM X ( σ ) = Map( F LM ( σ ) , X ) , where F LM is the restriction of F BM to ∆ / LM .3.2.12. Remark.
The operad
Ass X is an ∞ -categorical version of the construc-tion of Gepner-Hauseng [GH], 4.2.4, presented in the context of simplicial cate-gories. Our picture (53) should be compared to the formula in loc. cit. A detailedcomparison of two definitions is given in the recent paper by A. Macpherson,see [M].Note the following connection between the different variants of the construc-tion. In the following lemma, as usual, π : BM → Ass is the natural projection.3.2.13.
Lemma.
One has a natural equivalence π ∗ ( Ass X ) = BM X × BM ( BM X op ) rev . Proof.
Both constructions are represented by a functor ∆ / BM → Cat , with valuesin posets. The left-hand side is described by the composition∆ / BM π → ∆ / Ass F → Cat , whereas the right-hand side is described by G : ∆ / BM → Cat given by the formula G ( σ ) = F BM ( σ ) ⊔ F BM ( σ op ) op . We will construct the equivalence of G with F ◦ π as follows. We use that bothfunctors take values in posets and both satisfy the dual Segal conditions. We willfirst present a one-to-one correspondence between the sets Ob G ( σ ) and Ob F ◦ π ( σ ) functorial in σ . Then, in order to verify that the presented correspondenceis compatible with the partial orders, we will verify this in the case where | σ | = 1and then we will use the dual Segal condition which ensures that the partialorder on Ob F ◦ π ( σ ) and Ob G ( σ ), with σ = ( w f → w → . . . f n → w n ), is uniquelydetermined by the partial orders on F ◦ π ( f i ) and G ( f i ), respectively.So, it is sufficient to compare Ob F ◦ π ( σ ) with Ob G ( σ ) and verify, for | σ | = 1,the isomorphism of the corresponding posets.Let us first establish the bijection for σ : [0] → BM . We will use the notationof 3.2.2. For σ presented by the word a k m α b l , the poset F ◦ π ( σ ) is a (discrete)set of 2( k + α + l ) black and white points, each letter among the k + α + l definingone black and one white point (drawn from right to left), see the first line of (51).For the same σ , the poset G ( σ ) is also a discrete set of 2( k + α + l ) black and whitepoints, as F BM ( σ ) gives 2 k points corresponding to a -letters and one black pointto the m letter (if α = 1), see the second line of (51), whereas F BM ( σ op ) op gives a white point for m and pair of points for each b -letter. Note that passage σ σ op interchanges a ’s with b ’s, whereas the passage F BM ( F BM ) op interchanges blackand white points.Since Ob G ( σ )) = ⊔ Ob G ( w i ) and Ob F ◦ π ( σ )) = ⊔ Ob F ◦ π ( w i ), the corre-spondence for zero-dimensional simplices extends functorially to all σ .Now, the pictures presented in 3.2.4 show that for σ : [1] → BM our one-to-onecorrespondence is compatible with the partial order. (cid:3) BM X is a flat operad. Let X ∈ Cat . In this subsection we prove that BM X defined in (61), is a flat BM -operad. This will also imply that Ass X is a flat planaroperad.First of all, one has the following.3.3.1. Lemma. BM X is a category over BM .Proof. We have to verify that BM X is Segal and complete. It is Segal becauseof the dual Segal condition of F BM , see(50). To verify completeness, one hasto calculate the fibers. Fix w = a k m α b l . The fiber of F BM at w carries [ n ] to([ n ] op ⊔ [ n ]) k ⊔ ([ n ] op ) α . Thus, the fiber of F BM X at w is ( X op × X ) k × ( X op ) α .This is obviously a complete Segal space. (cid:3) Cocartesian arrows over conventional categories.
Let B be a conventionalcategory with no nontrivial isomorphisms, and let a category X over B be pre-sented by a functor F : (∆ /B ) op → S . We wish to formulate a condition for anarrow a ∈ F ( α ), α : [1] → B , to be cocartesian. Let a : x → y and α : ¯ x → ¯ y .Recall that a is cocartesian if the diagram(64) X y/ −−−→ X x/ y y B ¯ y/ −−−→ B ¯ x/ induces an equivalence X y/ → B ¯ y/ × B ¯ x/ X x/ of left fibrations over X .In what follows we use the presentation of categories by simplicial spaces, sothat X n = Map([ n ] , X ). A map of left fibrations over X is an equivalence if andonly if it induces an equivalence of the fibers.This means that it is sufficient to verify that the diagram (64) induces a carte-sian square of the corresponding maximal subspaces(65) { y } × X X −−−→ { x } × X X y y { ¯ y } × B B −−−→ { ¯ x } × B B . [ n ] op is, of course, isomorphic to [ n ]. But it has a different functoriality in n . B is discrete. Therefore, to verify that(64) is cartesian, it is sufficient to verifythe equivalence of the fibers of the vertical arrows. To formulate the criterion wewill use the following notation.Given an arrow β : ¯ y → ¯ z in B we denote by ( α, β ) the 2-simplex ¯ x α → ¯ y β → ¯ z in B . Proposition.
Let B be a conventional category with no nontrivial isomorphisms.Let a map f : X → B in Cat be given by a functor F : (∆ /B ) op → S . Thenan arrow a : x → y is f -cocartesian iff for any β : ¯ y → ¯ z in B the map { y } × F (¯ y ) F ( β ) → { x } × F (¯ x ) F ( β ◦ α ) defined as a composition { y }× F (¯ y ) F ( β ) a → { x }× F (¯ x ) F ( α ) × F (¯ y ) F ( β ) ∼ ← { x }× F (¯ x ) F ( α, β ) d → { x }× F (¯ x ) F ( β ◦ α )) , is an equivalence. (cid:3) Proposition.
The map p : BM X → BM presents BM X as a BM -operad.Proof. We have to prove that BM X is fibrous over BM .1. Let us verify that inert arrows in BM have cocartesian liftings. Let f : w ′ → w be inert, with w ′ = a k ′ m α ′ b l ′ and w = a k m α b l . The space F BM X ( f ) is the product of2 k + α copies of X (as in (52)). The description of the functor F BM X presenting BM X (61), (56) together with Proposition 3.3.2 assert that an object in the productpresents a cocartesian arrow iff all its components are equivalences. This givesa recipe to construct a cocartesian lifting: an object in F X ( w ′ ) is given by acollection of 2 k ′ + α ′ elements in X , and its cocartesian lifting is obtained byreplacing those corresponding to the 2 k + α positions retained by w , with theirdegeneracies in X .2. Let w = a k m α b l . The next step is to verify that k + α + l cocartesian liftingsof the inerts w → x , x ∈ { a, m, b } , give rise to the equivalence( BM X ) w → ( BM X ) ka × ( BM X ) αm × ( BM X ) lb . This also follows from the description of cocartesian liftings as presented bycollections of equivalences.3. The last condition means that, given f : w → u in BM and objects x ∈ ( BM X ) w , y ∈ ( BM X ) u , with a choice of cocartesian arrows ρ i ! : y → y i over theinerts ρ i : u → u i decomposing u , one has an equivalence(66) Map f ( x, y ) → Y i Map ρ i ◦ f ( x, y i ) . The space Map f ( x, y ) is, by definition, the fiber of the restrictionMap BM ([1] , BM X ) → Map BM ( ∂ [1] , BM X ) at ( x, y ), with the map [1] → BM given by f . In other words, Map f ( x, y ) is thefiber of the restriction map(67) Map( F BM ( f ) , X ) → Map( F BM ( w ) ⊔ F BM ( u ) , X ) . Now recall that, for f : w → u in BM , F BM ( f ) is presented by a graph, whoseset of vertices is precisely F BM ( w ) ⊔ F BM ( u ). Thus, the pair ( x, y ) is given by acollection of objects in X numbered by the set F BM ( w ) ⊔ F BM ( u ), and the fiberof (67) is given by the product of the mapping spaces in X between all pairs ofvertices in the graph, connected by an arrow in F BM ( f ).The same calculation can be applied to ρ i ◦ f . The space Map ρ i ◦ f ( x, y i ) is simi-larly defined by the arrows of the graph F BM ( ρ i ◦ f ). According to the descriptionof the arrows given in 3.2.4, each arrow from F BM ( f ) appears in precisely one of F BM ( ρ i ◦ f ). This proves that (66) is an equivalence. (cid:3) Recall 2.3 that a functor f : C → D assigns to each α : d → d ′ in D acolimit preserving map f α : P ( C d ′ ) → P ( C d ). Flatness of f , according to 2.3.2,means that this assignment is compatible with the compositions. We will usethis approach to prove that p : BM X → BM is a flat BM -operad. First of all, wewill give an explicit description of the map p α : P (( BM X ) w ) → P (( BM X ) v ) for any α : v → w in BM .3.3.4. Fix α : v → w in BM . We will describe p α : P (( BM X ) w ) → P (( BM X ) v ) interms of the graph presenting F BM ( α ), see, for instance, (54) as illustration.The categories F BM ( w ) and F BM ( v ) are discrete, consisting of points of x -type(denoted as ◦ ) and of y -type (denoted as • ). We assign to each x -type pointthe category P ( X ) and to any y -type point the category P ( X op ). To a disjointunion of points we assign the tensor product of the corresponding P ( X ) and P ( X op ) . So we get the categories P (( BM X ) w ) and P (( BM X ) v ). The functor p α will be defined as the tensor product of functors numbered by the connectedcomponents of F BM ( α ).All components consist of one vertex or one arrow. Taking into account thetype of the vertices, one has six different components.(1) An x -type single vertex.(2) An y -type single vertex.(3) An upward arrow.(4) A downward arrow.(5) A lower horizontal arrow.(6) An upper horizontal arrow.To each component F of one of the types above, we assign an arrow φ F as follows. The tensor product of categories with colimits, see [L.HA], 5.8.1. (1) For F of type (1) φ F : S → P ( X ) is the colimit preserving map carrying ∗ to the final object.(2) For F of type (2) φ F : S → P ( X op ) is the colimit preserving map carrying ∗ to the final object.(3) For F of type (3) φ F : P ( X ) → P ( X ) is the identity.(4) For F of type (4) φ F : P ( X op ) → P ( X op ) is the identity.(5) For F of type (5) φ F : P ( X op × X ) → S is the colimit preserving functorextending the Yoneda X op × X → S .(6) For F of type (6) φ F : S → P ( X × X op ) is the colimit preserving mapcarrying ∗ to the final object.We are now ready to formulate the result.3.3.5. Lemma.
Let α : v → w be an arrow in BM . The functor p α : P (( BM X ) w ) → P (( BM X ) v ) is naturally equivalent to the tensor product of φ F described above,over the connected components F of the graph F BM ( α ) .Proof. Denote by F α the composition F α : ∆ / [1] α → ∆ / BM F BM → Cat , where the functor F BM is described in 3.2.Note that F α is uniquely defined by its value F BM ( α ) at id [1] .The functor F αX : ∆ op / [1] → Cat given by the formula F αX ( s ) = Map( F α ( s ) , X )defines the base change BM αX = BM X × BM [1] as a category over [1].Recall that F α has the following properties.(D1) It has values in disjoint unions of totally ordered posets.(D2) For s ∈ ∆ / [1] with | s | = 0, F α ( s ) is discrete.(D3) F α satisfies a dual Segal condition.A functor F : ∆ / [1] → Cat satisfying the properties (D1)–(D3) listed abovewill be called distinguished . Any distinguished functor F determines a category T ( F ) over [1] represented by the simplicial space over [1] given by the formula s Map( F s , X ). Thus, it defines a colimit preserving functor φ F : P ( T ( F ) ) → P ( T ( F ) ), so that in our new notation p α = φ F α . Disjoint union F ⊔ G of dis-tinguished functors gives rise to the product of spaces over [1], so to the ten-sor product φ F ⊗ φ G of the corresponding colimit preserving functors. Thus, φ F α is the tensor product of φ F where F runs through indecomposable subfunc-tors of F α . These are determined by the connected components of the category F BM ( α ) = F α (id : [1] → [1]).The six possible types of components of F BM ( α ) are described in 3.3.4. Itremains to verify the formulas for φ F in each case presented there. In all cases apart from (5) the corresponding category over [1] is a cocartesianfibration; this can be verified, for instance, using Proposition 3.3.2. This givesthe formulas for φ F in the cases (1)–(4) and (6).The case (5) is more interesting. The fibers at 0 and at 1 of the total category Z := T ( F ) are ∗ and X × X op respectively. We have to describe the colimitpreserving functor φ F : P ( X op × X ) → S . It is uniquely defined by its restriction φ : X op × X → S .It remains to show that φ is just the canonical Yoneda map carrying ( x, y ) toMap( x, y ). This is verified as follows. For any category Z over [1] with fibers Z and Z over { } and { } respectively the classifying map Z op0 × Z → S is definedby the bifibration Fun [1] ([1] , Z ) → Z × Z , see [L.T], 2.4.7.10. In our case Z is a point and Z = X op × X so the abovebifibration becomes a left fibration Fun [1] ([1] , Z ) → X op × X . It remains toidentify Fun [1] ([1] , Z ) with Tw( X ). Looking at these categories as simiplicialspaces, we see that Tw( X ) n = X n − and Fun [1] ([1] , Z ) n = Map [1] ([1] × [ n ] , Z ).Let us calculate this space. Let s m : [ k ] → [1] be given by k − m values ofzeros and m + 1 ones. Denote by [ k ] m the k -simplex over [1] defined by s m .Then Map [1] ([ k ] m , Z ) = X m +1 . The standard decomposition of the product ofsimplices [1] × [ n ] = [ n + 1] n ⊔ [ n ] n − [ n + 1] n − ⊔ [ n ] n − . . . ⊔ [ n + 1] , see, for instance, [GZ], II.5.5, is an equivalence of categories over [1]. which yieldsFun [1] ([1] , Z ) n = X n +1 × X n − X n − × . . . × X X = X n +1 . It is easy to see that these identifications are compatible with the faces andthe degeneracies in ∆, which finally identifies Fun [1] ([1] , Z ) with Tw( X ). (cid:3) Proposition.
The map p : BM X → BM is a flat BM -operad.Proof. Let σ : [2] → BM be given by a pair of composable arrows α and β , with β active. We can use Lemma 3.3.5 to calculate the maps p α , p β and p β ◦ α . Themap p β ◦ α is determined by the connected components of F BM ( β ◦ α ), whereas thecomposition p α ◦ p β is determined by the components of F BM ( σ ). The componentsof F BM ( β ◦ α ) are determined by the components of F BM ( σ ) in Lemma 3.2.10. Theonly component of F BM ( σ ) making an impact different from the correspondingcomponent of F BM ( β ◦ α ), is a horizontal segment x ′ → y ′ defined by [0] → [1] → [0]. Since β is active, such component does not appear. (cid:3) Quivers and their action. M we define the planar quiver operad Quiv X ( M ) =Funop Ass ( Ass X , M ).We will also use LM and BM -versions of quiver operads. For an LM -operad M wedefine Quiv LM X ( M ) = Funop LM ( LM X , M ).Similarly, for a BM -operad M we define Quiv BM X ( M ) = Funop BM ( BM X , M ).3.4.2. Let M = ( M a , M m ) be an LM -operad, with an Ass -component M a ∈ Op Ass and an m -component M m ∈ Cat . Then, by (2.8.8), the
Ass -component of the LM -operad Quiv LM X ( M ) is Quiv X ( M a ) and its m -component is Fun( X, M m ).Similarly, for a BM -operad M = ( M a , M m , M b ), the components of Quiv BM X ( M )are (Quiv X ( M a ) , Fun( X, M m ) , M b ).3.4.3. Remark.
As our notation suggests, we consider Quiv X ( M ) as a primaryobject, and introduce Quiv LM and Quiv BM only to describe the action of Quiv X ( M )on various objects.3.4.4. Proposition.
Let X be a space. Then the category of associative algebras Alg
Ass (Quiv X ( M )) is equivalent to the category Alg ∆ op X ( M ) of Gepner-Haugseng [GH] .Proof. By definition,
Alg
Ass (Quiv X ( M )) = Fun Ass ( Ass X , M ). D. Gepner andR. Haugseng define M -enriched precategories with the space of objects X as ∆ X -algebras in M , where ∆ X is a certain generalized planar operad. They provein [GH], 4.2.7, that the image of ∆ X under the localization functor to planaroperads is naturally equivalent to O X , defined in [GH], 4.2.4, which is just aversion of our Ass X , described in terms of simplicial categories. For a detailedcomparison of two definitions see A. Macpherson [M]. (cid:3) Dependence on X and M . We have just assigned to any pair ( X, M ),where X ∈ Cat and M ∈ Op Ass , a planar operad Quiv X ( M ), such that associativealgebras in it are M -enriched precategories. We would like to be able to define anenriched precategory without specifying the space of objects or even the planaroperad M .This can be done using the formalism of operad families.3.5.1. The BM -operad family Quiv BM . We have a functor(68)
Cat op × Op BM → Op BM carrying ( X, M ) to Quiv BM X ( M ).This functor is a composition of(69) BM : Cat → Op BM carrying X to BM X , and the bifunctor(70) Funop : Op op BM × Op BM → Op BM adjoint to the product in Op BM .The functor (68) defines a bifibered BM -operad family over Cat × Op BM . Wedenote it as Quiv BM .A map j : X → Y in Cat defines j ! : Quiv BM Y ( M ) → Quiv BM X ( M ). A map f : M → N of BM -operads defines f ! : Quiv BM X ( M ) → Quiv BM X ( N ).Given j : X → Y and f : M → N as above, and P ∈ Quiv BM X ( M ) , Q ∈ Quiv BM Y ( N ) the space Map j,f Quiv BM ( P, Q ) of maps over ( j, f ) is equivalent to Map
Quiv BM X ( N ) ( f ! P, j ! Q ).Similarly to the above, one defines the families Quiv LM and Quiv.3.5.2. Multiplicativity.
The functor (69) preserves products as it is corepresentable,when considered as a functor (∆ / BM ) op → S and as the embedding Op BM → Fun((∆ / BM ) op , S ) preserves limits by 2.6.7. The functor (70) is lax symmetricmonoidal, with respect to the cartesian symmetric monoidal structure on Op BM as it is right adjoint to the direct product functor which is symmetric monoidal,see [H.R], A.5.3.Therefore, the functor Quiv BM : Cat op × Op BM → Op BM , defined in (68), is laxsymmetric monoidal.The latter implies, in particular, that one has a canonical operad map(71) µ : Quiv BM X ( M ) × Quiv BM Y ( N ) → Quiv BM X × Y ( M × N ) . In the same way, one defines lax symmetric monoidal functors Quiv LM : Cat op × Op LM → Op LM and Quiv : Cat op × Op Ass → Op Ass .One can reformulate the above multiplicativity property in terms of familiesof operads.The lax symmetric monoidal functor (68) defines, by adjunction, a map ofoperads(72) quiv BM : Op BM → Funop(
Cat op , Op BM ) , where Op BM is endowed with its cartesian SM structure. By Proposition 2.8.10, theright-hand side identifies with the category FamOp BM endowed with the cartesianSM structure. This yields a lax SM functorquiv BM : Op BM → FamOp BM carrying a BM -operad M to the family Quiv BM ( M ) of BM -operads. In the sameway, one has lax symmetric monoidal functors quiv LM : Op LM → FamOp LM andquiv : Op Ass → FamOp
Ass .This implies the following.3.5.3.
Corollary.
Let M ∈ Alg O ( Op Ass ) . Then the category of M -precategories PCat ( M ) has a canonical O -monoidal structure. In particular, if M is an E n -monoidal category, PCat ( M ) is a E n − -monoidal. Proof.
By the additivity theorem [L.HA], 5.1.2.2, M can be seen as an E n − -algebra object in Op Ass . Since the functor quiv : Op Ass → FamOp
Ass is lax symmet-ric monoidal, Quiv( M ) is an E n − -algebra in FamOp
Ass .Now, the functor
Alg : FamOp
Ass → Cat assigning to a family of planar operads M the category Alg
Ass ( M ), preserveslimits, so it carries E n − -algebras to E n − - algebras. (cid:3) Fix M ∈ Op BM .3.5.4. Lemma.
The contravariant functor X Quiv BM X ( M ) from Cat to Op BM carries κ -filtered colimits to limits, for a certain regular cardinal κ .Proof. The functor is a composition of several functors. • The functor X BM X considered as a functor Cat → P (∆ BM ) preserves κ -filtered colimits as the corepresenting objects F BM ( σ ) are κ -compact cat-egories (for any κ ). • The embedding Op BM → P (∆ BM ) reflects equivalences. It is a composition Op BM g → Cat + BM ♮ j → Cat BM h → P (∆ BM ) , with g, h accessible full embeddings and j colimit preserving. Thus, thisembedding preserves κ -filtered colimits for some κ . • Finally, we compose the above with the functor Funop( , M ) carryingcolimits to limits. (cid:3) Folding of BM . A - B bimodules are the same as left A ⊗ B op -modules. A BM -monoidal category C consists of a pair of monoidal categories C a and C b acting from the left and from the right on a category C m . The same action canalternatively be presented by a left C a × C rev b action on C m , defining, therefore, an LM -monoidal category. This fact, very well-known for the conventional categories,requires a justification in the context of ∞ - categories. In this subsection we generalize and prove the above observation to operads,constructing a folding functor φ : Op BM → Op LM . Furthermore, φ induces an equivalence of the corresponding categories of alge-bras, in particular, an equivalence between A - B bimodules and left A ⊠ B op -modules.This construction assigns to a BM -operad P with components ( P a , P m , P b ) an LM -operad with components ( P a × P rev b , P m ).This functor “folds” BM into LM , similarly to folding the simply-laced Dynkindiagram A n − into B n . This subsection is not formally connected to the rest ofthis section. We will use it in Section 6 to construct the Yoneda embedding.3.6.1. Folding.
The functor φ will be expressed via the functor(73) ψ : ∆ / LM → P (∆ / BM )defined in 3.6.2. The functor ψ induces a functor(74) φ : P (∆ / BM ) → P (∆ / LM ) , right adjoint to the colimit preserving extension ˆ ψ : P (∆ / LM ) → P (∆ / BM ). Thisfunctor carries F ∈ P (∆ / BM ) to φ ( F ) defined by the formula φ ( F )( σ ) = Map P (∆ / BM ) ( ψ ( σ ) , F ) . We present below the definition of (73) and, after that, we verify that thefunctor (74) induced by ψ , carries Op BM to Op LM .3.6.2. Functor ψ . Recall that BM = (∆ / [1] ) op and LM is the full subcategory of BM spanned by the s : [ n ] → [1] satisfying s (0) = 0 and having at most one value 1.The order-inverting functor op : ∆ → ∆ induces op : BM → BM (carrying s : [ n ] → [1] to s op : [ n ] op → [1] op ∼ → [1]) interchanging the subcategories Ass − with Ass + and LM with RM . Ass − is a full subcategory of LM ; denote by LM − the full subcategory of LM spanned by the objects that are not in Ass − . There are no arrows in LM from anobject of Ass − to an object of LM − .The category LM − is isomorphic to the subcategory of BM consisting of theobjects and the arrows invariant with respect to op. Any object w ∈ BM satisfying w = w op is given by s : [2 n − → [1] with s ( n −
1) = 0 , s ( n ) = 1; thecorresponding object v of LM − is given by the composition [ n ] i → [2 n − s → [1].We write w = v ∗ in this case; the same * notation is used for the arrows of LM − .Let us describe ψ ( σ ) for σ : [ n ] → LM . Let σ : v f → . . . f n → v n . Assume that the objects v , . . . , v m , m ≥ −
1, are in LM − , and the rest of v i arein Ass − . We will define ψ ( σ ) by the cocartesian square(75) ψ ( σ ≤ m ) z z ttttttttt $ $ ❏❏❏❏❏❏❏❏❏ ψ − ( σ ) % % ❏❏❏❏❏❏❏❏❏ ψ + ( σ ) y y ttttttttt ψ ( σ )of representable presheaves in P (∆ / BM ) where ψ ( σ ≤ m ) : v ∗ → . . . → v ∗ m , ψ − ( σ ) : v ∗ → . . . → v ∗ m → v m +1 → . . . → v n , ψ + ( σ ) : v ∗ → . . . → v ∗ m → v op m +1 → . . . → v op n ,and the map v ∗ m → v m +1 (resp., v ∗ m → v op m +1 ) is given as the composition of f m +1 (resp., f op m +1 ) with the inert v ∗ m → v m (resp., v ∗ m → v op m ). In particular, if m = n ,that is, if σ is a simplex in LM − , ψ ( σ ) is obtained by applying the functor ∗ to σ .On the contrary, if m = − ψ ( σ ) is the coproduct of two representables, σ and σ op .Note the following dual Segal condition for ψ . Lemma.
Let σ : [ n ] → LM be as above, < k < n . Denote by σ ≤ k and σ ≥ k thetwo halves of σ of dimensions k and n − k respectively. Then ψ ( σ ) = ψ ( σ ≤ k ) ⊔ ψ ( v k ) ψ ( σ ≥ k ) . (cid:3) φ carries categories over BM to categoriesover LM . A presheaf F ∈ P (∆ / BM ) represents a category over BM iff it satisfies thecompleteness and the Segal condition. Let us verify that φ ( F ) satisfies the Segalcondition. The latter means that φ ( F )( σ ) → φ ( F )( σ ≤ k ) × φ ( F )( v k ) φ ( F )( σ ≥ k ) is anequivalence for all k , 0 < k < n . This immediately follows from the dual Segalcondition for ψ .Completeness of φ ( F ) can be verified pointwise (see 3.1.4). If v ∈ Ass − , thecorresponding fiber is F ( v ) × F ( v op ). Otherwise, the fiber is F ( v ∗ ). In any case,this is a complete Segal space.3.6.4. Cat -enrichment.
Let B be a category. For X, Y ∈ Cat /B we denote asFun B ( X, Y ) the category representing the functor K Map
Cat /B ( X × K, Y ) . there are no arrows from Ass − to LM − Let K ∈ Cat be defined by a simplicial space K . Then BM × K as an object of Cat / BM is defined by the presheaf (∆ / BM ) op → ∆ op K → S . Therefore, φ ( BM × K ) isdefined by the functor carrying σ ∈ ∆ / LM as in (75) to K n × K m K n .This yields a canonical morphism φ ( X ) × K → φ ( X ) × φ ( BM × K ) = φ ( X × K ).This implies that φ : Cat / BM → Cat / LM preserves this Cat -enrichment, that is,induces a map Fun BM ( X, Y ) → Fun LM ( φ ( X ) , φ ( Y )) for any pair X, Y ∈ Cat / BM .3.6.5. φ carries operads to operads. It remains to verify that φ ( F ) is fibrous if F is fibrous. The first condition is the existence of cocartesian liftings of the inerts.To verify it, we will use Proposition 3.3.2.Let α : u → v be an inert arrow in LM . In the case u, v ∈ Ass − φ ( F )( α ) = F ( α ) × F ( α ) op , so we choose a pair of cocartesian liftings in F ( α ) and F ( α op )separately. In the case u, v ∈ LM − , α ∗ : u ∗ → v ∗ is also inert and we choose itscocartesian lifting in F ( α ∗ ).In the remaining case, α : a n m → a k , one has φ ( F )( α ) = F ( a n mb n → a n ) × F ( a n mb n ) F ( a n mb n → a n ) , and we choose a pair of inerts in F ( a n mb n → a k ) and in F ( a n mb n → b k ) havingthe same source in φ ( F )( u ) = F ( a n mb n ). The liftings chosen are cocartesian byProposition 3.3.2.Segal condition for φ ( F ) is pretty clear.It remains to verify the property (Fib3): given an arrow a : v → w in LM , x ∈ φ ( F )( v ) and y ∈ φ ( F )( w ), the mapMap a ( x, y ) → Y i Map ρ i ◦ a ( x, y i )defined by cocartesian liftings y → y i of ρ i : w → w i decomposing w , is anequivalence. The claim directly follows from the definition of φ ( F ) and from theproperty (Fib3) for the BM -operad defined by the functor F .3.6.6. φ carries BM -monoidal categories to LM -monoidal categories. Let F ∈ P (∆ / BM )represent a BM -monoidal category. Then φ ( F ) ∈ Op LM is an LM -monoidal category.In fact, we have to verify that any arrow α : u → v in LM has a cocartesian liftingin φ ( F ). We already know this for α inert. For α active, either α belongs to Ass − , or to LM − . In the first case ψ ( α ) = α ⊔ α op , whereas in the second case ψ ( α ) = α ∗ . In any case the candidate for a cocartesian lifting of α comes froma cocartesian lifting of α, α op or α ∗ in F . Proposition 3.3.2 allows one to verifythat the candidate is in fact a cocartesian lifting. Remark.
Note that the functor φ restricted to BM -monoidal categories has analternative (simpler) description. A monoidal BM -category is given by a functor F : BM → Cat satisfying Segal condition. The functor φ ( F ) : LM → Cat can be defined as the composition LM Ψ → P ( BM ) F ′ → Cat , where F ′ is the colimit preserving extension of F and Ψ carries v ∈ Ass − to v ⊔ v op and v ∈ LM − to v ∗ .3.6.7. The functor φ : Op BM → Op LM preserves Cat -enrichment and carries BM to LM . Therefore, it defines a canonical map(76) Φ : Alg BM ( C ) → Alg LM ( φ ( C )) . We have
Proposition. Φ is an equivalence.Proof. Both categories of algebras are cartesian fibrations over
Alg
Ass ( C a ) × Alg
Ass ( C b ) by 2.12.2. Thus, to prove Φ is an equivalence, it is sufficient, forany choice of A ∈ Alg
Ass ( C a ) , B ∈ Alg
Ass ( C b ), to prove that the mapΦ A,B : A BMod B ( C m ) → LMod A ⊠ B op ( C m )is an equivalence. Look at the commutative diagram(77) A BMod B ( C m ) Φ A,B / / G BM % % ▲▲▲▲▲▲▲▲▲▲▲ LMod A ⊠ B op ( C m ) G LM x x ♣♣♣♣♣♣♣♣♣♣♣♣ C m F LM F BM e e of solid arrows (the arrows G BM and G LM are the forgetful functors).We will first verify the claim in the case C is a BM -monoidal category. In this casethe functors G BM and G LM admit left adjoint functors F BM and F LM of free ( A, B )-bimodule and of free left A ⊠ B op -module respectively. Commutativity of the soliddiagram provides a map F LM → Φ A,B ◦ F BM , and, therefore, G LM ◦ F LM → G BM ◦ F BM .This is an equivalence: if V ∈ C m and u : V → F BM ( V ) is the canonical mapin C m presenting F BM ( V ) as a free ( A, B )-bimodule, the same map will present F BM ( V ) as a free left A ⊠ B op -module. Finally, according to [L.HA], 4.7.3.16, thisimplies that Φ A,B is an equivalence.To prove the claim for general C ∈ Op BM , we will find a fully faithful BM -operadmap C → D into a BM -monoidal category. We proceed as follows. Recall that BM → BM ⊗ is a strong approximation. Let C ′ ∈ Op BM ⊗ correspond to C underthe equivalence 2.7.2. Let D ′ = Env BM ⊗ ( C ′ ) be the BM ⊗ -monoidal envelope of C ′ . According to [L.HA], 2.2.4.10, one has a fully faithful morphism C ′ → D ′ of BM ⊗ -operads. The equivalence Op BM ⊗ → Op BM is given by a base change, so itcarries the fully faithful map C ′ → D ′ to a fully faithful map C → D in Op BM . Now, the fully faithful embedding C → D gives rise to a cartesian square A BMod B ( C m ) / / G BM (cid:15) (cid:15) A BMod B ( D m ) G BM (cid:15) (cid:15) C m / / D m for A BMod B ( C m ), and a similar cartesian square for LMod A ⊠ B op ( C m ). This reducesthe assertion for general C ∈ Op BM to that for D . (cid:3) A ∈ Alg
Ass ( C a ) and for B ∈ Alg
Ass ( C b ) , one has an equiv-alence A BMod B ( C m ) = LMod A ⊠ B op ( C m ) . Let us now assume C is a SM category.Then C a = C b = C m = C and the action of C × C on C factors through thebifunctor µ : C × C → C defined by the SM structure on C . This implies that,according to 2.12.14, one has an equivalence LMod A ⊠ B op ( C ) = LMod A ⊗ B op ( C ). Wehave verifies the following. Corollary.
Let
A, B be associative algebras in a symmetric monoidal category C . Then one has a natural equivalence A BMod B ( C ) = LMod A ⊗ B op ( C ) . In particular, for C = Cat we justify the ∞ -categorical version of the claimmade in 3.6.0.3.6.9. The (symmetric) operads LM ⊗ and RM ⊗ are equivalent: there is an obviousequivalence of LM and RM -algebras with values in any C ∈ Op , carrying an LM -algebra ( A, M ) to the RM -algebra ( M, A op ). Similarly, the operad BM ⊗ governingtriples ( A, M, B ) consisting of two algebras and a bimodule, is equivalent to theoperad
LLM governing a pair of algebras
A, C acting on the left on M , so that theactions commute with each other. The equivalence carries a triple ( A, M, B ) thetriple (
A, B op , M ).Using these equivalences, we can deduce from (37) and Corollary 3.6.8 thefollowing. Corollary.
Let M be left-tensored over monoidal categories A and B , so that thetwo actions commute. Let A be an algebra in A , B an algebra in B , and denote A ⊠ B the corresponding algebra in A × B . Then there is a canonical equivalenceof the categories of modules LMod A ⊠ B ( M ) = LMod A ( LMod B ( M )) . X be a fixed category and let an operad P ∈ Op BM be given by thefunctor P ( σ ) = Map( Y ( σ ) , X ) for a certain functor Y : ∆ / BM → Cat (the operads BM X and the relatives constructed above have this form). Then φ ( P ) is definedby the formula φ ( P )( σ ) = Map( Z ( σ ) , X ) where Z is the composition∆ / LM ψ → P (∆ / BM ) Y → Cat , where Y is extended from ∆ / BM to preserve colimits.3.6.11. Proposition. The functor φ : Op BM → Op LM preserves limits. For P ∈ Op BM one has φ ( P ) = φ ( P rev ) . φ ( BM X ) = LM X .Proof. The functor φ : Cat / BM → Cat / LM has a left adjoint, and so preserveslimits. The embedding Op LM → Cat / LM creates limits as it is conservative and is acomposition of two right adjoint functors, Op LM → Cat + / LM ♮ → Cat / LM , the second being the functor forgetting the markings. The second claim is animmediate consequence of the definition. To prove the third claim, we use Remark3.6.10. The functor φ ( BM X ) is represented by the composition∆ / LM ψ → P (∆ / BM ) F BM → Cat , which is easily seen to coincide with F LM . (cid:3) Corollary. φ ( BM X × BM ( BM Y ) rev ) = LM X × Y . In particular, φ ( π ∗ ( Ass X )) = LM X × X op .Proof. The first claim directly follows from Proposition 3.6.11. The second claimis a special case of the first, with Y = X op , joined with Lemma 3.2.13. (cid:3) φ preserves products, one has a canonical arrow(78) φ (Funop BM ( P , Q )) → Funop LM ( φ ( P ) , φ ( Q )) . Applying this to P = π ∗ ( Ass X ) and Q = π ∗ ( M ), and taking into account 3.6.12(2), we get a canonical map(79) φπ ∗ (Quiv X ( M )) → Quiv LM X × X op ( φπ ∗ ( M ))of LM -operads. Note that the map 79 is a morphism of functors from Cat op × Op Ass to Op LM . 4. Quiv X ( M ) when M is a monoidal category In reasonably good cases Quiv BM X ( M ) is a BM -monoidal category. In this section(Theorem 4.4.8) we will show that this happens when M itself is a BM -monoidalcategory, with the monoidal structure behaving well with respect to certain col-imits. A similar result holds for the LM -version. Furthermore, in Section 4.5 weidentify, for a monoidal category M with colimits, Quiv X ( M ) with the monoidalcategory of endomorphisms of Fun( X, M ) considered as a right M -module. We will proceed as follows. First of all, we describe the category of colors ofQuiv BM X ( M ), that is the fibers of p : Quiv BM X ( M ) → BM at the objects of BM = { a, m, b } .In order to prove the theorem, we describe local cocartesian liftings of theactive arrows in BM for q : Quiv BM X ( M ) → BM . This will allow us to see that, undercertain conditions on M , such local cocartesian liftings exist and commute withthe compositions.4.1. Colors of
Quiv BM X ( M ) . Let us describe the fibers of Quiv BM X ( M ) for M ∈ Op BM at a, m, b ∈ BM .The fibers of BM X at a, m and b in BM respectively are X op × X , X , and [0].By definition, Quiv BM X ( M ) a is Alg
Ass ◦− / BM (Quiv BM X ( M )) = Alg
Ass ◦− × BM BM X / Ass ( Ass − × BM M ) = Alg
Ass ◦ X / Ass ( Ass − × BM M ) , see 2.8.7. The latter easily yields4.1.1. Lemma.
Quiv BM X ( M ) a = Fun( X op × X, M a ) . (cid:3) In the same way one obtains4.1.2.
Lemma.
Quiv BM X ( M ) m = Fun( X, M m ) . Quiv BM X ( M ) b = M b . (cid:3) The same formulas describe the colors of Quiv LM X ( M ) for M ∈ Op LM .4.1.3. Lemma.
Quiv LM X ( M ) a = Fun( X op × X, M a ) . Quiv LM X ( M ) m = Fun( X, M m ) . (cid:3) Spaces of active maps.
Our next step is to describe the mapping spaces inQuiv BM X ( M ) over active arrows of BM . The description is given in Proposition 4.2.1.We proceed as follows. We assume that the active arrow α : w → u in BM lies over h n i → h i in Ass , as in general our mapping spaces will be products ofmapping spaces over such α .Fix an object g in Quiv BM X ( M ) u and an object f = ( f , . . . , f n ) in Quiv BM X ( M ) w .We wish to describe the space(80) Map α Quiv BM X ( M ) ( f, g )of arrows from f to g over α .Denote A = α ∗ ( BM X ), M = α ∗ ( M ). These are categories over [1]. Our first(quite straightforward) step will be to identify (80) with the fiber of the restrictionmap(81) Fun [1] ( A, M ) → Fun( A , M ) × Fun( A , M ) , at ( f, g ). Here A i , M i , i = 0 , , are the fibers of A, M at the ends of [1]. In order to calculate this fiber, we will find a special presentation of the category A over [1] as a certain colimit. This is done as follows. For each α we find acategory C with a pair of maps p : C → A , q : C → A , so that one has anequivalence(82) Θ : A ⊔ C ( C × [1]) ⊔ C A → A. Presentation of A as colimit (82) is given in 4.3.Proposition 4.2.1 below easily follows from this presentation.4.2.1. Proposition.
Let α : w → u be an active arrow in BM over h n i → h i . Let f ∈ Quiv BM X ( M ) w and g ∈ Quiv BM X ( M ) u . Then (83) Map α Quiv BM X ( M ) ( f, g ) = Map Fun(
C,M ) ( p ◦ f, q ◦ g ) , where, as above, M = α ∗ ( M ) , and ( C, p, q ) are described in 4.3.1. The samedescription holds for the space of active arrows in Quiv LM X ( M ) and M ∈ Op LM .Proof. Taking into account the presentation (82), the formula (81) can be rewrit-ten as the fiber of the restrictionFun [1] ( C × [1] , M ) → Fun(
C, M ) × Fun(
C, M )at ( f ◦ p, f ◦ q ). This fiber is easily identified with the right-hand side of theformula (83). (cid:3) C n denotes the free planar operad generated by one n -ary operation. This means that the colors of C n are numbered by 1 , . . . , n, C n ((1 , . . . , n ) , α in BM defines a unique map C n → BM which we denote as e α . The operadic map C ◦ n → Quiv BM X ( M ) is given by a choice of a pair of objects in Quiv BM X ( M ), one over h n i and another over h i . Thus, our mapping space (80) can be described as thefiber of the restriction map(84) Alg C n (Quiv BM X ( M )) → Alg C ◦ n (Quiv BM X ( M ))at ( f, g ).We can replace C n in the above formula with its strong approximation Q n → C n , and C ◦ n with Q ◦ n , see 2.9.6. Since Quiv BM X ( M ) = Funop BM ( BM X , M ), this allowsone to rewrite (84) as the fiber of(85) Alg Q n × BM BM X /Q n ( Q n × BM M ) → Alg Q ◦ n × BM BM X /Q ◦ n ( Q ◦ n × BM M )at ( f, g ). µ : [1] → Q n be the only active arrow in Q n . A Q n -operad is uniquelydescribed by its base change X → [1] with respect to µ , together with a decom-position of X , the fiber at { } ∈ [1], into a product X = Q ni =1 X ,i . This impliesthat, given two Q n -operads presented by categories A and M over [1], the spaceof active maps from f : A → M to g : A → M is given by the fiber of themap Fun [1] ( A, M ) → Fun( A , M ) × Fun( A , M )at ( f, g ).We now apply the above reasoning to A = α ∗ ( BM X ) and M = α ∗ ( M ). Wededuce the description of (80) as the fiber at ( f, g ) of the map (81).4.3. A as a colimit. In this subsection we construct an equivalence (82) for allvalues of α . We distinguish four different cases for α listed in 4.3.1. In eachone of the cases we provide formulas for C and for the maps p : C → A and q : C → A .To get the equivalence (82), we use the fact that both sides are described byformulas “universal in X ”. The latter means the following. Both sides of theequivalence are categories over [1] and so can be presented by a functor F :(∆ / [1] ) op → S . It is of the form F ( σ ) = Map( Y ( σ ) , X ) where Y : ∆ / [1] → Cat hasvalues in conventional categories (presented by finite posets), and is independentof X . So, the task of construction of equivalence (82) reduces to a comparison offinite posets.4.3.1. The active arrow α : w → u appearing in the definition of A , is uniquelydefined by its source which is an object of BM h n i . We distinguish below thefollowing cases.(w0) w is presented by σ : [ n ] → [1], n >
0, having the constant value 0.(w00) w is presented by σ : [0] → [1], σ (0) = 0.(w1) w is presented by σ : [ n ] → [1] having the constant value 1.(w2) w is presented by σ : [ n ] → [1] such that σ (0) = 0 , σ ( n ) = 1. In this caselet k be such that σ ( i ) = 0 for i ≤ k and σ ( i ) = 1 for i > k ( k < n ).We will now present formulas for C , p : C → A and q : C → A for each ofthe types of α separately. The case (w0) . In this case w = a n ∈ BM h n i and u = a ∈ BM h i .We have A = ( X op × X ) n , A = X op × X .We define the category C = X op × (Tw( X ) op ) n − × X and a pair of arrows p : C → A , q : C → A , as follows.The map p is induced by the n − X ) op → X × X op , see 3,whereas q is the projection to the first and the last factors. The case (w00) . Here we have A = [0], A = X op × X , C = Tw( X ). The map q : C → A is the canonical map (2) defining Yoneda embedding. The case (w1) . Here A = A = [0] and we put C = [0]. The case (w2) . In this case w = a k mb n − k − ∈ BM h n i and u = m ∈ BM h i . We have A = X × ( X op × X ) k and A = X . We define C = (Tw( X ) op ) k × X .The map p is induced by the k projections Tw( X ) op → X × X op , whereas q isthe projection to the last factor.4.3.2. The category A is described by a functor F A : (∆ / [1] ) op → S which is arestriction of BM X , with F A ( σ ) = Map( F BM ( σ ) , X ) where F BM ( σ ) are presented bythe diagrams (59) and (60).We will calculate the functor F : (∆ / [1] ) op → S describing the colimit A ⊔ C ( C × [1]) ⊔ C A and compare it to F A .In all cases appearing in 4.3.1, apart from (w00), the map p : C → A is aright fibration. In the case (w00), the map q : C → A is a left fibration. This iswhat makes the calculation easy.4.3.3. The calculation of F . Let(86) A p ←− C q −→ A be a diagram with p a right fibration, and let B = A ⊔ C ( C × [1]) ⊔ C A . Denote D = ( C × [1]) ⊔ C A , so that B = A ⊔ C D . We denote by A m , A m , C m the m -components of the presentation of A , A , C as simplicial spaces. The map D → [1] is a cocartesian fibration, so it is easy to describe its representative F D in Fun((∆ / [1] ) op , S ). For σ : [ m ] → [1] the pullback [ m ] × [1] D is the iteratedcylinder corresponding to the sequence C id → C → . . . → C q → A → . . . id → A , so that(87) F D ( σ ) = C m , σ ( i ) = 0 for all i,A m , σ ( i ) = 1 for all i,C a × A A b , σ = { a +1 b } , m = a + b, see [H.L], 9.8.6.Let us now describe B = A ⊔ C D . In general, a colimit in Cat / [1] can beexpressed as a colimit in presheaves on ∆ / [1] , followed by the localization functor L : Fun((∆ / [1] ) op , S ) → Cat / [1] .Therefore, B = L ( F B ′ ) where F B ′ = A • ⊔ C • F D is the colimit in Fun((∆ / [1] ) op , S ).It is very easy to calculate F B ′ . One has(88) F B ′ ( σ ) = A m , σ ( i ) = 0 for all i,A m , σ ( i ) = 1 for all i,C a × A A b , σ = { a +1 b } , m = a + b. Fortunately, in the case where p : C → A is a right fibration, F B ′ satisfiescompleteness and Segal conditions. Therefore, no localization is needed and B = B ′ .In the case of diagram (86) with q a left fibration, we proceed dually, presentingthe colimit in question as B = D ⊔ C A , where D = A ⊔ C ( C × [1]). In this case D → [1] is a cartesian fibration and we can repeat the above calculation. As aresult, we get(89) F B ′ ( σ ) = A m , σ ( i ) = 0 for all i,A m , σ ( i ) = 1 for all i,A a × A C b , σ = { a b +1 } , m = a + b. p : C → A isa right fibration. In the case (w00) the map q is a left fibration. Thus, we canapply formulas (88) and (89) to the calculation of the colimits.We will make a calculation separately for different values of α .4.3.5. The case (w0) . Here A = ( X op × X ) n , A = X op × X , C = X op × Tw( X ) n − × X . The formula (88) yields(90) F ( σ ) = ( X op m × X m ) n , σ ( i ) = 0 for all i,X op m × X m , σ ( i ) = 1 for all i, ( X m ) op × Tw( X ) n − a × X m , σ = { a +1 b } , m = a + b. The case (w00) . Here A = [0], A = X op × X , C = Tw( X ). The formula(89) yields(91) F ( σ ) = [0] , σ ( i ) = 0 for all i,X op m × X m , σ ( i ) = 1 for all i, Tw( X ) b , σ = { a b +1 } , m = a + b. The case (w1) . Here obviously F ( σ ) = [0].4.3.8. The case (w2) . Here A = ( X op × X ) k × X op , A = X op , C = X op × Tw( X ) k . The formula (88) yields(92) F ( σ ) = ( X op m × X m ) k × X op m , σ ( i ) = 0 for all i,X op m , σ ( i ) = 1 for all i, ( X m ) op × Tw( X ) ka , σ = { a +1 b } , k = a + b. Conclusion.
The formulas (90), (91), (92) have form F ( σ ) = Map( Y ( σ ) , X )where Y : ∆ / [1] → Cat takes values in finite posets.Comparing these formulas withthe pictures (59) and (60), we see that Y ( σ ) = F BM ( σ ) and, therefore, F = F A . Categories with colimits.
In what follows we will need a few basic factsabout categories with colimits which can be found in [L.T] and [L.HA]. Wepresent them below.4.4.1. We fix a set of categories K . The category Cat K is defined as the subcat-egory of Cat whose objects are categories having K -indexed colimits, and mor-phism preserving these colimits. The category Cat K has a symmetric monoidalstructure induced from the cartesian structure on Cat , see [L.HA], 4.8.1.3, 4.8.1.4.The latter means that tensor product C = ⊗ ni =1 C i is defined by a universal map F : C × . . . × C n → C preserving K -indexed colimits along each argument.Let C be a small category. The category P K ( C ) is defined as the smallestfull subcategory of P ( C ) containing the image of C and closed under K - indexedcolimits, see [L.T], 5.3.6.2 (our P K ( C ) is P KR ( C ) with R = ∅ ). The embedding Y : C → P K ( C ) is fully faithful and it is universal among functors from C tocategories having K -indexed colimits.The unit object in Cat K is S K = P K ([0]). This is the smallest full subcategoryof spaces containing [0] and closed under K -colimits.In what follows we need a certain smallness property of categories.4.4.2. Definition.
Let K be a set of categories. A category C is strongly K -smallif • Map C ( x, y ) ∈ S K for all x, y ∈ C . • For any F ∈ Fun( C op , S K ) the total category of F considered as rightfibration over C , is in K . • For any n the category Tw( C ) n is in K .4.4.3. Lemma.
Let C be strongly K -small. Then P K ( C ) = Fun( C op , S K ) .Proof. The Yoneda lemma yields a functor Y : C op × C → S whose essential imageis in S K . This can be interpreted as a functor C → Fun( C op , S K ). The universalproperty of P K yields a canonical functor j : P K ( C ) → Fun( C op , S K ). The compo-sition of this with the obvious embedding to P ( C ) is a full embedding. Therefore, j is a full embedding. It remains to verify that j is essentially surjective. Let F : C op → S K . Grothendieck construction converts F into a right fibration F over C which belongs to K . F is the colimit of the composition F → C → P K ( C ),which proves j is essentially surjective. (cid:3) The following result is standard; it is a minor modification of [L.HA], 4.6.1.6.4.4.4.
Lemma.
Let C be a closed monoidal category and let B ∈ C . Put C =hom( B, ) and let e : B ⊗ C → be the canonical evaluation map. Then thefollowing properties are equivalent. e is a counit of the duality. For any A ∈ C the map A ⊗ B → hom( C, A ) adjoint to A ⊗ B ⊗ C e → A ⊗ = A , is an equivalence. Property 2 is valid for A = C .Proof. This is a direct consequence of the proof of [L.HA], 4.6.1.6. Note thatProperty 2 for A = C provides the unit of the duality c : → C ⊗ B . (cid:3) The symmetric monoidal category
Cat K is closed, with the internal mappingobject assigning to ( A , B ) the full subcategory Fun K ( A , B ) of Fun( A , B ) spannedby the functors preserving K -indexed colimits, see [L.HA], 4.8.1.6. In particular,for B = P K ( C ) one has C := Fun K ( B, S K ) = Fun( C , S K ) = P K ( C op ).4.4.5. Lemma.
Let C be strongly K -small. Then P K ( C ) is a dualizable object in Cat K whose dual is P K ( C op ) .Proof. We will verify that the map e : P K ( C ) ⊗ P K ( C op ) → S K defined abovesatisfies 4.4.4. It is sufficient to verify the condition 2 of of the lemma for A = P K ( D ).We have to verify that the map P K ( D ) ⊗ P K ( C ) → Fun K ( P K ( C op ) , P K ( D )) = Fun( C op , P K ( D )) = P K ( D × C )is an equivalence.This is a colimit-preserving map and its restriction to D × C easily identifieswith the Yoneda embedding. This proves the assertion. (cid:3) The counit of the duality(93) e : P K ( C ) ⊗ P K ( C op ) → S K can be described as the colimit-preserving functor induced by the Yoneda map C × C op → S K for C op . The unit of the duality is the map c : S K → P K ( C op ) ⊗ P K ( C )preserving colimits and the terminal object.4.4.6. Examples. • Let α be a regular uncountable cardinal. We say that a space has size α if it can be presented by a simplicial set with less than α simplices. Acategory X has size α iff the spaces X n have size α for all n . Let K consistof all categories of size α . In this case X is strongly K -small iff X ∈ K . • The same is true if K is the collection of all spaces of size α .4.4.7. Let O be an operad and C → O a strong approximation. A C -algebraobject in Cat K will be called a C -monoidal category with K -indexed colimits. Here is our first important result. Theorem.
Let M be a BM -monoidal category with K -indexed colimits andlet X be strongly K -small as defined in 4.4.2. Then Quiv BM X ( M ) is a BM -monoidalcategory. The same claim holds with BM replaced by LM or Ass .Proof.
We will verify that any active arrow in BM has a locally cocartesian lifting,and that locally cocartesian liftings are closed under composition.It is sufficient to study the active arrows α : w → u over h n i → h i in Ass .Given f ∈ Quiv BM X ( M ) w , g ∈ Quiv BM X ( M ) u , the space Map α ( f, g ) is given bythe formula (83) of Proposition 4.2.1. This formula implies that the locallycocartesian lifting of α is given by a left Kan extension of p ◦ f with respect tothe projection q : C → A .If M has K -indexed colimits, this Kan extension exists. If the monoidal struc-ture on M preserves K -indexed colimits in each argument, the composition of lo-cally cocartesian liftings will be locally cocartesian. This proves the theorem. (cid:3) Note that Quiv BM X ( M ) is a BM -monoidal category with K -colimits.4.4.9. Corollary.
Let M ∈ Alg BM ( Cat K ) and let X be strongly K -small. Then Quiv BM X ( M ) ∈ Alg BM ( Cat K ) .Proof. The fibers Quiv BM X ( M ) w , w ∈ BM , obviously have K -colimits. The monoidalstructure is defined via left Kan extensions which preserve colimits as they havea right adjoint. (cid:3) More functoriality.
The assignment ( X, M ) Quiv BM X ( M ) is functorial.This means that maps j : X ′ → X and f : M → M ′ yield a map of operads f ! j ! :Quiv BM X ( M ) → Quiv BM X ′ ( M ′ ). Let us now assume the conditions of Theorem 4.4.8are fulfilled, so that Quiv BM X ( M ) and Quiv BM X ( M ′ ) are monoidal. We claim that, if f : M → M ′ is monoidal and preserves K -indexed colimits, the induced functor f ! : Quiv BM X ( f ) : Quiv BM X ( M ) → Quiv BM X ( M ′ ) is also monoidal. In fact, we have toverify that f ! preserves cocartesian liftings of active arrows in BM . The descriptionof cocartesian liftings as left Kan extension proves the claim.In the following subsection we will describe the monoidal structure on Quiv X ( M )by a universal property.4.5. Quiv X ( M ) as the category of endomorphisms. Assume that M is amonoidal category with K -indexed colimits. Recall that π : BM → Ass is thestandard projection. The BM -operad Quiv BM X ( π ∗ M ) is a BM -monoidal category.Its Ass − -component is Quiv X ( M ), Fun( X, M ) is its m -component, and M isthe Ass + - component of Quiv BM X ( π ∗ M ). This means that Quiv X ( M ) acts on theleft on the right M -module Fun( X, M ). In this subsection we will prove thatQuiv X ( M ) is the endomorphism object of Fun( X, M ) considered as right M -module. Moreover, the family of monoidal categories Quiv( M ) will turn out to be the endomorphism object of the family of right M -modules Fun( , M ), see4.5.3 and 4.5.5.4.5.1. Recall the setup of the endomorphism objects presented in [L.HA], 4.7.2.Let C be a monoidal category and let A be left tensored over C . For a ∈ C theendomorphism object End C ( a ) in C is defined as the one representing the functor(94) Map C ( c, End C ( a )) = Map A ( c ⊗ a, a ) . Endomorphism object does not necessarily exist; but if it does, it is uniquelydefined. If it exists, it automatically acquires a structure of associative algebraobject in C (see [L.HA], 4.7.2.40). Moreover, for any associative algebra c in C there is a canonical equivalence(95) Map Alg
Ass ( C ) ( c, End C ( a )) → { c } × Alg
Ass ( C ) Alg LM ( C , A ) × A { a } from the space of algebra maps c → End C ( a ) to the space of left c -module struc-tures on a (here we denote as ( C , A ) the LM -monoidal category defined by the left C -module A ).Here is a short description of the construction. By definition, End C ( a ) is theterminal object in the category C [ a ] whose objects are pairs ( c, f : c ⊗ a → a ). Itturns out that C [ a ] has a canonical structure of monoidal category such that theforgetful functor C [ a ] → C is monoidal. Then by a general result 3.2.2.5, [L.HA],the terminal object of a monoidal category acquires automatically an algebrastructure.4.5.2. We will apply the above construction as follows. We assume M is amonoidal category with K -colimits and X is strongly K -small.We set C = Cat K and A = RM M ( C ). We choose a = Fun( X, M ) (considered asa right M -module). The BM -monoidal category Quiv BM X ( π ∗ M ) defines an action ofthe monoidal category Quiv X ( M ) on the right M -module a = Fun( X, M ). Hereis our second main result of this section.4.5.3. Proposition.
The action of
Quiv X ( M ) on Fun( X, M ) ∈ RM M ( Cat K ) presents Quiv X ( M ) as the endomorphism object.Proof. In short, universality of the action of Quiv X ( M ) on Fun( X, M ) followsfrom duality between P K ( X ) and P K ( X op ). The details are below.1. The action of Quiv X ( M ) on Fun( X, M ) is described in Theorem 4.4.8 asthe compositionQuiv X ( M ) ⊗ Fun( X, M ) = Fun( X op × X, M ) ⊗ Fun( X, M ) → (96) Fun( X × X op × X, M ⊗ M ) µ → Fun( X × X op × X, M ) → Fun( X × Tw( X ) op , M ) κ → Fun( X, M ) , with µ defined by the monoidal structure of M and κ defined by the left Kanextension along the projection X × Tw( X ) op → X .Let us rewrite the composition (96) replacing all functor categories with presheaves.We get P K ( X op × X × X op ) ⊗ M ⊗ M µ → P K ( X op × X × X op ) ⊗ M → (97) P K ( X op × Tw( X )) ⊗ M κ → P K ( X op ) ⊗ M , The composition of the last two arrows comes from the composition P K ( X × X op ) → P K (Tw( X )) colim −→ S K which in turn is equivalent to the counit map (93) for X := C op .2. Let C ∈ Cat K . A map of right M -modules C ⊗ P K ( X op ) ⊗ M → P K ( X op ) ⊗ M is uniquely defined by its restriction C ⊗ P K ( X op ) → P K ( X op ) ⊗ M . Using duality between P K ( X ) and P K ( X op ), we can rewrite the latter as amap(98) C → P K ( X ) ⊗ P K ( X op ) ⊗ M = Fun( X op × X, M ) . In the case C = Quiv X ( M ), the map (98) is an equivalence. This means thatQuiv X ( M ) is a terminal object of C [ a ], in the notation of4.5.2. (cid:3) The identification of Quiv X ( M ) with the endomorphism object of Fun( X, M )is functorial in X . To show this, we will describe the whole family Quiv( M )of monoidal categories, based on cat K , the category of strongly K -small cate-gories , as the endomorphisms of the family X Fun( X, M ). For this oneneeds to properly define the categories involved.4.5.4. Endomorphisms of
Fun( , M ) . We replace the category
Cat K from 4.5.2with the category of families C := Cat K , cart /B , with B = cat K . This is the sub-category of Cat cart /B , see 2.11.2, spanned by the families X → B classified by thefunctors B op → Cat K , with morphisms X → X ′ over B inducing colimit pre-serving functors X X → X ′ X for each X ∈ B . The category C has a symmetricmonoidal structure induced from the cartesian structure on Cat cart /B : if the fami-lies X and Y over B are classified by the functors ˜ X , ˜ Y : B op → Cat K , the tensorproduct of X and Y is classified by the functor X ˜ X ( X ) ⊗ ˜ Y ( X ).We define A = RM M ( C ), and, as before, we have a left C -action on A . Thecanonical cartesian family of categories X Fun( X, M ) gives an object a ∈ A . Note the difference between cat K and Cat K , the category of categories with K -indexedcolimits. The family Quiv( M ) is a cartesian family of monoidal categories, and Quiv LM ( M )defines a left action of Quiv( M ) on a .In order to prove universality of the action of Quiv( M ) on a = Fun( , M ),we will repeat the second part of the proof of Proposition 4.5.3, working withfamilies over cat K .The functor cat op K → Cat K carrying X to P K ( X ) and f : X → Y to f ∗ : P K ( Y ) → P K ( X ), defines a cartesian family which will be denoted as P K ( ).Similarly, the assignment X P K ( X op ) defines a cartesian family P K ( op ). Thecanonical map P K ( op ) ⊗ M → Fun( , M ) is a map of cartesian families, inducingequivalence of the fibers. Therefore, it is an equivalence. In order to repeat theargument of the proof of 4.5.3, Step 2, it is enough to construct the duality datafor the families P K ( ) and P K ( op ). The counit of the duality e : P K ( ) ⊗ P K ( op ) → S K × cat K is defined by the Yoneda map, exactly as for a single X , see (93). Let us constructthe unit of the duality.Denote by p : P → cat K the cartesian fibration classified by the functor X P K ( X × X op ). Let Q be the full subcategory of P spanned by essentiallyconstant presheaves. Since for any f : X → Y in cat K the map f ∗ : P K ( Y × Y op ) → P ( X × X op ) preserves essentially constant presheaves, Q is also a cartesianfibration. Moreover, it is obviously equivalent to S K × cat K . This gives therequired unit map c : S K × cat K → P . The compositions (id ⊗ e )( c ⊗ id) and( e ⊗ id)(id ⊗ c ) are maps of cartesian families, whose fibers at any X ∈ cat K come from the duality data for P K ( X ). So they are equivalences.We have verified the following result.4.5.5. Proposition.
The action of
Quiv( M ) on Fun( , M ) ∈ RM M ( Cat K , cart / cat K ) presents Quiv( M ) as the endomorphism object. Multiplicativity.
We will now reconsider once more the multiplicativityproperty 3.5.2. Recall that for a pair of BM -operads M , N and a pair of categories X, Y one has a canonical map (71) of BM -operads µ : Quiv BM X ( M ) × Quiv BM Y ( N ) → Quiv BM X × Y ( M × N ) . We now assume that, for a collection of categories K , M , N ∈ Alg BM ( Cat K ) and X, Y and X × Y are strongly K -small. The BM -operads of quivers, under thisassumption, are BM -monoidal categories with K -indexed colimits. In this case wecan strengthen our claim.4.6.1. Proposition. The map µ (71) is a BM -monoidal functor. The composition (99) Quiv BM X ( M ) × Quiv BM Y ( N ) → Quiv BM X × Y ( M ⊗ N ) is also a BM -monoidal functor. The functor (99) preserves K -indexed colimits in each component and itinduces an equivalence of BM -monoidal categories (100) µ : Quiv BM X ( M ) ⊗ Quiv BM Y ( N ) ≃ → Quiv BM X × Y ( M ⊗ N ) . Proof.
1. We have to verify that the functor (71) preserves cocartesian liftings ofthe active arrows. These are described in Theorem 4.4.8 as left Kan extensions.Let α : w → u be an active arrow in BM over the active map h n i → h i . Followingthe calculation from 4.3, we denote as ( BM X ) w p X ← C X q X → ( BM X ) u ,( BM Y ) w p Y ← C Y q Y → ( BM Y ) u and ( BM X × Y ) w p X × Y ← C X × Y q X × Y → ( BM X × Y ) u the dia-grams described in 4.3.1 for the arrow α and the category X , Y , and X × Y respec-tively. One has α ∗ ( BM X ) × α ∗ ( BM Y ) = α ∗ ( BM X × Y ) and α ∗ ( M ) × α ∗ ( N ) = α ∗ ( M × N ).Furthermore, the formulas 4.3.5 — 4.3.8 show that the the diagram ( p X × Y , q X × Y )is equivalent to the product of ( p X , q X ) with ( p Y , q Y ). Now the claim follows asKan extensions commute with (this type of ) products.2. Note that the second claim does not immediately follow from 4.4.10 as thefunctor M × N → M ⊗ N does not preserve colimits. However, the left Kanextension of a functor C X × C Y → α ∗ ( M ) × α ∗ ( N ) along ( q X , q Y ) : C X × C Y → ( BM X ) u × ( BM Y ) u can be calculated as a composition of two left Kan extensionspreserved by the functor M × N → M ⊗ N , see Lemma 4.6.3. This implies theclaim.3. To verify the claim, we can forget about the monoidal structure of thecategories involved. Then the claim follows from Lemma 4.6.2 below. (cid:3) Lemma.
Let
X, Y be in cat K , M , N ∈ Cat K . Then the composition (101) Fun( X, M ) × Fun( Y, N ) → Fun( X × Y, M × N ) → Fun( X × Y, M ⊗ N ) is universal among the maps preserving K -indexed colimits in each argument.Proof. It is straightforward to see that 101) preserves K -indexed colimits in eachvariable. We will now prove the universality. First of all, let M = N = S K .Yoneda embedding X op → Fun( X, S K ) is determined by the functor X op × X → S K that classifies the left fibration Tw( X ) → X op × X , see (2). The diagram(2) is functorial in X and, moreover, preserves products. Therefore, Yonedaembedding for X op × Y op is equivalent to the composition X op × Y op → Fun( X, S K ) × Fun( Y, S K ) → Fun( X × Y, S K × S K ) → Fun( X × Y, S K ) , where the last map is induced by the product in S K . This is precisely the claimof the lemma for M = N = S K .We will now verify the claim for arbitrary M , N ∈ Cat K . We will first make anumber of simple observations.1. Let a functor f : X × Y → Z be adjoint to ˆ f : X → Fun(
Y, Z ) and let f ′ : X ′ × Y ′ → Z ′ be adjoint to ˆ f ′ : X ′ → Fun( Y ′ , Z ′ ). Then the map ˆ h : X × X ′ → Fun( Y × Y ′ , Z × Z ′ ) adjoint to the product f × f ′ is thecomposition of ˆ f × ˆ g with the product mapFun( Y, Z ) × Fun( Y ′ , Z ′ ) → Fun( Y × Y ′ , Z × Z ′ ) .
2. For arbitrary Z ∈ cat K , C ∈ Cat K , the map Z op × C → Fun( Z, C ),universal among the maps to an object in Cat K preserving the K -colimitsin C , is determined by the composition Z op × Z × C → S K × C → S K ⊗ C = C , where the first arrow is induced by the Yoneda map.3. Let X ∈ cat K , M , N ∈ Cat K . The composition X op × M × N → X op × M ⊗ N → Fun( X, M ⊗ N )is universal among the maps to an object in Cat K preserving the K -colimits in M and in N .We now apply Observation 2 in three instances, for ( Z, C ) = ( X, M ) , ( Y, N ) and( X × Y, M ⊗ N ). The maps y X : X op × X × M → M and y Y : Y op × Y × N → N induced by the Yoneda maps, are adjoint to the universal maps e X : X op × M → Fun( X, M ) and e Y : Y op × N → Fun( Y, N ) preserving colimits in M and N respectively. According to Observation 1, the map adjoint to their product(102) X op × Y op × X × Y × M × N → M × N is given by the composition of the product e X × e Y with the map(103) Fun( X, M ) × Fun( Y, N ) → Fun( X × Y, M × N ) . The last claim will remain true if we compose both (102) and (103) with M × N → M ⊗ N . This precisely means that the composition of (101) with X op × Y op × M × N → Fun( X, M ) × Fun( Y, N )is a universal map to an object of Cat K preserving colimits on M and N . (cid:3) Lemma.
Let F i : D i → M i be left Kan extensions of f i : C i → M i along u i : C i → D i for i = 1 , . Assume M i have K -colimits and C i , D i and C × C , D × D are strongly K -small. Then F ⊗ F : D × D → M ⊗ M defined asthe composition of ( F , F ) with the canonical map M × M → M ⊗ M , is aleft Kan extension of f ⊗ f . Proof.
We will prove that all compositions from the upper left corner to the lowerright corner of the diagram below are equivalent.Fun( C , M ) × Fun( C , M ) m (cid:15) (cid:15) Fun( C × C , M × M ) ≃ (cid:15) (cid:15) / / Fun( C × C , M ⊗ M ) ≃ (cid:15) (cid:15) Fun( C , Fun( C , M × M )) (cid:15) (cid:15) / / Fun( C , Fun( C , M ⊗ M )) (cid:15) (cid:15) Fun( C , Fun( D , M × M )) (cid:15) (cid:15) / / Fun( C , Fun( D , M ⊗ M )) (cid:15) (cid:15) Fun( D , Fun( D , M × M )) ≃ (cid:15) (cid:15) / / Fun( D , Fun( D , M ⊗ M )) ≃ (cid:15) (cid:15) Fun( D × D , M × M ) / / Fun( D × D , M ⊗ M ) . Here m is the product and the rest of nontrivial vertical arrows are left Kanextensions. Two middle squares are not commutative as the functor M × M → M ⊗ M do not preserve colimits. However, they preserve colimits in each of thearguments, so these squares commute on the image of Fun( C , M ) × Fun( C , M ). (cid:3) Example: S -enrichment of a category. In this subsection M = S is thecategory of spaces. We are working with the category Cat L of categories withsmall colimits (it is big and even locally big).We will show that any category C gives rise to an S -enriched (pre)category.4.7.1. Unit in
Quiv X ( S ) . Let X be a category. Since the monoidal categoryQuiv X ( S ) is identified with the category of colimit preserving endomorphisms of P ( X ), the unit in Quiv X ( S ) is given by the identity id : P ( X ) → P ( X ).The identification of endomorphisms of P ( X ) with Fun( X op × X, S ) identifiesid with the Yoneda map Y : X op × X → S assigning Map( x, y ) to the pair ( x, y ) ∈ X op × X .4.7.2. Let C be a category. Denote by j : C eq → C the maximal subspace of C .One has a lax monoidal functor j ! : Quiv C ( S ) → Quiv C eq ( S ) . The functor j ! carries algebras in Quiv C ( S ) to algebras in Quiv C eq ( S ). Theimage of the unit described above is called the S -enriched (pre)category corre-sponding to C . As expected, it has C eq as its space of objects, and it assigns thespace Map( x, y ) to a pair ( x, y ) ∈ ( C eq ) op × ( C eq ).4.7.3. Unit in
Quiv X ( M ) . Let now M ∈ Alg
Ass ( Cat L ) be arbitrary. The canoni-cal functor i : S → M (the unit of M considered as an associative algebra in Cat L )is monoidal and preserves colimits, therefore (see 4.4.10) it induces a monoidalfunctor i ! : Quiv X ( S ) → Quiv X ( M ) . In particular, the unit in Quiv X ( M ) is described by the composition Y : X op × X h −−−→ S i −−−→ M of the Yoneda map for X and the embedding i : S → M .4.8. Completeness: advertisement.
Associative algebras in Quiv X ( M ) arenot called enriched categories — but precategories — for two reasons.The first is that we want the category X of objects in an enriched categoryto be a space. The second reason is more important and it has to do with thecompleteness property.We will see in Section 5 that, if M = S is the category of spaces and if X is aspace, then the category of associative algebras in Quiv X ( S ) is equivalent to thatof Segal spaces.Let now C ∈ Quiv X ( M ) be an M -enriched precategory. Let j : M → S be thefunctor right adjoint to the canonical embedding S → M . The functor j is laxmonoidal, so j ! ( C ) is an S -enriched precategory, that is, corresponds to a Segalspace.We will say that C is an M -enriched category if X is a space and if j ! ( C ),considered as a Segal space, is complete.We will show later that M -enriched categories form a localization of the cat-egory of M -enriched precategories. This means that the full embedding of M -enriched categories into M -enriched precategories admits a left adjoint functor.5. The case M is cartesian Let M be a category with colimits and finite limits satisfying certain weakenedtopos conditions, see Definition 5.4.1 below. We endow M with the cartesianmonoidal structure.In this section we study precategories enriched over M in the sense of 3.1.1having a space of objects X . We prove that such precategories can be equivalentlydescribed as simplicial objects A • in M satisfying the Segal condition, such that A = X . The comparison of these two notions of enriched category proceeds intwo steps. Step 1.
We show that for any X ∈ M the category M /X × X has a canonical struc-ture of monoidal category. We show further that the associative algebrasin this monoidal category are precisely simplicial objects A • in M satis-fying the condition A = X together with the Segal condition. Step 2.
After that, assuming M satisfies the properties of Definition 5.4.1, weidentify the family of monoidal categories M /X × X for varying X ∈ S ⊂ M with the family of monoidal categories Quiv X ( M ) defined in Section 3.5.1. Non-symmetric cartesian structures.
Recall that
Ass = ∆ op ; we de-note h n i = [ n ] op . If M ∈ Op Ass , the fiber M is contractible; we will denote by ∗ an object of M .The following definition is a non-symmetric version of [L.HA], 2.4.1.5.1.1. Definition.
Let q : M → Ass be an object of
Fib ( Ass ♮, ∅ ); that is, a mapadmitting cocartesian lifting of the inerts .(NC1) A functor F : M → N to a category with fiber products N is called a laxNC structure if for each commutative diagram in Ass of form(104) h k + l i a / / b (cid:15) (cid:15) h k i t (cid:15) (cid:15) h l i s / / h i with t and s given by { k } ∈ [ k ] and { } ∈ [ l ] respectively, a correspondingto the embedding [ k ] → [ k + l ] as the initial segment, and b correspondingto the embedding [ l ] → [ k + l ] as the terminal segment, and for each x ∈ M k + l , the diagram(105) F ( x ) −−−→ F ( a ! ( x )) y y F ( b ! ( x )) −−−→ F ( t ! b ! ( x )) . is cartesian.(NC2) f is a weak NC structure if it is a lax NC structure, q : M → Ass is amonoidal category, and f carries cocartesian liftings of the active arrowsto equivalences.(NC3) f is an NC structure if it is a weak NC structure and the natural functor(106) M → N /F ( ∗ ) × F ( ∗ )21 This includes planar operads p : M → Ass , as well as, for instance, BM -operads, see Remarkin 2.9.7. NC is short for “non-symmetric cartesian”. induced by F and by the cocartesian liftings of two maps h i → h i in Ass , is an equivalence.5.1.2.
Remarks.
1. The original version of the theory developed by J. Luriein [L.HA], 2.4.1, aims to prove that an ∞ -category M with products hasan essentially unique structure of symmetric monoidal ∞ -category withmonoidal structure defined by the product; moreover, algebras with valuesin this symmetric monoidal category can be described by monoids in theoriginal ∞ -category M .2. In the original (symmetric) setup a cartesian structure exists and is essen-tially unique for a SM ∞ -category M satisfying the following properties. • The unit of M is a terminal object. • For any x, y ∈ M the pair of maps x ⊗ y → x ⊗ = x, x ⊗ y → ⊗ y = y, yields a cartesian diagram. The following NC version of the abovecharacterisation seems very plausible. A plausible claim.
Let M be a SM category with a terminal object ∗ . Then M has an NC structure iff for any x, y ∈ M the diagram x ⊗ y / / (cid:15) (cid:15) x ⊗ ∗ (cid:15) (cid:15) ∗ ⊗ y / / ∗ ⊗ ∗ is cartesian.Our non-symmetric analog will prove the following.First of all, for any ∞ -category M with fiber products and for any object X ∈ M we will construct an explicit monoidal category M ∆ X with an NC structure F : M ∆ X → M such that F ( ∗ ) = X .Further, we will prove that for any monoidal category q : C → Ass the com-position with F establishes an equivalence between the category of monoidalfunctors Fun ⊗ Ass ( C , M ∆ X ) and the category Fun weak ( C , M ) X of weak NC structures C → M carrying ∗ ∈ C to X ∈ M . This proves, in particular, that any monoidalcategory C with a NC structure F : C → M is equivalent to one of the M ∆ X .The family of monoidal categories M ∆ X appears as a small fragment of a two-parametric family M ⊙ X,Y of BM -monoidal categories constructed in 5.2; the sub-family M ⊙ X, ∗ is also very important to us; it is denoted as M = X . The family ofmonoidal categories M ∆ X is just the Ass − -component of M = X .Finally, this will allow us to describe algebras over a BM -operad O in M ⊙ X interms of lax functors O → M . As a consequence, associative algebras in M ∆ X will That is, we will construct a monoidal structure on M /X × X . be identified with simplicial objects Y • in M satisfying the condition Y = X andthe Segal condition.The details are below.5.2. A canonical family of monoidal categories.
In this subsection we willconstruct, for any category M with fiber products and a terminal object, a familyof BM -monoidal categories M ⊙ → M × M × BM parametrized by M × M , so thatits fiber M ⊙ X,Y at (
X, Y ) ∈ M × M is the triple ( M /X × X , M /X × Y , M /Y × Y ) witha certain monoidal category structure on M /X × X and M /Y × Y so that M /X × Y isleft-tensored over M /X × X and right-tensored over M /Y × Y .Especially important for us is the family M = obtained from M ⊙ by base changealong M → M × M , carrying X ∈ M to the pair ( X, ∗ ), ∗ being a terminal objectof M .Assuming some extra properties for M , we will be able to identify M = X withQuiv BM X ( M ), see 5.5.1.In our construction of the family M ⊙ → M × M × BM , we follow the originalconstruction of [L.HA], 2.4.1. First of all, we describe a functor E : BM op → Cat to (conventional) categories. This functor allows one to assign to M , in a wayclose to [L.HA], 2.4.1, a category ¯ M ⊙ over M × M × BM ; the family of BM -monoidalcategories M ⊙ will be defined as a full subcategory of ¯ M ⊙ .5.2.1. The functor E : BM op → Cat . Here is the rationale for our choice of E .Recall that BM op = ∆ / [1] . Given an object s : [ n ] → [1] in BM op , we expect thefiber of M ⊙ X at s to be equivalent to the product(107) ( M /X × X ) α × ( M /X × Y ) µ × M β/Y × Y , where s = a α m µ b β , α + µ + β = n , in the presentation of s described in 2.9.2.In order to have a functorial dependence on s in (107), we will describe thiscategory as the category of functors E ( s ) → M satisfying certain properties.We will use the following notation. For s : [ n ] → [1] we denote | s | = n . BM has two objects, ∅ L and ∅ R , corresponding to the maps [0] → [1] with the image0 and 1 respectively.The objects of E ( s ) are the pairs ( i, j ) such that 0 ≤ i ≤ j ≤ | s | , as well as twoextra objects, L and R . Let E be the poset with ( i, j ) ≤ ( i ′ , j ′ ) iff i ≤ i ′ ≤ j ′ ≤ j .Define, in addition, a groupoid E with the objects L, R, (0 , , . . . , ( n, n ), andwith a unique isomorphism from ( i, i ) to L if s ( i ) = 0 and to R if s ( i ) = 1. Thecategory E ( s ) is defined as the colimit of the diagram(108) E ←− { (0 , , . . . , ( n, n ) } −→ E . A map f : s → s ′ in BM op is given by [ n ] → [ n ′ ] → [1]. It defines f : E ( s ) → E ( s ′ )carrying ( i, j ) to ( f ( i ) , f ( j )), and preserving L and R . Example.
The category E ( ∅ L ) looks as follows(109) L (0 , ∼ o o R and E ( ∅ R ) is similar, with (0 ,
0) isomorphic to R .There are three non-isomorphic objects of BM lying over h i . These are s =(00) , (01) , (11).The category E (00) looks as follows.(110) (0 , { { ✇✇✇✇✇✇✇✇ ●●●●●●●● (0 , ∼ $ $ ❍❍❍❍❍❍❍❍❍ (1 , ∼ { { ✈✈✈✈✈✈✈✈✈ RL The category E (11) looks the same, with replacement of R and L .Finally, the category E (01) is as follows.(111) (0 , { { ✇✇✇✇✇✇✇✇ ●●●●●●●● (0 , ∼ (cid:15) (cid:15) (1 , ∼ (cid:15) (cid:15) L R
Distinguished squares in E ( s ) . The following commutative squares in E ( s )will be called distinguished squares . We will use them in order to define M ⊙ as a full subcategory of ¯ M ⊙ . Let s : [ n ] → [1]. For 0 ≤ i ≤ j ≤ k ≤ n thecommutative diagram in E ( s )(112) ( i, k ) { { ①①①①①①①① ●●●●●●●● ( i, j ) ❋❋❋❋❋❋❋❋ ( j, k ) { { ✇✇✇✇✇✇✇✇ ( j, j )will be called distinguished .For any map f : s → s ′ in BM op the corresponding functor f : E ( s ) → E ( s ′ )preserves distinguished arrows. M ∈ Cat . The functor E : BM op → Cat described above defines afunctor(113) Fun( E , M ) : BM → Cat , carrying s ∈ BM op to Fun( E ( s ) , M ). There are morphism of functors ι L , ι R : [0] → E from the constant functor with value [0] defined by the objects L, R ∈ E ( s ),respectively. This yields a morphism of functors Fun( E , M ) → Fun([0] ⊔ [0] , M ) = M × M . Let ¯ M ⊙ be the cocartesian fibration classified by the functor (113). Themorphisms ι L , ι R induce a map q = ( q L , q R ) : ¯ M ⊙ → M × M .We have defined a map ( q, p ) : ¯ M ⊙ → M × M × BM which will be shown to bea family of cocartesian fibrations based on M × M . The fiber of q at ( X, Y ) isdenoted as ¯ M ⊙ X,Y .We will need one more map, ¯ m : ¯ M ⊙ → M , defined as follows. The functor E : BM op → Cat is classified by a cartesian fibration e E → BM . The cocartesianfibration ¯ M ⊙ can be then described as an internal mapping object in the category Cat / BM , Fun Cat / BM ( e E , M × BM ), see [L.T], Corollary 3.2.2.13 and [GHN], 7.3.One has a canonical section top : BM → e E carrying s ∈ BM to the object (0 , | s | )of E ( s ). This map induces, by functoriality of Fun BM , a map ¯ M ⊙ → M × BM over BM . We denote by ¯ m : ¯ M ⊙ → M the projection to the first factor.5.2.5. The objects of ¯ M ⊙ over ( X, Y, s ) ∈ M × M × BM are the functors φ : E ( s ) → M , endowed with a pair of equivalences φ ( L ) ≃ X , φ ( R ) ≃ Y .We now define M ⊙ as the full subcategory of ¯ M ⊙ spanned by the objects φ : E ( s ) → M carrying the distinguished squares of E ( s ) to cartesian squares of M . We denote by M ⊙ X,Y the fiber of q : M ⊙ → M × M at ( X, Y ).5.2.6. The fiber of M ⊙ at ( X, Y, (00)) is equivalent to M /X × X , the fiber at( X, Y, (01)) is equivalent to M /X × Y and the fiber at ( X, Y, (11)) is equivalent to M /Y × Y .The following theorem is a direct analog of Theorem 2.4.1.5, [L.HA]. In theassertion (2) of the theorem we denote by E ( α ) : E ( s ′ ) → E ( s ) the functorcorresponding to α : s → s ′ in BM .5.2.7. Theorem. The map p : ¯ M ⊙ → BM is a cocartesian fibration. A morphism φ → φ ′ over α : s → s ′ is a cocartesian lifting iff for every e ∈ E ( s ′ ) the map φ ( E ( α )( e )) → φ ′ ( e ) is an equivalence in M . For any ( X, Y ) ∈ M × M the restriction of p : ¯ M ⊙ → BM to M ⊙ X,Y definesa cocartesian fibration to BM . If M has fiber products and a terminal object, the projection M ⊙ X,Y → BM defines on the triple ( M /X × X , M /X × Y , M /Y × Y ) a structure of BM -monoidalcategory. Proof.
The assertions (1) and (2) are immediate consequences of the construction.(2) implies that p : M ⊙ → BM is a cocartesiain fibration. Also by (2), p -cocartesianarrows in M ⊙ are sent to equivalences by q . According to Lemma 5.2.8 below,this implies that the fibers M ⊙ X,Y → BM are cocartesian fibrations. This provesassertion (3).It remains to verify that, if M admits fiber products and a terminal object,the map p : M ⊙ X,Y → BM is fibrous. We already know that p is a cocartesianfibration. Let s ∈ BM be of dimension n and let ρ i : s → s i , i = 1 , . . . , n be theinerts decomposing s . We have to verify that the map(114) M ⊙ X,Y,s → Y i M ⊙ X,Y,s i is an equivalence. Here M ⊙ X,Y,s i = M /X × X , M /X × Y , M /Y × Y for s i = a, m or b respectively. By definition, M ⊙ s is the subcategory of Fun( E ( s ) , M ) carryingdistinguished squares of E ( s ) to cartesian squares. Let E ( s ) ◦ denote the fullsubcategory of E ( s ) spanned by the objects ( i, j ) with j ≤ i + 1, L and R . Thefunctors in M ⊙ X,Y,s are right Kan extensions of functors E ( s ) ◦ → M carrying L to X and R to Y . By [L.T], 4.3.2.15, M ⊙ X,Y,s is the subcategory of Fun( E ( s ) ◦ , M )carrying L to X and R to Y . Now, E ( s ) ◦ decomposes into a colimit of diagramscorresponding to each ( i − , i ); this proves (114) is an equivalence. The factthat the ρ i form a p -product diagram is automatic for cocartesian fibrations, see[L.HA], 2.1.2.12. (cid:3) Lemma.
Given ( q, p ) : M → A × B such that p : M → B is a cocartesianfibration. Assume that q carries p -cocartesian arrows in M to equivalences in A .Then the the map ( q, p ) is also a cocartesian fibration.Proof. We will show that any p -cocartesian arrow f in M is also ( q, p )-cocartesian.The arrow f : x → y in M is cocartesian iff the diagram M y/ / / (cid:15) (cid:15) M x/ (cid:15) (cid:15) B p ( y ) / / / B p ( x ) / is cartesian. Since q carries i ( f ) to an equivalence, the diagram M y/ / / (cid:15) (cid:15) M x/ (cid:15) (cid:15) A q ( y ) / × B p ( y ) / / / A q ( x ) / × B p ( x ) / is also cartesian. (cid:3) Subfamilies.
The base change of M ⊙ with respect to the embedding M → M × M carrying X to ( X, ∗ ), ∗ being the terminal object of M , gives a one-parametric family p : M = → M × BM of BM -monoidal categories. Its fiber at X , M = X , is the triple ( M /X × X , M /X , M ).The Ass − -component of M = is the family of monoidal categories(115) p : M ∆ → M × Ass , whose fiber at X ∈ M is a monoidal structure on M /X × X .The functor m : M ⊙ → M is the restriction of ¯ m : ¯ M ⊙ → M defined in 5.2.4.We denote by the same letter the restriction of m to M ∆ .The result below is an immediate consequence of the construction and of theassertion 4 of 5.2.7.5.2.10. Proposition.
Let M admit products and fiber products over X . Then themap m : M ∆ → M restricted to M ∆ X , yields a NC structure. (cid:3) Proposition.
Let O ∈ Op BM and let p : M ⊙ → M × M × BM be the familyof BM -monoidal categories constructed in 5.2.5. Then the composition with m : M ⊙ → M gives rise to an equivalence (116) θ : Alg O ( M ⊙ ) → Fun laxNC ( O , M ) , where Fun laxNC ( O , M ) ⊂ Fun( O , M ) denotes the full subcategory spanned by thelax NC structures on O with values in M .Proof. First of all, we will describe the category of O -algebras in M ⊙ as a certainsubcategory of Fun( O , M ), for a specially designed category O . An O -algebra in M ⊙ is a commutative square O / / (cid:15) (cid:15) M ⊙ (cid:15) (cid:15) BM / / M × M × BM , where the lower horisontal map makes a choice of ( X, Y ) ∈ M × M , and theupper map preserves inerts. We keep the notation of5.2.4. Fun BM ( O , M ⊙ ) is afull subcategory of Fun BM ( O , ¯ M ⊙ ) = Fun( ˜ O , M ), with ˜ O := O × BM ˜ E (see [L.T],3.2.2.13 and [GHN], 7.3), spanned by the functors carrying distinguished squaresin E ( s ) to cartesian squares.We will denote by O ⊲ the colimit O × [1] ← O × { } → [0]. For x ∈ O thecanonical arrow from x to the cone point will be denoted by t x . We define O as the colimit of the following diagram.(117) O ⊲ ⊔ O ⊲ O ⊔ O o o ( ι L ,ι R ) / / ˜ O . By definition of O , Alg O ( M ⊙ ) identifies with the full subcategory of Fun( O , M )spanned by the functors F : O → M satisfying the following properties.(i) For every object x in O over s ∈ BM the functor F carries the distinguishedsquares in E ( s ) to cartesian squares in M .(ii) For every inert morphism a : x → y in O over s → t in BM and for any e ∈ E ( t )) the map F ( x, a ∗ ( e )) → F ( y, e ) is an equivalence.(iii) F carries the arrows t x in both copies of O ⊲ to equivalences.The section top : BM → ˜ E induces a map top : O → ˜ O which, composed withthe canonical map ˜ O → O , defines a restriction map(118) Fun( O , M ) → Fun( O , M ) . We wish to get the equivalence (116) as the one induced from (118).Let O ′ be the full subcategory of ˜ O spanned by the objects ( x, e ) where x ∈ O s and e ∈ E ( s ) is either (0 , | s | ) or ( i, i ) , L or R . Let us show that the embedding O ′ → ˜ O admits a left adjoint. Given ( x, e ) ∈ ˜ O s , ( y, f ) ∈ O ′ t , a map ( x, e ) → ( y, f )is given by a collection of the following data: • α : s → t in BM . • a : x → y in O over α . • u : e → α ∗ ( f ) in E ( s ).In the case e = ( i, j ) with i < j , there is a unique inert map β : s → s ′ with s ′ ∈ BM defined by [ j − i ] → [ n ], such that β ∗ ((0 , j − i )) = e . We choose v = idand b = β ! and the triple ( β, b, v ) defines a universal map from ( x, e ) to an objectin O ′ .In the case e is ( i, i ) or L or R the universal map is obviously the identity.We define O ′ as the colimit of the following diagram similar to (117)(119) O ⊲ ⊔ O ⊲ O ⊔ O o o ( ι L ,ι R ) / / O ′ . A functor F : O → M is a right Kan extension of its restriction to O ′ if andonly if F carries the universal maps ( x, ( i, j )) → ( β ! ( x ) , (0 , j − i )) to equivalencefor all x ∈ O s and all i < j ≤ | s | (here β : s → s ′ is the inert map mentionedabove). Any F satisfying the condition (ii) is therefore a right Kan extension.By [L.T], 4.3.2.15, Fun( O ′ , M ) identifies with the full subcategory of Fun( O , M )spanned by the functors that are right Kan extensions of their restrictions to O ′ .Thus, Alg O ( M ⊙ ) identifies with the full subcategory of Fun( O ′ , M ) spanned bythe functors whose right Kan extension satisfies (i)—(iii). The composition O top → O ′ → O ′ induces a map Fun( O ′ , M ) → Fun( O , M ); thismap establishes an equivalence of Fun laxNC ( O , M ) with the full subcategory ofFun( O ′ , M ) spanned by the functors F satisfying the following conditions.(1) The composition O → O ′ → M is a lax functor.(2) F carries the arrows t x in both copies of O ⊲ to equivalences.Let us verify that the properties (1) , (2) of a functor F ′ : O ′ → M are equiv-alent to the properties (i)—(iii) of F : O → M obtained from F ′ by right Kanextension. Recall that F ( x, e ) is defined by the equivalence F ( x, e ) → F ′ ( x ′ , e ′ )where ( x, e ) → ( x ′ , e ′ ) is the universal map from ( x, e ) to an object ( x ′ , e ′ ) of O ′ .Taking this into account, (i) is equivalent to (1), (ii) follows from the descriptionof F ( x, e ) in terms of ( x, e ) → ( x ′ , e ′ ), and condition (iii) is equivalent to (2). (cid:3) Here is a similar claim for planar operads. We deduce it from Proposition 5.3.1.5.3.2.
Proposition.
Let O be a planar operad and let p : M ∆ → M × Ass be thefamily of monoidal categories constructed in 5.2.9. Then the composition with m : M ∆ → M gives rise to an equivalence (120) θ : Alg O ( M ∆ ) → Fun laxNC ( O , M ) , where Fun laxNC ( O , M ) ⊂ Fun( O , M ) denotes the full subcategory spanned by thelax NC structures on O with values in M . Moreover, if O is a monoidal category, θ restricts to an equivalence (121) θ : Fun ⊗ Ass ( O , M ∆ ) → Fun weak ( O , M ) . Proof.
The planar operad O is obtained by base change from an operad map O ⊗ → Ass ⊗ . Denote Ass ′ = Ass ⊗ × BM ⊗ BM . This is a strong approximation of Ass that is a BM -operad. Denote O ′ = Ass ′ × Ass ⊗ O ⊗ . According to Proposition 5.3.1,one has an equivalence θ ′ : Alg O ′ ( M ⊙ ) → Fun laxNC ( O ′ , M ) . Note that
Ass ′ = Ass ⊔ {∅ R } and O ′ = O ⊔ {∅ R } . Making the base changewith respect to the map M → M × M carrying X to ( X, ∗ ), we get the requiredequivalence.The second part of the theorem claiming the equivalence (121), is straightfor-ward. (cid:3) Corollary.
Let p : C → Ass be a monoidal category and let f : C → M be an NC structure. Let X ∈ M be the image of ∗ ∈ C h i . Then the monoidalfunctor ˆ f : C → M ∆ X , corresponding via (121) to f , is an equivalence.Proof. The h i -component of the monoidal functor C → M ∆ X is, by definition,an equivalence. A monoidal functor which is an equivalence of the underlyingcategories, is a monoidal equivalence. (cid:3) Multiplicativity.
The functor M M ⊙ is corepresentable, therefore, itcommutes with limits. In particular,( M × M ′ ) ⊙ = M ⊙ × BM M ′⊙ . Prototopoi.
From now on we impose some extra conditions on a category M which allow for an analog of Grothendieck construction interpreting functors X → M as objects of an overcategory M /X .5.4.1. Definition. An ∞ -category M ∈ Cat K is called K -prototopos if it satisfiesthe following properties.(PT1) M has finite limits.(PT2) Finite products in M commute with K -indexed colimits.(PT3) For any space X ∈ S K the functor colim : Fun( X, M ) → M establishes anequivalence Fun( X, M ) → M /X . n - Cat of ( ∞ , n ) categories as defined in [L.G] or [Rz], sat-isfies the above properties with K the collection of all small categories. Lemma.
The category M = n - Cat is a prototopos.Proof.
Condition (PT1) is obvious and (PT2) follows from cartesian closednessof n - Cat , see [Rz].Let us check the condition (PT3). For X a point there is nothing to check. Ingeneral both Fun( X, M ) and M /X considered as functors of X carry colimits tolimits. This is obvious for Fun( X, M ) and follows from property (4) of Definition1.2.1, [L.G], for X M /X , as n - Cat is an absolute distributor, [L.G], 1.4. (cid:3)
Functoriality.
For a prototopos M the equivalence (PT3) has good func-torial properties. An arrow f : X → Y in S K gives rise to an adjoint pair f ! : M /X −→←− M /Y : f ∗ with f ! defined by the composition and f ∗ be the base change. Similarly, f givesrise to adjoint pair f ! : Fun( X, M ) −→←− Fun( Y, M ) : f ∗ with f ! defined by the left Kan extension and f ∗ by the composition with f . Thefunctors f ! commute with (PT3), so the adjoints also commute.5.4.4. Convolution.
Given three arrows f i : T → U i , i = 1 , , and g : T → V in S K , one defines an operation M /U × M /U → M /V as the composition(122) M /U × M /U × → M /U × U ( f × f ) ∗ −→ M /T g ! → M /V . The equivalence (PT3) allows one to rewrite this operation in terms of the func-tors, as Fun( U , M ) × Fun( U , M ) → Fun( U × U , M × M ) → (123) Fun( U × U , M ) ( f × f ) ∗ −→ Fun( T, M ) g ! → Fun( V, M ) . U = X × Y , U = Y × Z , V = X × Z , T = X × Y × Z , with the obvious choice of the arrows, we get maps(124) M /X × Y × M /Y × Z → M /X × Z , (125) Fun( X × Y, M ) × Fun( Y × Z, M ) → Fun( X × Z, M ) , generalizing the action of M /X × X on M /X given by Theorem 5.2.7 and the actionof Quiv X ( M ) on Fun( X, M ), see (96) .5.5. Identifying
Quiv( M ) with M ∆ . Let M be a K -prototopos. We consider M as a category in Cat K with the cartesian monoidal structure. We will nowidentify the family of monoidal categories Quiv X ( M ), X ∈ S K , with the family M ∆ = { M ∆ X } X ∈ M , restricted to S K ⊂ M .According to Proposition 4.5.5, Quiv( M ) is the endomorphism object of thecartesian family Fun( , M ) over S having the fiber Fun( X, M ) over X ∈ S . Since M is a prototopos, the family Fun( , M ) is equivalent to the cartesian family ofright M -modules X M /X . Since the family of monoidal categories M ∆ acts onthe M -module M / , this yields a canonical monoidal functor θ : M ∆ → Quiv( M )of cartesian families. In order to verify that this functor is an equivalence, wecan forget the monoidal structure. Comparison of the formulas (124) and (125)shows that θ is an equivalence.We have proved the following result.5.5.1. Proposition.
Let M be a prototopos. We consider it as a symmetricmonoidal category with cartesian structure. The family M = of BM -monoidal cat-egories is canonically equivalent to Quiv BM ( M ) . (cid:3) The equivalence above commutes with products. In fact, given two prototopoi, M and M ′ , the product M ∆ × M ′ ∆ acts on the right on the family of M × M ′ -modules M / × M ′ / . This gives a canonical map M ∆ × M ′ ∆ → Quiv BM ( M × M ′ ).It is easy to see this map is an equivalence. Thus,5.5.2. Corollary.
The equivalence M = → Quiv BM ( M ) commutes with products. (cid:3) Here X is a space and so X = X op = Tw( X ). M -enriched precategories, cartesian case. Applying Proposition 5.3.2to O = Ass , and using the identification of Quiv X ( M ) with M ∆ X , we immediatelyget the following.5.6.1. Corollary.
Let M be a K -prototopos and X ∈ S K . Then the category of M -enriched precategories with the space of objects X identifies with the categoryof simplicial objects A ∈ Fun(∆ op , M ) satisfying the Segal condition and having A = X . (cid:3) This result was previously obtained by R. Haugseng [Hau1], 7.5.6.
Enriched presheaves and the Yoneda lemma
Let M be a monoidal category with colimits. In this section we construct, foran M -enriched precategory A , a category of M -presheaves P M ( A ); we constructa Yoneda embedding A → P M ( A ) and prove it is fully faithful.Note that P M ( A ) does not necessarily have a structure of M -enriched precat-egory. It has another type of M -enrichment: it is just a left M -module.It turns out that the Yoneda lemma is precisely about the interplay of thesetwo types of enrichment. In this section we define and study functors from an M -enriched precategory to a category left-tensored over M .This approach to the Yoneda lemma was described, for conventional enrichedcategories, in our note [H.Y]. As in [H.Y], the central notion here is the notionof a functor from an M -enriched precategory to a left M -module.We fix a collection of categories K . Throughout this section M is a monoidalcategory with K -colimits. A left M -module is, by definition, a category B in Cat K with the left M -action commuting with K -colimits in each argument.6.1. Functors.
Let M be a monoidal category in Cat K . Let A be an M -precategorywith a strongly K -small category of objects X and let B be a left M -module.Let us first recall the conventional setup. An M -functor from A to B is givenby a map f : Ob( A ) → Ob( B ), and a compatible collection of maps(126) Hom A ( x, y ) ⊗ f ( x ) → f ( y ) , see [H.Y], 3.2.In our context, a functor from A to B will be given by a map f : X → B ,together with an extra structure which will correspond to (126). We will nowdescribe this extra structure.6.1.1. An M -module structure on B yields an LM -monoidal category which wedenote by ( M , B ). Applying to it the functor Quiv LM X , see 3.4.2, we get a leftQuiv X ( M )-module structure on the category Fun( X, B ). X ( M )-action on Fun( X, B ) is given byProposition 4.2.1 and the formulas 4.3.1 (our case is w = am ). Let A ∈ Quiv X ( M ) = Fun( X op × X, M ) and F ∈ Fun( X, B ). Then A ⊗ F is the col-imit of the functor Tw( X ) op → Fun( X, B ) carrying φ : x → y ∈ Tw( X ) to A ( y, ) ⊗ F ( x ) .6.1.3. We are now ready to give our key definition. Definition.
Let M be a monoidal category with K -indexed colimits, A ∈ Alg
Ass (Quiv X ( M ))be an M -enriched precategory and let B be a left M -module in Cat K . An M -functor F : A → B is a left A -module in Fun( X, B ).Taking into account the description of the Quiv X ( M )-action given in 6.1.2,an A -module structure on F ∈ Fun( X, N ) determines a compatible collection ofarrows(127) A ( x, y ) ⊗ F ( x ) → F ( y ) . M -functors from A to a left M -module B form a category denoted as Fun M ( A , B ).This is always a category with K -indexed colimits (as it is a category of modules,see [L.HA], 4.2.3.5).6.1.4. The category Fun M ( A , B ) has the expected functoriality in A and in B .To see this, look at the bifibered family of LM -operads Quiv LM over Cat × Op LM ,see 3.5.1. For a fixed M , this gives a bifibered family Quiv LM ( M , ) of LM -operadsover Cat × LMod M ( Cat K ) which we prefer to see as a cofibered family of LM -operads.Applying 2.12.3 and 2.12.4, we deduce that the category of LM -algebras in it is abifibered family of the categories Fun M ( A , B ) over PCat ( M ) × LMod M .6.1.5. Remark.
We know from Section 5 that an S -enriched precategory A havinga space of objects is nothing but a Segal space. Furthermore, any category B with colimits is a left S -module. We will see in 6.3.5 that the notion of M -functor F : A → B in this case coincides with that of a morphism of Segal spaces.6.1.6. Digression: functors to operads, functors to algebras.
Let R (cid:11) be a categorywith decomposition, let p : C → R be in Fib ( R (cid:11) ), and let X be a category. Wedefine Fun R ( X, C ) as the fiber product Fun( X, C ) × Fun( X, R ) R , with the diagonalmap δ : R → Fun( X, R ).The object Fun R ( X, C ) is fibrous over R , with the fiber Fun R ( X, C ) x = Fun( X, C x )at x ∈ R .Let now µ : P (cid:11) × Q (cid:11) → R (cid:11) be a universal bilinear map, so that R = P ⊗ µ Q .One has the following. this is, of course, the expected coend formula. Lemma.
One has a canonical equivalence in
Fib ( P (cid:11) )(128) Alg µ Q / R (Fun R ( X, C )) ∼ → Fun P ( X, Alg µ Q / R ( C )) . Proof.
Both objects represent the functor carrying K ∈ Cat / P to the space of themaps Map Cat + / P ♮ ( X ♭ × K ♭ × Q ♮ , C ♮ ) . Here the map X ♭ × K ♭ × Q ♮ → P ♮ is induced by µ . (cid:3) Multiplicative property.
Recall that, given monoidal categories M , M ′ ∈ Alg
Ass ( Cat K ) and K -strongly small X, X ′ , one has a canonical map (100)(129) µ : Quiv X ( M ) ⊗ Quiv X ′ ( M ′ ) → Quiv X × X ′ ( M ⊗ M ′ ) . For A ∈ Quiv X ( M ) and A ′ ∈ Quiv X ′ ( M ′ ) we denote by A ⊠ A ′ the image ofthe pair ( A , A ′ ) in Quiv X × X ′ ( M ⊗ M ′ ).Let now B be a left M ⊗ M ′ -module. According to 3.6.7, we can think of B as an M ′ - M rev -bimodule. This makes Fun( X ′ , B ) a Quiv X ′ ( M ′ )- M rev -bimodule,or, equivalently, an M -Quiv X ′ ( M ′ ) rev -bimodule. Applying the functor Quiv BM X ,we get a Quiv X ( M )-Quiv X ′ ( M ′ ) rev -bimodule structure on Fun( X, Fun( X ′ , B )),which, in turn, can be equivalently described as a structure of a left Quiv X ( M ) ⊗ Quiv X ′ ( M ′ )-module on Fun( X, Fun( X ′ , B )). We will see that this structure fac-tors through (129). Moreover, the following result holds. Proposition.
There is a canonical equivalence (130) Fun M ⊗ M ′ ( A ⊠ A ′ , B ) = Fun M ( A , Fun M ′ ( A ′ , B )) . Proof.
The triple T = ( M , B , M ′ rev ) is a BM -monoidal category. We haveFunop BM ( BM X × BM rev X ′ , T ) = Funop BM ( BM X , Funop BM ( BM rev X ′ , T )) =Quiv BM X (Quiv BM X ′ ( T rev ) rev ) . (131)This BM -monoidal category describes the category Fun( X, Fun( X ′ , B )) as Quiv X ( M )-Quiv X ′ ( M ′ ) rev -bimodule. We apply the folding functor φ to get a structure ofleft Quiv X ( M ) ⊗ Quiv X ′ ( M ′ )-module on Fun( X, Fun( X ′ , B )). Applying Corol-lary 3.6.12 and 3.6.13, we get a canonical mapFun( X, Fun( X ′ , B )) → Fun( X × X ′ , B )from the left Quiv X ( M ) ⊗ Quiv X ′ ( M ′ )-module to the left Quiv X × X ′ ( M ⊗ M ′ )-module.This is an equivalence of left modules over the equivalence (129) of monoidalcategories.This implies, together with 3.6.9, the equivalence of the corresponding cate-gories of modules LMod A ⊠ A ′ (Fun( X × X ′ , B )) = LMod A ( LMod A ′ (Fun( X, Fun( X ′ , B )))) . Taking into account Lemma 6.1.6, the right-hand side of the equivalence can berewritten to yield(132)
LMod A ⊠ A ′ (Fun( X × X ′ , B )) = LMod A (Fun( X, LMod A ′ (Fun( X ′ , B )))) . This is precisely our claim. (cid:3)
Pre-enrichment of left M -module. Let, as above, B be a left M -module.For any pair of objects b, c ∈ B one defines a functor(133) hom B ( b, c ) : M op → S to spaces as a composition of ⊗ b : M → B and the presheaf B op → S representedby c ∈ B .We will refer to the collection of presheaves hom B ( b, c ) on M as a pre-enrichmentof B .Given an M -functor F : A → B , where A is an M -precategory and B a left M -module, for any two objects x, y ∈ A we have a map of presheaves on M (134) A ( x, y ) → hom B ( F ( x ) , F ( y ))defined by the map A ( x, y ) ⊗ F ( x ) → F ( y ), see (127).The map (134) is defined uniquely up to equivalence.6.1.9. Definition. An M -functor F : A → B is called M -fully faithful if (134) isan equivalence of presheaves for each pair x, y ∈ A .The functor i : S K → M preserving K -indexed colimits and carrying the ter-minal object to the unit in M , is the unit of M considered as an algebra in Cat K .Thus, it is monoidal. It induces an LM -monoidal functor i LM : ( S K , B ) → ( M , B ),as a cartesian lifting of i : S K → M , see 2.12.3. The functor i LM admits a rightadjoint j LM : ( M , B ) → ( S K , B ) which is, by the general property [L.HA], 7.3.2.7,lax LM -monoidal. It induces a functor j LM ! : Quiv LM X ( M , B ) → Quiv LM X ( S K , B ) thatcarries LM -algebras to LM -algebras.An M -functor F : A → B is just an LM -algebra ( A , F ) in Quiv LM X ( M , B ), soits image under j LM ! is an LM -algebra in Quiv LM X ( S K , B ). In other words, this is an S K -functor j ! ( A ) → B which we denote as j ! ( F ).The following result is immediate.6.1.10. Lemma.
Let F : A → B be an M -fully faithful functor from an enrichedprecategory A to a left M -module B . Then the S K -functor j ! F : j ! A → B is S K -fully faithful. (cid:3) Enriched presheaves.
First of all, we will define the opposite of an M -enriched precategory. Let A ∈ Alg
Ass (Quiv X ( M )). The opposite precategory A op will have X op as the category of objects, and will be enriched over M rev .Furthermore, the category M has a right M -module structure which can beinterpreted as a left M rev -module structure. This allows one to define the categoryof M -presheaves on A , P M ( A ), as Fun M rev ( A op , M ).Finally, we will see that P M ( A ) has a natural left M -module structure comingfrom the left M -module structure on M .This will allow us to define Yoneda map Y : A → P M ( A ) as an M -functor.Details are presented below.6.2.1. Opposite enriched category.
Since A ∈ Alg
Ass (Quiv X ( M )), one has an op-posite algebra A op ∈ Alg
Ass (Quiv X ( M ) rev ).One has a natural equivalenceQuiv X ( M ) rev = Funop( Ass X , M ) rev = Funop( Ass rev X , M rev ) . Lemma.
The planar operad
Ass rev X is naturally equivalent to Ass X op .Proof. The planar operad
Ass X is defined by the functor F X : ∆ op / Ass → S givenby the formula F X ( σ ) = Map( F ( σ ) , X )where the functor F : ∆ / Ass → Cat is described in 3.2. The planar operad
Ass rev X is therefore defined by the functor carrying σ to Map( F ( σ ◦ op) , X ). The claimwill follow from the functorial identification F ( σ ◦ op) = F ( σ ) op , which, taking into account that F ( σ ) are posets, is enough to define on theobjects, that is, on σ : [0] → Ass with the image h n i . In this case the objects of F ( σ ) are x i , y i , i = 1 , . . . , n , and the equivalence is given by “reading the sequenceof objects from left to right”, that is carrying x i to y n − i and y i to x n − i . (cid:3) The lemma identifies Quiv X ( M ) rev with Quiv X op ( M rev ). Thus, for any en-riched precategory A ∈ Alg
Ass (Quiv X ( M )) we can now assign its opposite A op ∈ Quiv X op ( M rev ).6.2.2. Enriched presheaves.
Given an enriched precategory A , we define P M ( A ) asthe category of M rev -functors from A op to M (considered as a left M rev -module).Thus,(135) P M ( A ) = LMod A op (Fun( X op , M )) , where Fun( X op , M ) has a canonical left Quiv X op ( M rev )-module structure de-scribed in 6.1.1. The category P M ( A ) has K -indexed colimits as Fun( X op , M )has K -indexed colimits and the action of Quiv X op ( M rev ) respects them. P M ( A ) is a left M -module. The left M -module structure on M yieldsa left M -module structure on P M ( A ). This follows from 6.1.7: the categoryFun( X op , M ) is a Quiv X op ( M rev )- M rev -bimodule, so P M ( A ), the category of A -modules in Fun( X op , M ), is a right M rev -module, which is the same as a left M -module.6.2.4. Yoneda map.
Yoneda map we present below is a special case of a verygeneral phenomenon — the structure of a left A ⊠ A op -module on A for anyassociative algebra A in a monoidal category C . Here is the construction.Let C be a monoidal category and let A ∈ Alg
Ass ( C ). The opposite algebra A op is an algebra in C rev . The algebra A defines an object in Alg BM ( π ∗ ( C )) — this is A considered as an A -bimodule in C (considered as a C -bimodule category). Wedenote by A ⊠ A op the algebra in the monoidal category C × C rev defined by thepair ( A, A op ). The category C is a left C × C rev -module and A is a module over A ⊠ A op , as shown in 3.6.7.We will apply this to C = Quiv X ( M ) and A ∈ Alg
Ass (Quiv X ( M )). We get astructure of a left C × C rev -module on C , and a left A ⊠ A op -module structure on A .The folding map (79) applied to the left A ⊠ A op -module A , yields an LM -algebrain Quiv LM X × X op ( φπ ∗ ( M )), whose Ass -component is A ⊠ A op , an associative algebraobject in Quiv X × X op ( M ⊗ M rev ), and whose m -component is Fun( X × X op , M ),where M is considered as a left M ⊗ M rev -module. In other words, we have definedan M ⊗ M rev -functor ˜ Y from A ⊠ A op to M .The Yoneda map is defined as the M -functor Y : A → P M ( A ) (from the M -precategory A to the left M -module P M ( A )) corresponding to ˜ Y via the equiva-lence(136) Fun M ⊗ M rev ( A ⊠ A op , M ) = Fun M ( A , P M ( A )) , obtained as the special case of (130) for M ′ = M rev and A ′ = A op .The enriched presheaves of the form Y ( x ), x ∈ X , are called representablepresheaves . We will write Y A ( x ) when we have to explicitly mention the enrichedprecategory A .6.2.5. Free presheaves.
Since enriched presheaves on A are just A op -modules withvalues in the category Fun( X op , M ), we have the notion of a free presheaf — thisis just a free left A op -module. We will now show that representable presheavesare free.Let ¯ h : X × X op → S K be the Yoneda map described in 2.2.4. For x ∈ X we define h x : X op → S K the presheaf represented by x . Applying the unit i : S K → M , we get a functor i ◦ h x ∈ Fun( X op , M ). We have the following result. Corollary.
The enriched presheaf Y A ( x ) on A , represented by x ∈ X , is afree A op -module generated by i ◦ h x .Proof. Denote F : Fun( X op , M ) −→←− P M ( A ) : G the adjoint pair of functors, G being the forgetful functor and F being the free A op -algebra functor. The functor G ( Y A ( x )) is just A ( , x ) ∈ Fun( X op , M ).Denote by X ∈ Quiv X ( M ) the unit of the monoidal structure. Accordingto 4.7.3, G ( Y X ( x )) identifies with i ◦ h x . Let i A : X → A be the unit map.One has a canonical map G ( Y X ( x )) → G ( Y A ( x )). This gives a functor i ◦ h x → G ( Y A ( x )), or, by adjunction, a map F ( i ◦ h x ) → Y A ( x ).To verify this is an equivalence, we can apply once more the forgetful functor G and prove the equivalence θ : A op ⊗ ( i ◦ h x ) → G ( Y A ( x )). The explicit formula6.1.2 for the tensor product expresses A op ⊗ ( i ◦ h x ) as the colimit(137) colim z → t { A ( , z ) × Map X ( t, x ) : Tw( X ) op → Fun( X op , M ) } . One has a canonical map j : A ( , x ) → A op ⊗ ( i ◦ h x ) defined by id x ∈ Tw( X )and id x ∈ Map X ( x, x ). Its composition with θ is an equivalence, as θ can bepresented is the composition A op ⊗ ( i ◦ h x ) → A op ⊗ G ( Y ( x )) → G ( Y ( x )) , and the composition θ ◦ j can be now factored as A ( , x ) → A ( , x ) ⊗ A ( x, x ) → A ( , x ) , where the first arrow is induced by the identity → A ( x, x ), and the second bycomposition in A . Now it remains to verify that j is an equivalence. We canrewrite (137) as the colimit(138) colim z → t → x { A ( , z ) : Tw( X ) op × X op ( X /x ) op → Fun( X op , M ) } of the functor assigning A ( , x ) to the object z → t → x of Tw( X ) op × X op ( X /x ) op ,where the colimit of the compositionTw( X ) op × X op ( X /x ) op π → X /x α → Fun( X op , M ) , with π carrying z → t → x to z → x and α carrying z → x to A ( , x ). Now ourclaim follows as π is cofinal by the Quillen’s Theorem A [L.T], 4.1.3.1 and X /x has a terminal object id x . (cid:3) The enriched Yoneda lemma.
Let x ∈ A and let F ∈ P M ( A ). Proposition.
The presheaf hom P M ( A ) ( Y ( x ) , F ) on M is represented by F ( x ) .Proof. We will construct an equivalence of presheaves(139) h F ( x ) → hom P M ( A ) ( Y ( x ) , F ) , where h F ( x ) is the presheaf on M represented by F ( x ). We will construct a canonical equivalence(140) Map M ( m, F ( x )) → Map P M ( A ) ( m ⊗ Y ( x ) , F ) , which is, since Y ( x ) is a free right A -module generated by i ( h x ), the same as(141) Map M ( m, F ( x )) → Map
Fun( X op , M ) ( m ⊗ i ( h x ) , F ) . An object m ∈ M defines an adjoint pair(142) L m : S −→←− M : R m , where L m ( S ) = i ( S ) ⊗ m and R m ( M ) = Map( m, M ). Applying to both partsthe functor Fun( X op , ), we get an adjoint pair(143) L m : P ( X ) −→←− Fun( X op , M ) : R m . The equivalence (141) is obtained from this adjunction and from the conventionalYoneda applied to Map M ( m, F ) ∈ P ( X ). (cid:3) As usual, we now have
Corollary.
The Yoneda embedding Y : A → P M ( A ) is M -fully faithful. (cid:3) Enrichment of a left-lensored category.
Let once more M be a monoidalcategory with colimits and let B be a left M -module.Assume that for all b, c ∈ B the weak enrichment functorhom B ( b, c ) : M op → S defined by (133), is representable. Then one would like to believe that B acquiresa canonical M -enrichment.This is in fact so, as Proposition 6.3.1 below asserts.Let X be a category with a functor F : X → B . Recall that the categoryFun( X, B ) is left-tensored over Quiv X ( M ), so it makes sense to look for an en-domorphism object of F ∈ Fun( X, B ) in Quiv X ( M ).6.3.1. Proposition.
Let M be a monoidal category in Cat K and B be a left M -module. Let F : X → B be a functor from a strongly K -small category X to B sothat, for any x, y ∈ X the functor hom B ( F ( x ) , F ( y )) : M op → S is representable.Then the endomorphism object A = End Quiv X ( M ) ( F ) exists; A is an M -enrichedprecategory and the corresponding M -functor e F : A → B , extending F : X → B ,is M -fully faithful.Proof. Let us recall the construction of the endomorphism object presented in [L.HA],4.7.2. For a monoidal category C , a left C -module F , and an object F ∈ F , amonoidal category C [ F ], whose objects are C ⊗ F → F, C ∈ C , is constructed.If C [ F ] has a terminal object, this is the endomorphism object of F in C , and itautomatically acquires an algebra structure.
00 VLADIMIR HINICH
We apply this construction to C = Quiv X ( M ) acting on F = Fun( X, B ) andto the object F ∈ F .Let us describe the terminal object of C [ F ]. The formulas 6.1.2 for the actionof Quiv X ( M ) on Fun( X, B ) identify Map F ( A ⊗ F, F ) with(144) lim ψ : z → z ′ ∈ Tw( X ) lim φ : x → y ∈ Tw( X ) Map( A ( y, z ) ⊗ F ( x ) , F ( z ′ )) =lim ψ : z → z ′ ∈ Tw( X ) lim φ : x → y ∈ Tw( X ) Map( A ( y, z ) , hom B ( F ( x ) , F ( z ′ ))) . This proves that the terminal object A : X op × X → M is given by the formula A ( x, y ) = hom B ( F ( x ) , F ( y )) . (cid:3) Let us study the functoriality of the above construction with respect to changeof M and X .Let a : N → M be a monoidal functor having right adjoint b : M → N ( b isautomatically lax monoidal). Given a left M -module B and a functor F : X → B satisfying the conditions of the previous proposition, we can construct twoendomorphism algebras, A = End Quiv X ( M ) ( F ) and A N = End Quiv X ( N ) ( F ). Onehas6.3.2. Lemma.
One has a canonical equivalence A N = b ! ( A ) .Proof. The adjoint pair a : N −→←− M : b induces an adjoint pair a ! : Quiv X ( N ) −→←− Quiv X ( M ) : b ! . Thus, for any N -enriched precategory A ′ one hasMap( A ′ , A N ) = Map( A ′ ⊗ F, F ) = Map( a ! ( A ′ ) ⊗ F, F ) =Map( a ! ( A ′ ) , A ) = Map( A ′ , b ! ( A )) . This implies A N = b ! ( A ). (cid:3) Let now f : X ′ → X be a map of spaces and F : X → B be as above. Assume F satisfies the conditions of Proposition 6.3.1, so that A = End Quiv X ( M ) ( F ) exists.Denote F ′ = f ! ( F ) = F ◦ f .6.3.3. Lemma.
The functor F ′ satisfies 6.3.1 and its endomorphism object is A ′ := f ! ( A ) .Proof. The condition of 6.3.1 is clearly satisfied. Let A ′ = End Quiv ′ X ( M ) ( F ′ ).Applying f ! to the pair ( A , F ), we get ( f ! A , F ′ ), so an algebra map f ! A → A ′ . Itremains to prove this is an equivalence. For this one can forget the multiplicativestructure and compare f ! A ( x, y ) with A ′ ( x, y ) for x, y ∈ X ′ . This proves theclaim. (cid:3) More precisely, A ( x, y ) is the object of M representing hom B ( F ( x ) , F ( y )). Corollary.
Let M be a monoidal category on Cat L , B be a left M -module in Cat L and let f : X → B be a functor, such that, for any x, y ∈ X hom B ( F ( x ) , F ( y )) isrepresentable. Then A = End Quiv X ( M ) ( f ) exists; A is an M -enriched precategoryand the M -functor ˜ F : A → B extending F : X → B , is M -fully faithful.Proof. Let κ be a cardinal satisfying 3.5.4. A large category X can be presented,after a change of universe, as a κ -filtered colimit of κ -compact categories X α , forwhich Propostion 6.3.1 can be applied. We get Quiv X ( M ) = lim α Quiv X α ( M )which implies the analogous expression for Alg
Ass (Quiv X ( M )). Denote i α : X α → X the canonical map. By Lemma 6.3.3 the endomorphism objects of F ◦ i α arecompatible, which gives the endomorphism object of F . (cid:3) The case M = S . Let now M = S . For a category B with colimits weapply 6.3.4 to the identity functor F := id B . The endomorphism object of id B , E := End Quiv B ( S ) (id) is an S -precategory with the category of objects B . The unitmap B → E in Quiv B ( S ) is an equivalence by the explicit description of E ( x, y )given in the proof of 6.3.1. One has a canonical map i : B → B defined by the B -module structure on id B . Functoriality (6.1.4) defines for any A ∈ Quiv X ( M )a map(145) Map PCat ( S ) ( A , B ) → Map S ( A , B ) = Fun S ( A , B ) eq fibered over Map Cat ( X, B ). It is an equivalence as it induces equivalences of thefibers.Let us now assume that X is a space. Then the embedding i : B eq → B induces an equivalence Map( X, B eq ) = Map( X, B ) and the equivalence (145) canbe rewritten as(146) Map PCat ( S ) ( A , i ! B ) = Map S ( A , B ) . Note that i ! B is precisely the S -enriched precategory we assigned to B in 4.7.2.As PCat ( S ) is equivalent to the category of Segal spaces, we deduce that Map S ( A , B )is the space of maps between the corresponding Segal spaces.6.3.6. Let M ∈ Alg
Ass ( Cat L ), A ∈ PCat ( M ), B ∈ LMod M ( Cat L ). Assume thatfor all b, c ∈ B the presheaf hom B ( b, c ) is representable.Denote B = End Quiv B ( M ) ( B ). By 6.1.4, one has a map of spaces(147) γ : Map PCat ( M ) ( A , B ) → Fun M ( A , B ) eq , deduced by functoriality from the canonical action of B on B ∈ Fun( B , B ). Lemma.
The map (147) is an equivalence.
02 VLADIMIR HINICH
Proof.
Let X be the category of objects of A . The source and the target of (147)are fibered over Map( X, B ). It is sufficient to verify that for any f : X → B the fiber γ f at f of (147) is an equivalence. By Lemma 6.3.3, the source of γ f identifies with Map Alg (Quiv X ( M )) ( A , End
Quiv X ( M ) ( f )), which, by the universalproperty of the endomorphism object [L.HA], 4.7.1.41, identifies with the targetof γ f . (cid:3) M ( A , B ),provided M is a presentably E -monoidal category, that is a E -algebra in thecategory of presentable categories and colimit preserving functors.In this case, according to [GH], 4.3.5 and 4.3.16, PCat ( M ) is presentablymonoidal. In particular, the tensor product in PCat ( M ) admits a right adjoint A , A ′ FUN ( A , A ′ ) . For any presentable category B left-tensored over M we denote B = End Quiv B ( M ) ( B ). Proposition.
Let M be presentably E -monoidal category, A a M -enriched pre-category, B presentable category left-tensored over M . Then one has an equiva-lence Fun M ( A , B ) = FUN ( A , B ) . Proof.
For any A ′ ∈ PCat ( M ) one hasMap PCat ( M ) ( A ′ , Fun M ( A , B )) = Fun M ( A ′ , Fun M ( A , B )) eq =(148)Fun M ⊗ M ( A ′ ⊠ A , B ) eq = Fun M ( A ′ ⊗ A , B ) eq = Map PCat ( A ′ ⊗ A , ¯ B ) =Map PCat ( A ′ , FUN ( A , ¯ B )) . (cid:3) Remark.
In particular, for M presentably E -monoidal the category ofenriched presheaves P M ( A ) can be defined entirely in the world of M -categoriesas FUN ( A op , M ). We are grateful to the referee for pointing this out.We will now use Proposition 6.3.1 to prove Proposition 2.8.9 describing theoperad Funop( C , D ) in the case where C , D are symmetric monoidal categories.6.3.9. Proof of Proposition 2.8.9.
Recall that the bifibration p = ( p , p ) : Ar → Cat × Cat is the composition of the embedding Ar → Cat / [1] with the restrictionto the ends of [1].Denote by q : U → Cat the universal cocartesian fibration. This is the co-cartesian fibration classified by the identity functor id :
Cat → Cat . There is acanonical evaluation map(149) ev : U × Cat Ar → U of cocartesian fibrations over Cat , defined by the family of evaluation maps P × Fun( P , Q ) → Q . We construct this map as follows .We apply Corollary 6.3.4 to M = Cat , X = Cat and f = id. We deducethe existence of A = End Quiv
Cat ( Cat ) (id), with A : Cat op × Cat → Cat givenby the formula A ( P , Q ) = Fun( P , Q ). This yields the evaluation map 149 asthe cocartesian fibration q : U → Cat is classified by id :
Cat → Cat and thebifibration p : Ar → Cat × Cat is classified by A .Note that (149) preserves products. Therefore, it induces a SM functor(150) ev × : U × × Cat × Ar × → U × . Let now C be a strong approximation of an operad and let P , Q be C -monoidalcategories. Making the base change of (150) with respect to the map C → Cat × × Cat × defined by the pair ( P , Q ), we get P × F P , QC → Q , which yields the map of C -operads(151) η C : F P , QC → Funop C ( P , Q ) . Note that η C commutes with the base change: for any q : D → C q ∗ ( η C ) is acomposition of η D with the equivalence described in 2.8.8.It remains to prove that η C is an equivalence of C -operads. The fiber of (150)at h i ∈ Com is (149). This gives, for any x ∈ C , an equivalence of both ( F P , QC ) x and Funop C ( P , Q ) x with Fun( P x , Q x ). Thus, η C induces an equivalence of thecategories of colors.It remains to verify the spaces of active maps in F P , QC and Funop C ( P , Q ). For n -ary operations, we will use the approximation Q n of the free operad C n , asin 2.9.6. Fix a map φ : Q n → C . We have to verify that φ ∗ ( φ C ) induces an equiv-alence of the spaces of Q n -algebras. Since η C commutes with the base change,the claim immediately reduces to the case C = Q n and φ = id. Q n -monoidal category P is just a collection P , . . . , P n of categories endowedwith a functor P × . . . × P n → P . Both spaces of algebras, Map Op Qn ( Q n , F P , Q Q n )and Map Op Qn ( Q n , Funop Q n ( P , Q )) = Map Op Qn ( P , Q ), identify with the space n Y i =1 Map( P i , Q i ) × Map( Q P i , Q ) (Map( Y P i , Q ) × [1]) × Map( Q P i , Q ) Map( P , Q ) Construction of the map 149 is the only reason for which we delayed the proof of Propo-sition 2.8.9 till Section 6.
04 VLADIMIR HINICH describing the 2-diagrams(152) P × . . . × P n / / { f i } (cid:15) (cid:15) P t | qqqqqqqqqqqqqqqqqqqqqq f (cid:15) (cid:15) Q × . . . × Q n / / Q , with the horizontal arrows defined by the operations in the Q n -monoidal cate-gories P and Q . 7. Completeness
In this section M is a monoidal category in Cat L . We will define here M -enriched categories as M -enriched precategories satisfying a completeness condi-tion.This material is not new as our notion of enriched precategory is equivalent,for X a space, to that of [GH] where the completeness issue has already beenaddressed, see [GH], 5.6. We present a new, very easy, proof of [GH], 5.6, basedon Proposition 6.3.1.7.1. According to Section 5, the category of S -enriched precategories is equiva-lent to the category of Segal spaces. This justifies the following definition.7.1.1. Definition.
1. An S -enriched precategory A ∈ Quiv X ( S ) is called an S -enriched category if X is a space and the Segal space defined by A iscomplete.2. An M -enriched precategory A is called an M -enriched category if its image j ! ( A ) ∈ Quiv X ( S ) with respect to a functor j : M → S defined as j ( M ) =Map( , M ), is an S -enriched category.7.1.2. In this section we denote PCat ( M ) = Alg
Ass (Quiv( M )) the category of M -enriched precategories, and Cat ( M ) the category of M -enriched categories.In 7.2 below we prove that the full embedding Cat ( M ) → PCat ( M ) admits aleft adjoint localization functor. The construction of localization makes use ofYoneda embedding.Let J denote the (conventional) contractible groupoid on two objects. Wewill denote by the same letter the corresponding simplicial space and the S -enriched precategory. We will also denote by ∗ the terminal object in PCat ( S )(the singleton).We keep the notation i : S −→←− M : j for the adjoint pair of functors. For an M -enriched precategory A completeness of j ! ( A ) means that the canonical map(153) Map PCat ( S )( ∗ , j ! ( A )) → Map
PCat ( S )( J , j ! ( A ))is an equivalence. The source of this map is X , the space of objects of A , whereasthe target can be rewritten using the adjunction. Defining J M := i ! ( J ), we get astandard characterization of completeness. Proposition. An M -precategory A with the space of objects X is complete iffthe natural map (154) X → Map
PCat ( M )( J M , A ) is an equivalence. Localization functor.
Given A ∈ Alg
Ass (Quiv X ( M )), we define X ′ to bethe subspace of representable functors in P M ( A ) eq .The embedding Y ′ : X ′ → P M ( A ) is tautological. The localization L ( A ) isdefined as the endomorphism object A ′ := End Quiv X ′ ( M ) ( Y ′ ) whose existence isguaranteed by Proposition 6.3.1.By definition, the Yoneda embedding Y : X → P M ( A ) factors through Y ′ ,yielding y : X → X ′ . The universality of A ′ yields a unique map a : A → A ′ over y : X → X ′ .To prove the completeness of A ′ , we use criterion 7.1.2. Let i : P M ( A ) eq → P M ( A ) be the obvious embedding. The space Map PCat ( S ) ( J , j ! A ′ ) identifies withthe subspace of Map PCat ( S ) ( J , j ! i ! P M ( A ) ) = Map Seg ( J , P M ( A )) , Seg being the category of Segal spaces, spanned by the functors with valuesin representable objects. Thus, completeness of P M ( A ) considered as a Segalspace implies completeness of A ′ . Note that the functor a : A → A ′ induces anequivalence Map PCat ( M ) ( J M , A ) = Map PCat ( M ) ( J M , A ′ ) , as both identify with the same subspace X ′ of P M ( A ). Therefore, if A is complete, X = X ′ and A = A ′ . This proves universality of the map A → A ′ .7.2.1. Corollary.
Let A , B be M -enriched precategories, so that B is complete.Then the composition with Yoneda embedding B → P M ( B ) identifies the space Map
PCat ( M ) ( A , B ) with the subspace of Fun M ( A , P M ( B )) = Fun M × M rev ( A ⊠ B op , M ) consisting of the maps F : A → P M ( B ) whose essential image is in the space ofrepresentable presheaves.Proof. Let A ∈ Quiv X ( M ), B ∈ Quiv X ′ ( M ). Since B is complete, the Yonedaembedding Y : B → P M ( B ) induces an equivalence of X ′ with the space ofrepresentable presheaves in P M ( B ). One has a commutative diagram of spaces(155) Map( A , B ) ' ' PPPPPPPPPPPP / / Fun M ( A , P M ( B )) rep u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ Map(
X, X ′ ) , where the superscript rep denotes the space of M -functors with essential image inrepresentable presheaves. In order to prove the horizontal arrow is an equivalence,
06 VLADIMIR HINICH it is enough to verify the equivalence of fibers over any f : X → X ′ . The fiberof the left-hand side is Map( A , f ! B ), whereas the fiber of the right-hand side isthe space of A -module structures on the composition F : X f → X ′ → P M ( B ).Since B is the endomorphism object of X ′ → P M ( B ), f ! ( B ) is the endomorphismobject of F , so that the right-hand side also identifies with Map( A , f ! B ). (cid:3) Correspondences
In the conventional category theory, a correspondence from C to D can bedefined as a functor C op × D → Set . Equivalently, it can be defined as a category X over [1], with a pair of equivalences C ∼ = X , D ∼ = X .A similar description for ∞ -categories is also well-known, see [L.T], 2.3.1,or [AF], 4.1. See also D. Stevenson, [S].In this Section we prove this result, using the techniques of Section 6, in agreater generality, including, for instance, ( ∞ , n )-categories.8.1. Correspondences, I.
Let M be a monoidal category with colimits, and let C , D ∈ Cat ( M ). A correspondence from C to D is, by definition, an M -functor D → P M ( C ).Equivalently, a correspondence can be defined as an M ⊗ M rev -functor from D ⊠ C op to M , or as an M rev -functor from C op to P M rev ( D op ).A correspondence is right-representable if the essential image of the correspond-ing M -functor D → P M ( C ) belongs to the space of representable presheaves.According to 7.2.1, a right-representable correspondence gives rise to a uniquelydefined functor D → C . Left-representable correspondences are defined simi-larly; they lead to functors C → D . An adjoint pair of functors in Cat ( M ) can bedefined as a correspondence which is simultaneously left and right representable.8.1.1. The category of M -enriched correspondences, Cor ( M ), classifies pairs C , D ∈ Cat ( M ), endowed with an object of Fun M ⊗ M rev ( D ⊠ C op , M ). This leadsto the following definition.(156) Cor ( M ) = Cat ( M ) × Alg
Ass (Quiv( M ⊗ M rev )) Alg LM (Quiv LM ( M ⊗ M rev , M )) , where the map Cat ( M ) → Cat ( M ⊗ M rev ) ⊂ Alg
Ass (Quiv( M ⊗ M rev )) carries apair ( C , D ) to D ⊠ C op .We will prove later that, for M prototopos 5.4.1, the described above cate-gory Cor ( M ) can be equivalently described as Cat ( M ) / [1] . We will start with adigression about Kan extensions, with the aim of proving Corollary 8.2.4.8.2. Kan extensions.
Let f : Y → X be a cocartesian fibration and let M bea category with colimits. In 8.2.2 below we present a convenient way to describethe left Kan extension f ! : Fun( Y, M ) → Fun( X, M ). π : M • → Cat classifying the functor e M : Cat op → Cat defined by the formula e M ( B ) = Fun( B, M ).We compareit to the trivial cartesian fibration M × Cat over
Cat classifying the constantfunctor c M : Cat op → Cat with value M . For each B ∈ Cat the diagonal functor δ B : M → Fun( B, M ) has a left adjoint, given by the colimit.The diagonal c M → e M yields a functor δ : M × Cat → M • which has a(relative) left adjoint by [L.HA], 7.3.2.6. We denote the functor left adjoint to δ by colim : M • → M × Cat , for an obvious reason.Denote M • X = X × Cat M • the base change, where the map X → Cat classifies f . The category M • X over X can be described as an internal mapping object, M • X = Fun Cat /X ( Y, M × X ), see [GHN], 7.3.One has a canonical equivalence θ f defined as the composition(157) Fun Cat ( X, M • ) ∼ → Fun X ( X, M • X ) ∼ → Fun X ( Y, M × X ) ∼ → Fun( Y, M ) . Lemma.
The composition
Fun( Y, M ) = Fun Cat ( X, M • ) colim → Fun( X, M ) is left adjoint to f ∗ : Fun( X, M ) → Fun( Y, M ) .Proof. The functor colim : Fun
Cat ( X, M • ) → Fun( X, M ) is left adjoint to thediagonal δ : Fun( X, M ) → Fun
Cat ( X, M • ). Now the claim follows from the factthat the composition of δ with θ f is equivalent to f ∗ : Fun( X, M ) → Fun( Y, M ). (cid:3) Lemma.
Assume f : Y → X is a left fibration and that M is a prototopos.Then the left Kan extension functor f ! : Fun( Y, M ) → Fun( X, M ) is a rightfibration.Proof. Since f is a right fibration, we can replace the base of the cartesian fibra-tion M • with S ⊂ Cat . Moreover, since M is a prototopos, M • can be equivalentlydescribed as classifying the functor S op → Cat carrying B → M /B . The functor(colim , π ) : M • → M × S in this interpretation becomes just the forgetful functorwhich is a right fibration as all M /B → M are right fibrations.Now the functor f ! is the compositionFun( Y, M ) = Fun S ( X, M • ) → Fun S ( X, M × S ) = Fun( X, M )which is a right fibration as a base change of a right fibration. (cid:3) We will apply the above result to X = ∆ op , Y = (∆ / [1] ) op and the forgetfulfunctor f : Y → X . Recall that Ass = ∆ op and BM = (∆ / [1] ) op . We have8.2.4. Corollary.
For a prototopos M , one has a natural equivalence Fun( BM , M ) = Fun( Ass , M ) / [1] , where [1] in the right hand side of the formula is interpreted as a composition ∆ op = Ass → S → M .
08 VLADIMIR HINICH
Proof.
According to 8.2.3, the left Kan extension f ! : Fun( BM , M ) → Fun(
Ass , M )is a right fibration. Since the left-hand side has a terminal object ∗ , it is equivalentto the overcategory over f ! ( ∗ ) = [1]. (cid:3) Categories over [1] . We now assume that M is a prototopos, in the senseof Definition 5.4.1.The category of M -categories, Cat ( M ), is equivalent to the full subcategoryof Fun( Ass , M ), spanned by the simplicial objects X • satisfying the followingproperties • X is a space (that is, it belongs to the essential image of the uniquecolimit-preserving functor S → M preserving the terminal object. • X • is complete and satisfies the Segal condition.Applying 8.2.4, we get the following.8.3.1. Lemma.
Let M be a prototopos. The category Cat ( M ) / [1] is naturallyequivalent to the full subcategory of Fun( BM , M ) spanned by the functors F : BM → M satisfying the following properties. • Segal conditions. • Completeness: the restrictions of F to Ass ± are complete. (cid:3) We now have everything we need to prove the main result of this subsection.8.3.2.
Proposition.
Let M be a prototopos. The the category of M -correspondences Cor M by the formula (156), has an equivalent description as Cat ( M ) / [1] where [1] ∈ Cat ( M ) is the image of the “genuine” segment [1] ∈ Cat under the canonicalmap
Cat = Cat ( S ) → Cat ( M ) .Proof. We use Lemma 8.3.1 to present
Cat ( M ) / [1] as the full subcategory ofFun laxNC ( BM , M ) spanned by the functors satisfying the completeness condition.According to Proposition 5.3.1 applied to the operad BM , Fun laxNC ( BM , M ) iden-tifies with Alg BM ( M ⊙ ).Using folding functor and Proposition 3.6.7, we deduce that Fun laxNC ( BM , M )is naturally equivalent to Alg LM ( φ ( M ⊙ ))).We will now present a description (156) in terms of LM -algebras.Using the fact that M is symmetric monoidal, we replace Fun M ⊗ M rev ( D ⊠ C op , M ) with Fun M ( D ⊗ C op , M ), where D ⊗ C op is the pushforward of D ⊠ C op with respect to SM functor M ⊗ M rev → M .We get the following symmetric monoidal version of (156).(158) Cor ( M ) = Cat ( M ) × Alg
Ass (Quiv( M )) Alg LM (Quiv LM ( M , M )) , where the map Cat ( M ) → Alg
Ass (Quiv( M )) carries a pair of algebras ( C , D ) to D ⊗ C op . Denote as i : Ass → LM and j : LM → BM the standard embeddings. Thefunctor i ∗ : Op LM → Op Ass has a right adjoint i ∗ carrying a planar operad O to thepair ( O , [0]). This functor carries monoidal categories to LM -monoidal categories.This simple trick allows one to describe Alg
Ass ( O ) as Alg LM ( i ∗ ( O )). Therefore,the expression (158) for Cor ( M ) can be rewritten as the full subcategory of LM -algebras in the family of LM -monoidal categories i ∗ Quiv( M ) × i ∗ Quiv( M ) Quiv LM ( M , M ) . According to Proposition 5.5.1, Quiv( M ) = M ∆ and Quiv LM ( M , M ) = j ∗ M = .Therefore, the above family of LM -monoidal categories is equivalent to(159) i ∗ ( M ∆ ) × i ∗ M ∆ j ∗ M = . The above formula may require explanation. The category
FamMon LM admitslimits which commute with the forgetful functor FamMon LM → Cat . Thus, thefiber product 159 is a family of monoidal categories over S × S , with the fiberover ( X, Y ) being i ∗ ( M ∆ X × M ∆ Y ) × i ∗ M ∆ Y × X op j ∗ M = Y × X op . In Subsection 8.4 below we construct, for M a prototopos, a canonical equivalenceof (159) with φ ( M ⊙ ). M -correspondences are precisely LM -algebras in either ofthese categories, satisfying completeness property. This proves the equivalenceof two descriptions. (cid:3) An equivalence of φ ( M ⊙ ) with (159). We will construct a compatiblepair of LM -monoidal functors. φ ( M ⊙ ) → i ∗ ( M ∆ ) and φ ( M ⊙ ) → j ∗ ( M = ) . This will give an LM -monoidal functor from φ ( M ⊙ ) to the fiber product (159)which will be shown to be an equivalence.8.4.1. The map φ ( M ⊙ ) → i ∗ ( M ∆ ) is adjoint to the equivalence ( M ∆ ) = i ∗ ( φ ( M ⊙ )), see Remark 3.6.6.The construction of the map φ ( M ⊙ ) → j ∗ ( M = ) is more tricky.Recall that the family M ⊙ is constructed using a functor E : BM op → Cat , withthe values in conventional categories, see 5.2.1. For each s ∈ BM the category E ( s ) is endowed with a collection of distinguished diagrams; the family M ⊙ isclassified by the functor F : BM → Cat carrying s ∈ BM to Fun ′ ( E ( s ) , M ), whereFun ′ denotes the full subcategory of functors carrying distinguished diagrams in E ( s ) to cartesian diagrams in M . We will present a similar description for φ ( M ⊙ )and for j ∗ ( M = ). We will define two functors E φ and E j from LM op to Cat , withvalues in conventional categories, and with families of distinguished diagrams, spanned by the objects for which the algebras C , D in Quiv( M ) are complete.
10 VLADIMIR HINICH so that φ ( M ⊙ ) classifies the functor F φ carrying s ∈ LM to Fun ′ ( E φ ( s ) , M ), and,similarly, j ∗ ( M = ) classifies the functor F j carrying s ∈ LM to Fun ′ ( E j ( s ) , M ).Then the functor φ ( M ⊙ ) → j ∗ ( M = ) will be induced by a map E j → E φ offunctors.By definition, E j : LM op → Cat is just the composition E ◦ j . The functor E φ will be constructed in two steps; we will first present an obvious choice E ′ φ for it,and then we will replace it with a more appropriate version E φ .We keep the notation of Remark 3.6.6. The family φ ( M ⊙ ) classifies the functor F φ : LM → Cat , F φ ( s ) = Fun ′ ( E ′ φ ( s ) , M ), for the following choice of E ′ φ .(160) E ′ φ ( s ) = ( E ( s ) ⊔ L ⊔ R E ( s op ) , if s ∈ Ass ⊂ LM , E ( s ∗ ) if s ∈ LM − Here the colimit is meant to be in the naive sense, identifying the pairs of objects L and R .Distinguished squares in E ′ φ ( s ) come from distinguished squares in E . Theformula F φ ( s ) = Fun ′ ( E ′ φ ( s ) , M ) is immediate. The choice of E ′ φ as a representingfunctor for F φ has, however, a drawback: there is no obvious morphism of functors E j → E ′ φ required for the construction to work. This is why we make a minoramendment.We will define a modified functor E φ : LM op → Cat and a collection of dis-tinguished diagrams in E φ ( s ) and a pair of full embeddings i : E j → E φ and i : E ′ φ → E φ so that the map i induces an equivalenceFun ′ ( E φ ( s ) , M ) → Fun ′ ( E ′ φ ( s ) , M ) . The functor i : E j → E φ will induce then the map F φ → F j we are going toconstruct.For s ∈ Ass we denote by ( i, j ) ′ the object of E ′ φ ( s ) coming from ( i, j ) ∈ E ( s )and ( i, j ) ′′ the one coming from E ( s op ).8.4.2. The categories E φ will be defined as total categories of correspondences .Let us recall a general setup. Given a pair of (conventional) categories A , B , anda functor Φ : A op → P ( B op ), a category C over [1] is defined, with fibers A and B over 0 and 1 respectively, and with the Hom-sets Hom( X, Y ) = Φ( X )( Y ) for X ∈ A , Y ∈ B . We will apply this construction for A := E ( s ) and B := E ′ φ ( s )and to the functor Φ defined by the formulas below. In what follows we denoteby h Y the copresheaf on B corepresented by Y ∈ B . Case I: s ∈ Ass ⊂ LM . The functor Φ is defined by the following formulas. • Φ( R ) is an empty copresheaf. • Φ( L ) = h L ⊔ h R . We are working now with conventional categories for which the equivalence between cor-respondences and categories over [1] is not a problem! • For 0 ≤ i ≤ j ≤ n , n = | s | , Φ( i, j ) = h ( i,j ) ′ ⊔ h ( n − j,n − i ) ′′ . Case II: s ∈ LM − . The correspondence is given by a functor Φ : E ( s ) op → P ( E ( s ∗ ) op ) assigning to e ∈ E ( s ) the copresheaf on E ( s ∗ ) defined as follows. • Φ( R ) = Φ( | s | , | s | ) is an empty copresheaf. • Φ( L ) = h L ⊔ h R . • For ( i, j ) , j < | s | Φ( i, j ) = h ( i,j ⊔ h ( n − j,n − i ) , n = | s ∗ | . • For j = | s | , Φ( i, j ) = h ( i, j − i − .Note that the formulas for Φ presented above, specifying the value of Φ one theobjects, uniquely extend to the arrows of E ( s ).The formulas above are functorial in s : a map s → s ′ defines a map of corre-spondences, and, therefore, a map of the total categories E φ ( s ′ ) → E φ ( s ). Thus,we have constructed E φ : LM op → Cat .8.4.3. We will now define distinguished diagrams in E φ ( s ). They are of twotypes:(1) Distinguished diagrams in E ′ φ ( s ) ⊂ E φ ( s ).(2) For each X ∈ E ( s ) such that Φ( e ) = ⊔ h Y i , the diagram X → Y i in E φ ( s ) .We define Fun ′ ( E φ ( s ) , M ) as the full subcategory of Fun( E φ ( s ) , M ) spannedby the functors carrying distinguished diagrams to cartesian diagrams in M .This means, for instance, that any distinguished diagram X → Y i induces anequivalence F ( X ) = Q F ( Y i ).8.4.4. Lemma.
The restriction map
Fun ′ ( E φ ( s ) , M ) → Fun ′ ( E ′ φ ( s ) , M ) is anequivalence.Proof. A functor F : E φ ( s ) → M is a right Kan extension of its restriction to E ′ φ ( s ) if and only if the distinguished diagrams of second kind are sent to cartesiandiagrams in M . The rest follows from [L.T], 4.3.2.15. (cid:3) φ ( M ⊙ ) → j ∗ ( M = ) as classifying the mor-phism of functors F φ → F j induced by the embedding E j → E φ . Note that if afunctor F : E φ ( s ) → M carries distinguished diagrams to cartesian diagrams in M , the composition E ( s ) → E φ ( s ) → M satisfies the same property.Therefore, the restriction map Fun ′ ( E φ ( s ) , M ) → Fun( E ( s ) , M ) has image inFun ′ ( E ( s ) , M ).Let us construct an equivalence between the two compositions φ ( M ⊙ ) → i ∗ M ∆ .Using the adjunction between i ∗ and i ∗ , we can deduce it from an equivalence oftwo adjoint maps i ∗ φ ( M ⊙ ) → M ∆ . Both are maps of cocartesian fibrations on Ass classifying functors to
Cat . Let us describe these functors. The restriction i ∗ φ ( M ⊙ ) has a more convenient presentation than the one via E φ : LM op → Cat . the set of indices is allowed to be empty.
12 VLADIMIR HINICH
Let E ◦ ( s ) ⊂ E ( s ), s ∈ Ass , be the full subcategory spanned by all objects exceptfor R .One has an isomorphism E ( s ) → E ( s op ), e e op , interchanging L with R andcarrying ( i, j ) to ( | s | − j, | s | − i ).We define E ◦ φ ( s ) = Λ × E ◦ ( s ) (recall that Λ has three objects, 0 , , f : E ◦ ( s ) → E φ ( s ) carrying (0 , e )to e ∈ E ( s ) ⊂ E φ ( s ), (1 , e ) to e ∈ E ( s ) ⊂ E ′ φ ( s ), and (2 , e ) to e op ∈ E ( s op ).The functior f is fully faithful, with the image consisting of all objects except R ∈ E ( s ) ⊂ E φ ( s ). Thus, with the obvious collection of distinguished dia-grams, the map Fun ′ ( E φ ( s ) , M ) → Fun ′ ( E ◦ φ ( s ) , M ) is an equivalence. Finally,Fun ′ ( E ◦ φ ( s ) , M ) = Fun ′ (Λ , Fun ′ ( E ◦ , M )), where now Fun ′ (Λ , ) denotes the fullsubcategory of functors, carrying Λ to a product diagram. In these terms theleft composition restricts F : Λ → Fun ′ ( E ( s ) , M ) to the pair (1 ,
2) and thentakes a product, and the right composition just restricts F to 0 ∈ Λ .8.4.6. It remains to verify that the map φ ( M ⊙ ) → i ∗ ( M ∆ ) × i ∗ M ∆ j ∗ M = is an equivalence. This is an LM -monoidal functor of cartesian families of LM -monoidal categories, so it is sufficient to fix a pair of spaces ( X, Y ) and restrictoneself to the fibers at s = (00) and (01). In both cases the fiber at (0 ,
0) is M ∆ X × M ∆ Y and the fiber at (01) is M /X × Y . References [AF] D. Ayala, J. Francis, Fibrations of ∞ -categories, preprint arXiv 1702.02681.[AFR] D. Ayala, J. Francis, N. Rozenblyum, Factorization homology I: Higher categories.Adv. Math. 333 (2018), 1042-1177.[BGS] C. Barwick, S. Glasman, J. Shah, Spectral Mackey functors and equivariant algebraicK-theory, Tunisian J. Math, 2 (2020), 97–146, arXiv: 1505.03098.[DK1] W. Dwyer, D. Kan, Simplicial localizations of categories. J. Pure Appl. Algebra 17(1980), no. 3, 267–284.[DK2] W. Dwyer, D. Kan, Calculating simplicial localizations. J. Pure Appl. Algebra 18(1980), no. 1, 17–35.[DK3] W. Dwyer, D. Kan, Function complexes in homotopical algebra. Topology 19 (1980),no. 4, 427–440.[GZ] P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory, Springer, 1967.[GH] D. Gepner, R. Haugseng, Enriched ∞ -categories via non-symmetric operads, Adv.Math., 279(2015), 575–716.[GHN] D. Gepner, R. Haugseng, T. Nikolaus, Lax colimits and free fibrations in ∞ -categories. Doc. Math. 22 (2017), 12251266.[GR] D. Gaitsgory, N. Rozenblyum, A study in derived algebraic geometry, Vol. I.[Hau1] R. Haugseng, Rectification of enriched ∞ -categories. Algebr. Geom. Topol. 15(2015), no. 4, 19311982.[Hau2] R. Haugseng, ∞ -operads via Day convolution, preprint arXiv:1708.09632. [H.loc] V. Hinich, Dwyer-Kan localization revisited, preprint arXiv:1311.4128, Homology,homotopy, applications, 18(2016), 27–48.[H.L] V. Hinich, Lectures on ∞ ∞ , n -categories, Geom. Topol. 14 (2010), no.1, 521-571.[S] D. Stevenson, Model structure for correspondences and bifibrations, preprintarXiv:1807.08226. Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838,Israel
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