The Lifting Theorem for Multitensors

The Lifting Theorem for Multitensors Mic hael Batanin, Denis-Charles Cisinski, and Mark W eb er Abstract. W e co ntin ue to dev elop the theory of [ 2 ] and [ 16 ] on monads and multite nsors. The cen tral result of this paper – the lif ting theorem for multite nsors – enables us to see the Gray tensor pro duct of 2-categories and the Crans tensor product of Gra y categories as part of our emerging framework. Moreov er we explain ho w our lifting theorem gives an alte rnative description of Da y con v olution [ 4 ] in the unenrich ed cont ext. Contents 1. Multitensors and functor op erads 1 2. The lifting theorem 3 3. Multitensor lifting made explicit 7 4. Gray and Crans tensor pro ducts 13 5. Contractibilit y 14 6. Standard conv olution 16 Appendix A. T ransﬁnite constructions in monad theory 18 Ac knowledgements 25 References 25 1. Multitenso rs and functor op erads This pap er contin ues the developments of [ 2 ] and [ 16 ] o n the in terplay b et ween monads a nd multitensors in the globular a pproach to higher categor y theory . T o take an impo rtan t example, a ccording to [ 16 ] there a re tw o related combinatorial ob jects which can b e used to describ e the notion of Gr a y categ ory . One has the monad A o n the categor y G 3 ( Set ) of 3-globular sets whose algebra s are Gray cat- egories, which was ﬁr st describ ed in [ 1 ]. On the other hand there is a multit enso r (ie a lax monoidal s tr ucture) on the category G 2 ( Set ) of 2 -globular sets, s uc h that categorie s enr ic hed in E a re exa ctly Gray catego ries. The theor y descr ibed in [ 16 ] explains how A and E ar e related as part of a genera l theo r y which applies to a ll op erads of the so rt deﬁned origina lly in [ 1 ]. How ever there is a third ob ject which is missing from this picture, namely , the Gray tensor pro duct o f 2 -categories. It is a simpler ob ject than A and E , and ca tegories enriched in 2-Cat for the Gray tenso r pro duct a re exactly Gray categorie s. The purp ose of this pa per is to exhibit the Gray tensor pro duct as 1 2 MICHAEL BA T ANIN, DENIS-CHARLES CISINSKI, AND MARK WEBER part of our e mer ging framework. This is done by means of the lifting theo r em for m ultitensor s – theorem(7) of this article. Recall [ 2, 16 ] that a multitensor ( E , u, σ ) on a ca tegory V consists of n-ary tensor pr oduct functors E n : V n → V , whose v a lues on o b jects are denoted in any of the following wa ys E ( X 1 , ..., X n ) E n ( X 1 , ..., X n ) E 1 ≤ i ≤ n X i E i X i depe nding on what is mo st conv enient, together with unit and substitution ma ps u X : Z → E 1 X σ X ij : E i E j X ij → E ij X ij for all X , X ij from V which are natural in their arguments and satisfy the obvious unit a nd asso ciativity axioms. It is also useful to think of ( E , u, σ ) mo re abstractly as a lax algebr a structure on V for the monoid monad M on CA T , and so to denote E as a functor E : M V → V . The basic exa mple to keep in mind is that of a monoidal structure o n V , for in this case E is given b y the n - a ry tensor pro ducts, u is the ident ity and the comp onen ts of σ ar e g iv en by coherence isomor phisms fo r the monoidal str ucture. A c ate gory enriche d in E consists of a V -enriched gra ph X together with com- po sition maps κ x i : E i X ( x i − 1 , x i ) → X ( x 0 , x n ) for all n ∈ N and sequences ( x 0 , ..., x n ) of ob jects of X , s atisfying the evident unit and asso ciativity axioms. W ith the evident no tio n of E -functor (see [ 2 ]), one has a category E -Cat o f E - categories and E -functors toge ther with a forgetful functor U E : E - Cat → G V . When E is a distributive multitensor, that is when E n commutes with copro ducts in ea ch v aria ble, one can construct a monad Γ E on G V ov er Se t . The ob ject map of the under lying endofunctor is g iv en by the formula Γ E X ( a, b ) = a a = x 0 ,...,x n = b E i X ( x i − 1 , x i ) , the unit u is use d to pr o vide the unit of the monad and σ is used to provide the m ultiplicatio n. The identiﬁcation o f the alg ebras of Γ E and catego ries enriched in E is witnessed by a cano nical is omorphism E -Cat ∼ = G ( V ) Γ E ov er G V . This con- struction, the senses in whic h it is 2-functorial, and its resp ect of v arious categorica l prop erties, is explained fully in [ 1 6 ]. W e use the no ta tion a nd ter minology o f [ 16 ] freely . If one r e stricts attention to unary op erations , then E 1 , u a nd the comp onents σ X : E 2 1 X → E 1 X provide the underlying endofunctor, unit, and multiplication for a monad on V . This monad is ca lled the u nary p art of E . W hen the unar y part of E is the identit y monad, the multitensor is a functor op er ad . This coincides with existing termino lo gy , see [ 14 ] for instance, except that we do n’t in this pap er consider any symmetric group actions. Since units for functor o pera ds are iden tities, we denote any such as a pair ( E , σ ), where a s for genera l multit enso rs E denotes the functor part a nd σ the substitution. By deﬁnition then, a functor o pera d is a multitensor. On the other ha nd, a s observed in [ 2 ] lemma(2.7), the una ry part of a multitensor E a c ts o n E , in the THE LIFTING THEOREM FOR MUL TITENSORS 3 sense that as a functor E fa c tors as M V V E 1 V / / U E 1 / / and in addition, the substitution maps are morphisms of E 1 -algebra s. More over a n E -ca tegory structure on a V -enriched graph X includes in particular an E 1 -algebra structure o n each hom X ( a, b ) of X with r espect to w hich the compo sition maps are mo r phisms of E 1 -algebra s. The s e obser v ations lead to Question 1 . Given a m ultitensor ( E , u , σ ) o n a category V can one ﬁnd a functor op erad ( E ′ , σ ′ ) on V E 1 such that E ′ -categor ies a re exa ctly E -catego ries? The ma in result of this pap er, theorem(7), says that question(1) has a nice answer: when E is dis tributiv e and a ccessible and V is co complete, one can indeed ﬁnd a unique distr ibutiv e access ible such E ′ . Mo reov er as we will see in se c tion(6), this constructio n generalis e s Day conv olution [ 4 ] and s ome of its lax ana logues [ 6 ]. Perhaps the ﬁr st app earance o f a ca se of o ur lifting theorem in the literatur e, that do es not inv olve co n volution, is in the work of Ginzburg and Kapranov on Koszul duality [ 8 ]. F ormula (1 .2 .13) of that pap er, in the case of a K -collectio n E coming from an op erad, implicitly inv olves the lifting o f the multitensor corre- sp onding (as in [ 2 ] example(2.6 )) to the given o pera d. F or instance, our lifting theorem g iv es a sa tisfying gener al explanation for w hy one must tensor ov er K in that for m ula. This pap er is o rganised in the following wa y . The lifting theorem is proved in section(2), using s ome tr ansﬁnite c onstructions fr om mo nad theory which ar e recalled in appendix (A ). The lifted functor op erad is unpack ed explicitly in sec - tion(3). In section(4) we explain how the Gr a y tenso r pro duct of 2-c ategories a nd Crans tensor pro duct of Gray categorie s is o btained as a lifting via theo rem(7). Part o f the in terplay b et ween monads and m ultitensor s describ ed in [ 16 ] cov ers contractible mult itenso rs a nd their relation to the contractible op erads of [ 1 ]. In section(5) w e extend this analysis to the lifted mu ltitenso r s, and in e xample(25) explain ho w this g ives a diﬀerent pro of of the co n tractibility of the op erad for Gray categorie s. In section(6) we explain how Da y c on volution, in the unenriched setting, can also b e obtained via our theor em(7). 2. The l ifting theorem The idea whic h enables us to answer ques tion(1) is the following. Given a distributive m ultitensor E on V one can consider also the multitensor f E 1 whose unary part is also E 1 , but whose no n-unary par ts are all cons ta n t at ∅ . This is clearly a sub-multit enso r of E , also dis tributiv e, and mor eo ver as we shall see one has f E 1 -Cat ∼ = G ( V E 1 ) ov er G V . Thu s from the inclusion f E 1 ֒ → E one induces the forgetful functor U ﬁtting in the co mm utative triangle G ( V E 1 ) E -Ca t G V o o U U E            G ( U E 1 ) ? ? ? ? ? ? F or suﬃcien tly nice V and E this forgetful functor has a left adjoint . The category of algebras of the induced mona d T will b e E -Cat s ince U is monadic. Th us pr oblem is reduced to that of establis hing that this mona d T aris e s fro m a multitensor on 4 MICHAEL BA T ANIN, DENIS-CHARLES CISINSKI, AND MARK WEBER V E 1 . Theorem(42) of [ 16 ] gives the prop erties that T m ust satisfy in or de r that there is such a m ultitensor , and gives an explicit formula for it in terms of T . In the interpla y be tw een m ultitensor s and monads descr ibed in [ 16 ] the con- struction E 7→ Γ E of a monad on G V over Set from a distributive multitensor provides the ob ject map o f 2-functors Γ : DISTMUL T → MND ( CA T / Set ) Γ ′ : OpDISTMUL T → O pMND( CA T / Set ) . That the monads ( S, η , µ ) on G V that arise from this construction a re “over Set ” means that fo r a ll X ∈ G V , the V -graph S X has the same ob ject s et a s X , and the c o mponents of the unit η and multip lica tion µ ar e identities on ob jects. The- orem(42) of [ 16 ] alluded to above characterises the monads on G V ov er Set o f the form Γ E as those which are distributive and p ath-like in the sense o f deﬁni- tions(41) and (38) of [ 16 ] res pectively . Note that the prop erties of distr ibutivity and path-likeness concern only the functor part of a g iven monad on G V over Set . On the way to the pro of of theorem(7) b elow, it is necessar y to hav e av aila ble these deﬁnitions fo r functor s over Set betw een ca tegories of enriched gr aphs. Suppo se that catego r ies V and W hav e co products . Recall that a ﬁnite sequence ( Z 1 , ..., Z n ) o f ob jects of V may be regar ded a s a V -g raph whose ob ject set is { 0 , ..., n } , hom from ( i − 1) to i is Z i for 1 ≤ i ≤ n , and other homs a re initial. Then a functor T : G V → G W ov er Set determines a functor T : M V → W whos e ob ject map is given by T ( Z 1 , ..., Z n ) = T ( Z 1 , ..., Z n )(0 , n ) . By deﬁnition T amoun ts to functor s T n : V n → W for each n ∈ N , a nd one may consider the v arious catego rical prop erties that such T may enjoy , as in the discussion o f [ 16 ] section(4 .3). Deﬁnition 2. Let V a nd W b e categ ories with co pro ducts. A functor T : G V → G W ov er Set is distributive w he n for e ac h n ∈ N , T n preserves copro ducts in each v ariable. Given a V -gra ph X and sequence x = ( x 0 , ..., x n ) of o b jects of X , one ca n deﬁne the morphism x : ( X ( x 0 , x 1 ) , X ( x 1 , x 2 ) , ..., X ( x n − 1 , x n )) → X whose ob ject map is i 7→ x i , and whos e hom map betw een ( i − 1) and i is the ident ity . F or all such sequences x one has T ( x ) 0 ,n : T i X ( x i − 1 , x i ) → T X ( x 0 , x n ) and so taking all seq uences x s tarting at a and ﬁnishing a t b one induces the canonical map π T ,X ,a,b : a a = x 0 ,...,x n = b T i X ( x i − 1 , x i ) → T X ( a, b ) in W . Deﬁnition 3. Let V a nd W b e categ ories with co pro ducts. A functor T : G V → G W ov er Se t is p ath-like when for all X ∈ G V and a, b ∈ X 0 , the ma ps π T ,X ,a,b are isomorphisms. THE LIFTING THEOREM FOR MUL TITENSORS 5 Clearly a monad ( T , η , µ ) on G V o ver Set is dis tributiv e (re s p. path-like) in the sense of [ 16 ] iﬀ the underly ing endo functor T is so in the sense just deﬁned. Lemma 4. L et V , W and Y b e c ate gories with c opr o ducts and R : V → W , T : G V → G W and S : G W → G Y b e fun ct ors. (1) If R pr eserves c opr o ducts t hen G R is distributive and p ath-like. (2) If S and T ar e distributive and p ath-like, then so is S T . Proof. (1): Since R preser v es the initial ob ject one has G R ( Z 1 , ..., Z n ) = ( RZ 1 , ..., RZ n ) and so G R : M V → W s e nds sequences of le ng th n 6 = 1 to ∅ , and its unary part is just R . Thus G R is distributive since R pr eserves copro ducts, and copro ducts of copies of ∅ are initial. The s umma nds o f the domain of π G R,X,a,b are initial unless ( x 0 , ..., x n ) is the sequence ( a, b ), thus π G R,X,a,b is clea rly an iso mor- phism, and so G R is pa th-lik e. (2): Since S a nd T are path-like and distributive one has S T ( Z 1 , ..., Z n )(0 , n ) ∼ = a 0= r 0 ≤ ... ≤ r m = n S 1 ≤ i ≤ m T r i − 1 j E (1) i

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