Algebras of higher operads as enriched categories

Algebras of higher operads as enric hed categories Mic hael Batanin and Mark W eb er Abstract. W e decribe the corresp onden ce b et ween normalised ω -op erads in the sense of [ 1 ] and certain lax monoidal str uctures on the category of globular sets. As with ordinary monoidal categories, one has a notion of category en- riched in a lax monoidal category . Within the aforement ioned corresp ondence, we prov ide also an equiv alence betw een the algebras of a given normalised ω - operad, and categories enri c hed in globular sets for the i nduce d lax monoidal structure. This is an imp ortan t step in reconciling the globular and si mplicial approac hes to higher category theory , because in the sim pl icial approac hes one proceeds inductively following the idea that a weak ( n + 1 )-category is some- thing like a category enri c hed in wea k n - categ ories, and in this pap er w e b egin to r eveal how such an intuition ma y b e formulate d i n terms of the machine ry of globular op erads. 1. In tro duction The sub ject of enr ic hed categor y theor y [ 7 ] w as brought to matur it y b y the eﬀorts of Max Kelly and his colla bor ators. Max also had a ha nd in the genes is of the study of op erads, and in [ 8 ] whic h for a long time w ent unpublished, he lay ed the categorical basis for their further analysis. It is with grea t pleasure that w e are able to pres en t the following paper , which r elates enriched category theo ry and the study of higher op erads, in dedica tion to a great mathematician. In the combinatorial approach to deﬁning and working with higher catego rical structures, one uses g lobular ope r ads to s a y what the str uc tur es of interest are in one go. Ho w ever in the simplicia l appro ac hes to higher catego ry theory , one pro ceeds inductively following the idea that a weak ( n + 1 )-category is so mething like a category enriched in weak n -ca tegories. This is the ﬁrst in a series of pap ers whose purp ose is to reveal and study the inductive asp ects hidden within the globular op eradic appro ac h. An ω -op erad in the sense of [ 1 ] ca n b e succinctly describ ed as a ca r tesian monad mor phism α : A →T , where T is the monad on the catego ry b G of globular sets whos e algebra s ar e strict ω - categories. The algebras of the given o pera d a re just the alg ebras of the monad A . Among the ω - oper ads, one can distinguish the normalise d ones, which don’t provide any structure at the ob ject le v el, so that one may regar d a glo bular set X and the g lobular set AX as having the same ob jects. F or exa mple, the op erad constructed in [ 1 ] to de ﬁne weak- ω -categ ories, and indeed any ω - oper a d th at has been constructed to give a deﬁnition of weak- ω -category , 1 2 MICHAEL BA T ANIN AND MARK WEBER is normalised. One of t he ma in r esults of this pa per, corollar y(7.9 ), pro vides t wo alternative views o f nor malised op erads: a s M T -op erads and as T -multitensors. The notion of T - oper a d, and mor e generally of T -multicategory , makes sens e for any cartesia n monad T on a ﬁnitely complete categ o ry V (s e e [ 10 ]). A T -o pera d can be deﬁned as a car tes ian monad mo r phism into T , in the same wa y as w e hav e already outlined in the ca se T = T ab ov e. Under c e rtain co nditio ns on V and T , one has a monad M o n V which is also cartesian and whose algebr as are monoids in V , this mona d distributes with T , a nd the co mp osite monad M T is also cartesian, so one can consider M T - oper a ds. All of this is so in the case T = T . On the other hand a mult itensor structure on a categor y V is just another name for the structure of a lax m onoida l c a tegory o n V . This gener al notio n ha s bee n discussed both in [ 6 ] within the framework of lax mo no ids, and in [ 2 ] wher e it is expressed in the la nguage of in ternal op erads. A multitensor is like a monoidal structure, except that the co herences a re not necessarily inv ertible, and one works in a n “unbiased” setting deﬁning an n -ary tensor product for all n ∈ N . J us t as with monoidal categories one can consider categor ies enr ic hed in a lax monoidal category . In particular if V has ca rtesian pr o ducts and T is a monad on V , one can deﬁne a cano nical multitensor T × on V , with the prop erty that ca tegories enriched in ( V , T × ) a re exactly catego ries enriched in T -Alg for the cartesian tensor pr oduct. When V is lextensive a nd T is a p.r.a monad in the sens e o f [ 13 ], one can deﬁne a T -multitensor in an analogo us way to the deﬁnition of T - oper ad: as a cartes ian m ultitensor morphism into T × . These a ssumptions on V and T a re a little s tronger than asking that T b e a ca rtesian monad, a nd are clearly sa tisﬁed for all exa mples of interest for us such as T = T . The cor respo ndence b et ween norma lised T -op erads and T - m ultitensors alr eady discussed a lso includes an impo rtant feature at the level o f alge br as. N amely the al- gebras of a given norma lised T -op erad α : A →T cor resp ond to categories enriched in the asso ciated T -multitensor. In this wa y , a n y higher categorica l str ucture de- ﬁnable b y a normalised T -op erad is expressed as a category enric hed in b G for a canonically deﬁned la x monoidal structure on b G . This pap er is or ganised as follows. In section(2) we r ecall the deﬁnition of a lax monoidal categor y a nd of categorie s enr ic hed therein, a nd give the example of T × . Multitensors, tha t is lax monoidal structures, generalis e non-symmetric op erads , and sectio ns (3) and (4) explain how ba sic o p erad theory gener alises to mult itensor s. In sectio n(3 ) we see how under certain conditions, one may reg ard multitensors as monoids for a certain monoidal s tructure, whic h generalises the subs titution tensor pro duct of collections familiar from the theory of op erads. Pr opo sition(3.3) is in fact a s pecial case o f pr opo sition(2.1) of [ 6 ]. Nevertheless we g iv e a s elf-con tained account of propo sition(3.3) and related notions, to k eep the ex p osition relatively self-contained and as elementary a s p ossible for our purp oses. In s e ction(4) w e explain how o ne can induc! e a monad from a multitensor. The theory of T -multitensors, which is the m ultitensoria l analogue of the theor y of T -o pera ds described in [ 1 0 ], is given in section(5), and it is at this level of g enerality that one sees the eq uiv alence betw een T -multitensors and M T -op erads. F rom this p oin t in the pap er we b egin working directly with the case T = T . In se c tion(6) we give a se lf-con tained inductive description o f the monad T . This is a very b eautiful mathema tical ob ject. It is a p.r.a monad and its funct or part ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 3 preserves coproducts . It has a nother crucial pr o perty , called tightness , which imp lies that for an y endofunctor A , if a cartes ian transfor mation α : A →T exists then it is unique. This prop ert y is very useful, for instance when building up a des cription of T one need not chec k the monad axioms b ecause these co me for free once one has given cartes ia n trans formations η : 1 →T a nd µ : T 2 →T . The inductive descr iption of T given here is c lo sely related to the wreath pr oduct of Clemens B erger [ 4 ]. In s ection(7) we g iv e the corres p ondence b etw een normalis e d T -op erads a nd T - m ultitensors , as well as the ident iﬁcation b etw een the algebr as of a g iv en normalis ed op erad a nd c a tegories enriched in the associa ted multitensor. In the ﬁna l s ection we explain how our results may b e a dapted to nor malised n -op erads, that is to ﬁnite dimensio ns, and then explain how the alg ebras of T , which we deﬁned as a combinatorial ob ject, really ar e strict ω -c ategories. This last fact is of co urse well-kno wn, but the simplicit y and canonicity of our pro of is a pleasant illustration of the theory developed in this pap er. The work discuss ed h ere is in a sense purely formal. Everything works at a high level of g enerality . Things b ecome mo r e interesting and subtle when we wish to lift the lax monoida l structures we obtain on b G , or one of its ﬁnite dimensio na l analogues , to the c ategory of algebr as o f a nother o pera d. F or example alr e ady in this pap e r, one can see that the lax monoidal structure T × on b G co r resp onds to cartesia n pro duct o f T -alg e bras, in the s e ns e that they giv e the same enriched categorie s. It is fr o m the general theory of s uch lifted lax monoida l str uctures tha t the Gray tensor pro duct and its many v ariants, and many other examples, will b e captured within o ur framework. These issues will be the sub ject of [ 3 ]. 2. Lax Monoi dal Categories In this section w e r ecall the notion of lax monoida l c ate gory , which is a gen- eralisatio n of the well-known concept of monoida l categ ory . As with monoidal categorie s, one ca n cons ide r categories enriched in a lax monoidal catego ry . Any monad T on a ﬁnitely complete c a tegory V deﬁnes a canonical la x monoidal struc- ture T × on V , and for this structure enr ic hed categories cor resp o nd to categ ories enriched in T -Alg regar ded a s monoidal via cartesian pro duct. Given a 2 - monad T on a 2-categor y K one may co nsider lax algebr as for T . A lax T -algebra structure on an o b ject A ∈ K is a triple ( a, u, σ ) consis ting of an action a : T A → A together with 2-cells A η A / / 1 A   @ @ @ @ @ @ @ T A a } } | | | | | | | | A u + 3 T 2 A µ A / / T a   T A a   T A a / / A σ + 3 satisfying so me well-known a xioms. See [ 9 ] for a complete descr iption of these axioms, a nd of the 2-categ ory Lax- T - Alg . When T is the identit y , la x algebras ar e just monads in K . The exa mple mo st imp ortant for us howev er is when T is the monoid mo nad M on CA T. Deﬁnition 2. 1. A mult itensor on a ca tegory V is a lax M -alg ebra structure ( E , u, σ ) on V . A category V equipp ed with a multitensor structure is called a lax monoidal c ate gory . When u is the iden tity the m ultitensor and lax monoidal structure are sa id to b e n ormal . 4 MICHAEL BA T ANIN AND MARK WEBER W e shall now unpack this deﬁnition. Since MV = ` n ≥ 0 V n a functor E : MV →V amounts to functor s E n : V n →V for n ∈ N . Before pro ceeding further we digres s a little o n notatio n. F or functor s of man y v ar iables we shall use some space saving notatio n: we deem that the following expressions E n ( X 1 , ..., X n ) E 1 ≤ i ≤ n X i E i X i are synon ymous, and we will frequent ly use the latter, often leaving t he “ n ” un- men tioned when no confusion would res ult. In par ticular for X ∈V and 1 ≤ i ≤ n , E i X denotes E n ( X, ..., X ). W e iden tify the num ber n with the or dered set { 1 , ..., n } and we re fer to elements of the o rdinal sum n • := n 1 + ... + n k as pa ir s ( i, j ) where 1 ≤ i ≤ k and 1 ≤ j ≤ n i . F ollowing these conv en tions E i E j X ij and E ij X ij are syno n ymous with E k ( E n 1 ( X 11 , ..., X 1 n 1 ) , ..., E n k ( X k 1 , ..., X kn k )) and E n • ( X 11 , ..., X 1 n 1 , ......, X k 1 , ..., X kn k ) resp ectively . W e will use m ultiply indexed expressions (like E i E j k E l X ij kl ) to more eﬃciently conv ey ex pressions that hav e m ultiple layers o f brack ets and a pplications of E ’s. The remaining data for a multitensor on V amounts to ma ps u X : X → E 1 X σ X ij : E i E j X ij → E ij X ij that ar e natural in the arguments and satis fy E i X i u E i / / 1   E 1 E i X i σ          E i X i = E i E j E k X ij k σ E k / / E i σ   E ij E k X ij k σ   E i E j k X ij k σ / / E ij k X ij k = E i E 1 X i σ   = = = = = = = E i X i 1   E i u o o E i X i = Thu s a m ultitensor is very m uch like a functor-op erad in the sense o f [ 11 ], except that there a r e no sy mmetr ic group actions with r e spect to which the subs titutions are equiv ariant 1 . An eq uiv alent formulation of deﬁnition(2.1), in the language o f [ 2 ], is that a multitensor on V is a non- symmetric o pera d internal to the endomorphism op erad of V . Example 2.2 . A norma l multitensor o n V such that σ is inv ertible is just a monoidal structure on V , with E n playing t he role of the n -fold tensor pro duct. In the ca s e where V is ﬁnitely co mplete and E n is n - fold cartesian pro duct and for the sake o f the next example, we denote the isomo rphism “ σ ” as ι : Q i Q j X ij → Q ij X ij 1 More precisely , functor-operads in the sense of [ 11 ] are norm al lax algebras for the symmetric monoidal category 2-m onad on CA T. ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 5 Example 2.3. Let T b e a mo nad on a ﬁnitely complete ca tegory V . Denote by k X i : T Q i X i → Q i T X i the cano nical maps which measure the ex ten t to which T preserves pro ducts. One deﬁnes a multitensor ( T × , u, σ ) as follows: T × k ( X 1 , ..., X k ) = Q 1 ≤ i ≤ n T ( X i ) u is the unit η X : X → T X of the monad, and σ is deﬁned as the comp osite Q i T Q j T X ij Q i kT / / Q i Q j T 2 X ij ιµ / / Q ij T X ij F or the remainder of this section let ( V , E ) b e a lax monoidal categor y . Deﬁnition 2.4. An E -c ate gory ( X , κ ), o r in other words a c ategory enriched in ( V , E ), consis ts of • a set X 0 of ob jects. • for all pair s ( x 0 , x 1 ) of elements of X 0 , an ob ject X ( x 0 , x 1 ) of V . These ob jects are called the homs of X . • for all n ∈ N and ( n +1)-tuples ( x 0 , ..., x n ) of elements o f X 0 , maps κ x i : E 1 ≤ i ≤ n X ( x i − 1 , x i ) → X ( x 0 , x n ) called the c omp ositions of X . satisfying unit a nd asso ciative laws, which say that X ( x 0 , x 1 ) E 1 X ( x 0 , x 1 ) u / / X ( x 0 , x 1 ) κ   id % % L L L L L L L L L L L E i E j X ( x ( ij ) − 1 , x ij ) E ij X ( x ( ij ) − 1 , x ij ) σ / / X ( x 0 , x mn m ) κ   E i X ( x ( i 1) − 1 , x in i ) E i κ   κ / / commute, where 1 ≤ i ≤ m , 1 ≤ j ≤ n i and x (11) − 1 = x 0 . Since a choice of i and j ref- erences an element of the ordinal n • , the pr edecessor ( ij ) − 1 o f the pair ( ij ) is well-deﬁned when i and j ar e no t b oth 1. An E -monoid is an E -categor y with one ob ject. Deﬁnition 2. 5 . Let ( X , κ ) and ( Y , λ ) b e E -catego ries. An E -functor f : ( X , κ ) → ( Y , λ ) consists of a function f 0 : X 0 → Y 0 , and for all pairs ( x 0 , x 1 ) from X 0 , arrows f x 0 ,x 1 : X ( x 0 , x 1 ) → Y ( f x 0 , f x 1 ) satisfying a functor ialit y axiom, which says that E i X ( x i − 1 , x i ) E i f / / κ   E i Y ( f x i − 1 , f x i ) λ   X ( x 0 , x n ) f / / Y ( f x 0 , f x n ) commutes. W e deno te by E -Cat the categor y o f E -catego ries and E -functors, a nd by Mon( E ) the full subca tegory of E -Cat consisting of the E -monoids. 6 MICHAEL BA T ANIN AND MARK WEBER Example 2.6. A no n-symmetric op erad ( A n : n ∈ N ) u : I → A 1 σ : A k ⊗ A n 1 ⊗ ... ⊗ A n k → A n • in a bra ided monoidal category V deﬁnes a multit ensor E on V via the formula E 1 ≤ i ≤ n X i = A n ⊗ X 1 ⊗ ... ⊗ X n with u and σ providing the str ucture maps in the o b vious wa y . The categ o ry Mon( E ) of E -monoids is the usua l c a tegory of a lgebras of A , and th us E -categ ories are a natural notion of “many ob ject algebr a” for a n o pera d A . Our notation for multitensors makes evident the a nalogy w ith monads and algebra s: a mult itensor E is analog ous to a mona d and an E -ca tegory is the analog ue of an algebra for E . In particular o bs erve that the following basic facts are instance s of the axioms for the lax monoidal category ( V , E ) and categor ie s enr ic hed ther e in. Lemma 2.7. (1) ( E 1 , u, σ ) is a monad on V . (2) The monad E 1 acts on E n for al l n ∈ N , that is σ : E 1 E i X i → E i X i is an E 1 -algebr a structur e on E i X i . (3) With r esp e ct to the E 1 -algebr a structur es of ( 2) al l of the c omp onents of σ ar e E 1 -algebr a morphisms. (4) Each hom of an E - c ate gory ( X , κ ) is an E 1 -algebr a, with the algebr a struc- tur e on X ( x 0 , x 1 ) given by κ : E 1 X ( x 0 , x 1 ) → X ( x 0 , x 1 ) . (5) With r esp e ct t o the E 1 -algebr as of (2) and (4), al l the c omp onents of κ ar e morphisms of E 1 -algebr as. Prop osition 2.8. L et T b e a monad on a ﬁnitely c omplete c ate gory V . Re gar ding T - Alg as a monoidal c ate gory via c artesian pr o duct one has T × -Cat ∼ = ( T - Alg)-Cat c ommu ting with the for getful funct ors into Set . Proof. Let X 0 be a set and for a, b ∈ X 0 let X ( a, b ) ∈ V . Suppo se tha t κ x i : Q i T X ( x i − 1 , x i ) → X ( x 0 , x n ) for each n ∈ N a nd x 0 , ..., x n in X 0 , are the str uc tur e maps fo r a T × -categor y structure. Then by lemma(2 .7) the κ a,b : T X ( a, b ) → X ( a, b ) are algebra structures for the homs , and for x ij ∈ X 0 with 1 ≤ i ≤ k a nd 1 ≤ j ≤ n i one ha s the inner regio ns ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 7 of T Q i X ( x i − 1 , x i ) Q i T X ( x i − 1 , x i ) Q i X ( x i − 1 , x i ) T Q i T X ( x i − 1 , x i ) T X ( x 0 , x n ) X ( x 0 , x n ) Q i T X ( x i − 1 , x i ) Q i T 2 X ( x i − 1 , x i ) Q i T X ( x i − 1 , x i ) T Q i η / / T κ / / κ   k   Q i κ   Q i η / / κ / / k          Q i T κ   + + + + + + + Q i ηT / / Q i µ , , X X X X X κ ' ' P P P P P P commutativ e, and the commutativ e outer regio n is the a sso ciativit y axiom for the comp osites κ ′ x i : Q i X ( x i − 1 , x i ) Q i η / / Q i T X ( x i − 1 , x i ) κ x i / / X ( x 0 , x n ) for each x 0 , ..., x n ∈ X 0 . T aking the pro duct structure on T - Alg as norma l, the unit ax io m for the κ ′ is clear ly satisﬁed, a nd so they ar e the structure maps for a ( T - Alg)-categor y structure. Conv ersely given algebr a structure s κ a,b and structure maps κ ′ x i one can deﬁne κ x i as the co mposite Q i T X ( x i − 1 , x i ) Q i κ x i − 1 ,x i / / Q i X ( x i − 1 , x i ) κ ′ x i / / X ( x 0 , x n ) and since the regio ns of Q i T Q j T X ( x ( ij ) − 1 , x ij ) Q ij T 2 X ( x ( ij ) − 1 , x ij ) Q ij T X ( x ( ij ) − 1 , x ij ) Q ij X ( x ( ij ) − 1 , x ij ) X ( x 0 , x n ) Q i X ( x i − 1 , x i ) Q i T X ( x i − 1 , x i ) Q i T Q j X ( x ( ij ) − 1 , x ij ) Q ij T 2 ( x ( ij ) − 1 , x ij ) k / / Q ij µ / / Q ij κ   κ ′   Q i T Q j κ   Q i T κ ′   Q i κ / / κ ′ / / k / / Q ij κ / / Q i κ ′ u u l l l l l l l l l l l l l Q ij T κ   commute, the commutativit y of the outside of this diagra m shows that the κ x i satisfy the asso ciativity condition o f a T × -categor y s tr ucture, and the unit axio m follows from the unit T - algebra axiom on the homs. The co r resp ondence just de- scrib ed is clear ly a bijection, and co mpletes the description of the isomorphism on ob jects ov er Set. Let f 0 : X 0 → Y 0 be a function, κ x i : Q i T X ( x i − 1 , x i ) → X ( x 0 , x n ) λ y i : Q i T Y ( y i − 1 , y i ) → Y ( y 0 , y n ) 8 MICHAEL BA T ANIN AND MARK WEBER be t he s tructure maps for T × -categor ies X and Y , κ ′ x i and λ ′ y i be the asso ciated ( T - Alg)-categor y structures, and f a,b : X ( a, b ) → Y ( f a, f b ) for a , b ∈ X 0 be ma ps in V . In the following display the diagram o n the left Q i X ( x i − 1 , x i ) Q i f / / Q i η   Q i Y ( y i − 1 , y i ) Q i η   Q i T X ( x i − 1 , x i ) Q i T f / / κ   Q i T Y ( y i − 1 , y i ) λ   X ( x 0 , x n ) f x 0 ,x n / / Y ( y 0 , y n ) Q i T X ( x i − 1 , x i ) Q i T f / / Q i κ   Q i T Y ( y i − 1 , y i ) Q i λ   Q i X ( x i − 1 , x i ) Q i f / / κ ′   Q i Y ( y i − 1 , y i ) λ ′   X ( x 0 , x n ) f x 0 ,x n / / Y ( y 0 , y n ) explains how the T × -functor axio m for the f a,b implies the ( T -Alg)-functor axiom, and the diag ram on the r ig h t shows the co n verse.  3. Distributive multitensors as monoids It is w ell-known that monads on a category V are monoids in the strict mono idal category End( V ) of endofunctors o f V who s e tensor pr o duct is given by comp osition. Given the a nalogy betw een mona ds a nd mult itensor s, o ne is led to ask under what circumstances are m ultitensors monoids in a certain monoida l category . O ne natural answer to this question, that we shall pres en t now, requir es that we r estrict attention to distributive m ultitensors to be deﬁned b elow. Throughout this section V is assumed to have copr oducts. Deﬁnition 3.1. A functor E : MV →V is distributive when for all n ∈ N , E n preserves c o pro ducts in each v ariable. W e denote by Dist( V ) the category whose ob jects ar e such functors MV →V , and whose morphisms a re natur al transfor ma - tions b et ween them. A multitensor ( E , u, σ ) (resp. lax mo noidal catego ry ( V , E )) is sa id to b e distributive when E is distr ibutive. Examples 3.2. In the cas e where ( V , ⊗ , I ) is a gen uine mono ida l categor y , V is distributive in the ab ov e sense iﬀ ( X ⊗− ) and ( −⊗ X ) preserve copro ducts for each X ∈ V . If in addition ⊗ is just cartes ian pro duct a nd T is a monad on V whose functor part pr eserves copro ducts, then the multitensor T × of example(2.3) is also distributive. When E is distributive we have E 1 ≤ i ≤ n a j ∈ J i X ij ∼ = a j 1 ∈ J 1 ... a j n ∈ J n E i X ij i for a n y doubly indexed family X ij of ob jects o f V . T o characterise distributivity via this for m ula w e must be more precise a nd say that a certain cano nical map betw een these ob jects is a n isomorphism. It is howev er mo re c on venien t to express all this in terms of copro duct co cones. T o state such an equation we m ust hav e for each 1 ≤ i ≤ n a family of maps ( c ij : X ij → X i • : j ∈ J i ) ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 9 which fo rms a copro duct co cone in V . Given a choice for each i of j ∈ J i , one obtains a map E i c ij : E i X ij → E i X i • , and distributivity s a ys that a ll such maps together for m a copro duct co cone. The morphisms that comprise this co cone are indexed b y elements of Q i J i in ag reemen t with the right hand s ide of the ab ov e for m ula. F or wha t will so on fo llo w it is worth r ecalling that the (obviously true) statement “a copr o duct of copro ducts is a copro duct” can b e describ ed in a similar wa y . That is, given c ij as above together with another copr oduct co cone ( c i : X i • → X •• : 1 ≤ i ≤ n ) , for e ac h choice of i and j one obtains a comp osite a rrow X ij c ij / / X i • c i / / X •• , and the collec tion of all such compo sites is a copro duct co cone. Deﬁne the unit I o f Dist( V ) by I 1 =1 V and fo r n 6 =1 , I n is constant at ∅ . The tensor pro duct E ◦ F of E a nd F in Dist ( V ) is deﬁned as : ( E ◦ F ) n = a k ≥ 0 a n 1 + ... + n k = n E i F n i and so for all k and n i ∈ N where 1 ≤ i ≤ k we hav e maps E i F j c ij / / E ◦ F ij which w e shall also deno te by c ( n 1 ,...,n k ) as conv enience dictates. F or all n ∈ N the set o f a ll such maps suc h that n • = n form a copro duct co cone. In the case where E = I one has I i F j ∼ = ∅ when k 6 =1, and so c ( n ) : F n → ( I ◦ F ) n is inv ertible, the inv erse of w hich we denote by λ . In the case where F = I one has E i I j ∼ = ∅ when not all the n i ’s a r e 1, and s o c (1 ,..., 1) : E n → ( E ◦ 1 ) n is invertible, the inv erse of which we deno te by ρ . Given E , F and G in Dist( V ), one ha s for a ll r ∈ N , m i ∈ N such that 1 ≤ i ≤ r , and n ij ∈ N for all i a nd 1 ≤ j ≤ m i , a co mposite E i F j G k c ij G k / / (E ◦ F ij ) G k c ( ij ) k / / (( E ◦ F ) ◦ G ) n •• and for all n ∈ N , the set of all such comp osites obtained from such choices with n •• = n forms a copro duct co cone (the co pro duct of copro ducts is a copro duct). F or a g iv en choice of r , m i and n ij as above one can als o form a comp osite E i F j G k E i c jk / / E i (F ◦ G j k ) c i ( jk ) / / ( E ◦ ( F ◦ G )) n •• and for all n ∈ N , the set of all such comp osites obtained from such choices with n •• = n fo rms a copro duct co cone b ecause E is distributive. Thus for each n , E , F 10 MICHAEL BA T ANIN AND MARK WEBER and G one has a unique isomorphism α , such that for all c hoices o f r , m i and n ij with n •• = n , the diag ram (1) E i F j G k c ij G k / / E i c jk # # F F F F F F F F F (E ◦ F ij ) G k c ( ij ) k / / (( E ◦ F ) ◦ G ) n α   E i (F ◦ G j k ) c i ( jk ) / / ( E ◦ ( F ◦ G )) n commutes. Prop osition 3. 3. The data ( I , ◦ , α, λ, ρ ) ju s t describ e d is a monoidal stru ctur e for Dist( V ) . The c ate gory Mon(Dist( V )) is isomorph ic to the c ate gory of distributive multitensors and morphisms t her e of. Proof. The ca se of (1) for which m i =1 amo unts to the commutativit y of the outside o f (E ◦ I i ) F k c ik / / (( E ◦ I ) ◦ F ) n α   E i F k ρ − 1 F k > > | | | | | c ik / / E i λ − 1 B B B B B E ◦ F ik ρ − 1 ◦ F u u : : u u u E ◦ λ − 1 I I I $ $ I I I E i (I ◦ F k ) c ik / / ( E ◦ ( I ◦ F )) n = = and the inner commutativities indicated here are obtained fro m the deﬁnition of the arrow ma p of “ ◦ ”. But the c ik : E i F k → E ◦ F ik for all c hoices with n •• = n for m a copr oduct co cone, a nd so the triangle in the above diagram, which is the unit cohere nce for Dist ( V ), must commute also . F or E , F , G and H in Dist( V ) we will now see that the cor respo nding a sso ciativit y pentagon commutes. F or each n and choice of r , p i for all 1 ≤ i ≤ r , m ij for all i and 1 ≤ j ≤ p i , ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 11 and n ij k for all i , j and 1 ≤ k ≤ m ij , such that n ••• = n , we get a diagram of the form: • • • • • / / / / /   / / / y y y y | | y y y y         E E E E " " E E E E • • •       • • •       • • • • 4 4   / / * * ? ? • • • o o o o o o • • • O O O O O O id / / 9 9 9 9 9 9 9 9 9 9 id   9 9 9 9 9 9 9 9 9 9 v v v v v v v v v v v v v v v v v id { { v v v v v v v v v v v v v v v v v           id             H H H H H H H H H H H H H H H H H id # # H H H H H H H H H H H H H H H H H      id        I I I I I I I I I I id $ $ I I I I I I I I I I / / 8 8 8 8 8 8 8 id   8 8 8 8 8 8 8 w w w w w w w w { { w w w w w w w w ◦ α α t α t α t ◦ α n where the inner-mos t p e ntagon what we ar e trying to prov e the co mm utativity of. The outer p entagon has all vertices equal to E i F j G k H l . The compo sites of the dotted paths of length 3, when taken ov er all choices, form co pr oduct co cones of each of the v ertices of the inner p ent ago n. F or instance for the top left vertex w e hav e E i F j G k H l (E ◦ F ij ) G k H l cGH / / ((E ◦ F) ◦ G ij k ) H l cH / / ((( E ◦ F ) ◦ G ) ◦ H ) n c / / and the tw o indicated paths inv olving the left most vertex are E i F j G k H l (E ◦ F ij ) G k H l cGH / / (E ◦ F ij )(G ◦ H kl ) E ◦ F c / / ((( E ◦ F ) ◦ G ) ◦ H ) n c / / and E i F j G k H l E i F j (G ◦ H kl ) E F c / / (E ◦ F ij )(G ◦ H kl ) cG ◦ H / / ((( E ◦ F ) ◦ G ) ◦ H ) n c / / and in a similar vein the reader will easily supply the details of the other dotted paths. The lab els of the regio ns of the diagram indicate wh y the cor r espo nding region co mm utes: “ α ” means the region co mm utes by the deﬁnitio n o f α , “n” indicates commutativit y b ecause of natur alit y , “ ◦ ” indicates c omm utativity b ecause of the deﬁnition of the arr o w map of ◦ , and “t” indicates tha t the region comm utes trivially . The outer p en tagon of course also commutes trivially . Since all this is true for a ll choices of the r , p i , m ij and n ij k , w e obtain the co mmutativit y of the inner pentagon since the top left dotted comp osites together exhibit ((( E ◦ F ) ◦ G ) ◦ H ) n as a copro duct. The statement a bout Mon(Dist( V )) follows immediately b y unpacking the deﬁnitions inv olved.  4. Monads from m ulti tensors Multitensors gene r alise non-s ymmetric op erads by example(2.6). Given cer ta in hypotheses on the amb ient bra ided monoidal categ ory V , a non- symmetric op erad therein gives rise to a monad on V whose algebras are those o f the original op erad. 12 MICHAEL BA T ANIN AND MARK WEBER Thu s one is led to as k whether one can deﬁne a monad from a m ultitensor in a similar w ay . Such a constr uc tio n is descr ib ed in the present sectio n, and we contin ue to assume throughout this s ection that V has co pro ducts. Deﬁne the functor Γ : Dist( V ) → E nd( V ) as Γ( E )( X ) = a n ≥ 0 E 1 ≤ i ≤ n X and so for each X in V w e get c n : E 1 ≤ i ≤ n X → Γ( E )( X ) for n ∈ N fo r ming a copr oduct co cone. By the deﬁnition of I the map c 1 : X → Γ( I )( X ) is an isomorphism, and we deﬁne that the inv erses o f these maps are the comp onents o f a n isomorphism γ 0 : 1 V → Γ( I ). F o r X in V and m and n i in N where 1 ≤ i ≤ m , we can consider comp o sites E i F j X E i Γ F X E i c j / / Γ( E )Γ( F ) X c m / / and since E is distr ibutiv e all such comp osites exhibit Γ( E )Γ( F ) X as a copr oduct. F or X , m and n i as above one also has comp osites E i F j X (E ◦ F ij ) X c ij / / Γ( E ◦ F ) X c n • / / and all such comp osites exhibit Γ( E ◦ F ) X as a copro duct. Thus there is a uniq ue isomorphism γ 2 making E i F j X (E ◦ F ij ) X Γ( E ◦ F ) X E i Γ( F ) X Γ( E )Γ( F ) X E i c j O O c m / / γ 2 * * T T T T T T T c ij / / c n • 4 4 j j j j j j j j commute, and γ 2 is clearly na tural in X . Prop osition 4.1. The data ( γ 0 , γ 2 ) make Γ into a monoidal fu n ctor. F or any distributive mu ltitensor E , one has an isomorphism Mon( E ) ∼ = Γ E -Alg c ommuting with the for getful functors into V . Proof. The deﬁnit ion of γ 2 in the case where E = I and the m =1 says that the outside of F j X (I ◦ F j ) X Γ( I ◦ F ) X Γ( F ) X Γ( I )Γ( F ) X c j O O γ 0 Γ F / / γ 2 * * T T T T T T T T λ − 1 / / c n 4 4 j j j j j j j j j Γ λ − 1 1 1 = commutes for all m ∈ N , and the regio n labelled with “= ” comm utes because of the deﬁnition of the arrow maps of ◦ . Thus the inner triangle, whic h is the left unit ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 13 monoidal functor coherence axiom, commu tes also. The deﬁnition o f γ 2 in the case where F = I and the n i ’s a r e all 1 s a ys that the outside of E i I j X (E ◦ I ij ) X Γ( E ◦ I ) X E i Γ( I ) X Γ( E )Γ( I ) X E i γ 0 O O c m / / γ 2 * * T T T T T T T T T T T T T ρ − 1 / / c n • 4 4 j j j j j j j j j j j j j j Γ( E ) X c m 6 6 m m m m m m m m Γ ρ − 1 a a a a a a 0 0 a a a a a a a a a a a a a a a a a Γ( E ) γ 0 ? ?         = = commutes for all m ∈ N , and the regio ns lab elled with “= ” commute b ecause of the deﬁnition of the arr o w maps of ◦ . Thus the inner triangle, which is the rig h t unit monoidal functor coher ence a xiom, commutes also . So it remains to verify that for E , F and G in Dist( V ), that (2) Γ( E )Γ( F )Γ( G ) Γ( E ◦ F )Γ( G ) Γ( E )Γ( F ◦ G ) Γ(( E ◦ F ) ◦ G ) Γ( E ◦ ( F ◦ G )) γ 2 Γ( G ) / / γ 2   : : : : : : : : Γ α w w n n n n n n n n n n n n n Γ( E ) γ 2           γ 2 ' ' P P P P P P P P P P P P P commutes. Now g iv en X in V a nd r , m i and n ij in N where 1 ≤ i ≤ r and 1 ≤ j ≤ m i , one o btains a diagra m of the for m • • • • • / / / / /   / / / y y y y | | y y y y         E E E E " " E E E E • • •       • • • / / / / / / • • • • w w       o o • • • / / / / / / • • • o o o o o o • • • O O O O O O id / / id 9 9 9 9 9 9 9 9 9 9   9 9 9 9 9 9 9 9 9 9 id v v v v v v v v v v v v v v v v v { { v v v v v v v v v v v v v v v v v id                 id         id H H H H H H H H H H H H H H H H H # # H H H H H H H H H H H H H H H H H id / / id ; ; ; ; ;   ; ; ; ; ; id           id H H H H H H H H H H H H # # H H H H H H H H H H H H w w w w w w w w { { w w w w w w w w         t γ 2 n t γ 2 α Γ t n γ 2 Γ E t γ 2 where the inner-mo st p entagon is (2) instantiated at X , and all the o uter vertices are E i F j G k X . The tw o 3-fold paths int o Γ( E )Γ( F )Γ( G )( X ) are the top-leftmost 14 MICHAEL BA T ANIN AND MARK WEBER path E i F j G k X E i F j Γ( G ) X E i F j c k / / Γ( E ) F j Γ( G ) X c i / / Γ( E )Γ( F )Γ( G )( X ) Γ( E ) c j / / and E i F j G k X E i F j Γ( G ) X E i F j c k / / E i Γ( F )Γ( G )( X ) E i c j / / Γ( E )Γ( F )Γ( G )( X ) c i / / and these are equal b ecause o f natura lit y . The compo sites so for med by ta king a ll choices of r , m i and n ij exhibit Γ( E )Γ( F )Γ( G )( X ) as a co pro duct b ecause E and F are distributive. The left-most dotted path in to Γ( E ◦ F )Γ( G )( X ) is E i F j G k X E i F j Γ( G ) X E i F j c k / / (E ◦ F ij )Γ( G ) X c ij / / Γ( E ◦ F )Γ( G )( X ) c ij / / , the other pa th int o Γ( E ◦ F )Γ( G )( X ) is E i F j G k X E ◦ F ij G k X c ij / / (E ◦ F ij )Γ( G ) X E ◦ F ij c k / / Γ( E ◦ F )Γ( G )( X ) c ij / / , and similarly the r eader will ea sily supply the deﬁnitions o f the other dotted paths in the ab ov e diag ram. The la belled reg ions of that diagram commute for the rea sons indicated b y the la b els as with the pro of of prop osition(3.3), the r e gion lab elled by “Γ” commutes by the deﬁnition o f the a r row map of Γ, and the region lab elled by “Γ E ” commutes by the deﬁnition of the a rrow ma p of Γ E . The outer diagram commutes trivia lly and s inc e this is all true for all c hoices of the r , m i and n ij , the inner pentagon co mmutes a s required. The statemen t a b out Mon( E ) f ollows immediately by unpacking the deﬁnitions inv olved.  Example 4.2. One can apply pr o positio n(4.1 ) to the ca se of example(2.6) when ( V , ⊗ , I ) is a distributive braided mo noidal categ ory , b ecause then the m ultitensor on V de ter mined by a non-sy mmetric op erad will also b e distributive. In this wa y one obtains the usua l constructio n of the monad induced by a non-symmetric op erad. Example 4.3. Applying prop osition(4.1) to the case of a distributive monoidal ca t- egory ( V , ⊗ , I ) a s in example(3.2), one recov ers the usual monoid monad M :=Γ( ⊗ ). In the case where ⊗ is cartesian pro duct and T preser v es copr o ducts, in view of Γ( T × )= M T one obtains a monad structure on M T , and th us a monad dis tributiv e law λ : T M → M T , and the algebr a s of M T are monoids in T -Alg b y prop osi- tion(2.8). In ter ms of Γ and T × one can des c ribe λ ex plicitly . The substitution for T × , describ ed in example(2.3), is a map µ × : T × ◦ T × → T × in Dist( V ), and λ is the comp osite T M ηT M η / / M T M T Γ µ × / / M T in End( V ). ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 15 5. Multitenso rs as o p e rads Given a car tesian monad T on a ﬁnitely c omplete ca tegory V one has the well- known no tion o f T - op er ad as desc ribed for example in [ 10 ]. There is an a na logous notion of T -multitensor and w e shall des c ribe this in the present section. U nder certain conditions the g iven monad T distr ibutes wit h the monoid monad M o n V a nd the c o mposite mona d M T is ag ain car tesian, in which case one has an equiv a le nc e of catego r ies b et ween T -mult itensor s and M T - o pera ds. The theory describ ed in this section r equires that T is a little mor e than cartesian, namely tha t it is p.r.a in the sense of [ 13 ], and tha t V is lextensive. Both no tio ns will b e reca lle d here for the rea ders’ conv enience. W e r ecall some asp ects of the theory parametric rig h t adjoin ts fr o m [ 13 ]. A functor T : A→B is a p ar ametric right adjoint (p.r .a) 2 when for all A ∈ A , the induced functors T A : A / A → B /T A given by applying T t o arrows have left adjoints, and when A has a termina l ob ject 1 , this is eq uiv alent to asking that T 1 has a left a djoin t. Right adjoints are clearly p.r.a a nd p.r.a functor s are closed under comp osition. Mor eo ver one has the following simple o bserv atio n whic h we shall us e often in this w ork . Lemma 5.1. Le t I b e a set and F i : A i →B i for i ∈ I b e a family of p.r.a functors. Then Q i A i Q i F i / / Q i B i is p.r.a. Proof. Given X i ∈ A i for i ∈ I , w e have ( Q i F i ) ( X i ) = Q i (( F i ) X i ), which as a pro duct of right adjoints is a r igh t a djoin t.  There is a mo r e explicit characteris ation of p.r .a functors which is sometimes useful. A map f : B → T A is T -generic whe n for any α , β , and γ making the outside of B α / / f   T X T γ   T A T β / / T δ < < T Z commute, there is a unique δ for which γ ◦ δ = β and T ( δ ) ◦ f = α . The alternative characterisation says that T is p.r.a iﬀ every map f : B → T A factor s as B g / / T C T h / / T A where g is generic, a nd suc h gener ic factorisatio ns are unique up to isomorphism if they e xist (see [ 1 3 ] for mor e details). One deﬁnes a monad ( T , η , µ ) o n a catego ry V to be p.r.a when T is p.r.a as a functor, and η and µ are c a rtesian transfor mations. One ha s the following corresp onding deﬁnition for multitensors. Deﬁnition 5.2. A m ultitensor ( E , u, σ ) on V is p.r.a when E : MV →V is p.r.a and u and σ are cartesian transforma tio ns. 2 W e reserve the right to use this abbreviation also as an adjective, as i n “ T is p arametrically r epresen t a ble”. 16 MICHAEL BA T ANIN AND MARK WEBER It is s tr aight -for w ard to observe t hat E is p.r.a iﬀ E n : V n →V is p.r .a for each n ∈ N . Example 5.3 . Let ( T , η , µ ) be a p.r.a monad on V a category with ﬁnite pro ducts. First no te that T × n is the compo site V n T n / / V n Q / / V and so is p.r.a. b y lemma(5.1) and the comp osability of p.r.a’s. F r om [ 13 ] lemma(2.14 ) the canonical maps k X i : T Q i X i → Q i T X i which mea sure the extent to which T prese rv es pro ducts ar e cartesian natura l in the X i . Thus T × is a p.r .a multitensor. F or a p.r .a monad ( T , η , µ ) on a catego ry V recall that a T -op er ad is ca rtesian monad morphism α : A → T . That is, A is a mona d on V , α is a natural transformation A → T whic h is compatible with the monad structures, and the naturality squares of α are pullbacks. The car tesianness of α and p.r.a ’ness of T implies that A is itself a p.r.a mo nad. F or instance when T = T the monad on the categ ory b G of globular sets whose algebra s ar e strict ω - categories, to b e recalled in detail in section(6), T - oper ads are the ω -op erads of Ba tanin [ 1 ]. By analogy one has the following deﬁnition for multitensors. Deﬁnition 5.4. Let ( T , η , µ ) be a p.r .a monad o n V a categor y with ﬁnite pro ducts. A T -mu ltitensor is a cartesia n m ultitensor mor phism ε : E → T × . Example 5.5. W e will now unpa c k this notion in the case where V = Set and T is the identit y monad. Beca use of the pullback squares E i X i Q i T X i ( T 1 ) n E n 1 ε X i / / Q i T t X i   E i t X i   ε 1 / / the data for ε amounts to a sequence of ob jects E n := E n 1 ∈ V fo r n ∈ N , tog ether with ma ps ε n,i : E n → T 1 for 1 ≤ i ≤ n . In this cas e T 1=1 so ε amounts to a sequence ( E n : n ∈ N ) of sets. In terms o f this data o ne has (3) E 1 ≤ i ≤ n X i = E n × Q i X i The unit of the multitensor amoun ts to an element u : 1 → E 1 , and the substitution σ amounts to functions σ n 1 ,...,n k : E k × E n 1 × ... × E n k → E n • for each ﬁnite s equence ( n 1 , ..., n k ) of natural num ber s. The multitensor axio ms for ( E , u, σ ) corresp ond to axioms that make ( E , u, σ ) a non-symmetric op erad in Set. ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 17 W e assume thro ughout this section that V is lextensive. Let us no w re call this notion. A catego r y V is lextensive 3 [ 5 ] when it has ﬁnite limits, copro ducts and for each fa mily o f ob jects ( X i : i ∈ I ) of V the functor Q i ∈ I V /X i → V /  ` i ∈ I X i  which se nds a family of maps ( h i : Z i → X i ) to their copro duct is an equiv a lence. This last prop ert y is equiv alent to saying that V has a strict initial ob ject a nd that copro ducts in V ar e disjoin t and stable. Ther e are man y examples of lextensive categorie s: for instance every Grothendieck top os is lextensive, as is CA T. Moreov er if T is a copro duct pr eserving monad on a lextensive catego ry V then T -Alg is also lextensive: for such a T the forgetful functor T -Alg →V c r eates ﬁnite limits and copro ducts, and so these exist in T - Alg and interact as nicely as they did in V . Thu s in particular the categor y of a lgebras of a n y higher oper ad is lextensive. Note in particular tha t lex tens ivit y implies distributivity (see [ 5 ]) a nd so the res ults of the previous t wo sections apply in this one. The next result summarises how lextensivity int era cts well with p.r.a’ness. Lemma 5.6. L et A and B b e lext ensive and I b e a set. (1) The functor ` : A I →A , which takes an I -indexe d family of obje cts o f A to its c opr o duct, is p.r.a. (2) If F i : A→B fo r i ∈ I ar e p.r.a functors, then ` i F i : A→B is p.r.a. (3) If F i : A→B for i ∈ I ar e functors and φ i : F i → G i ar e c artesian tr ans- formations, then ` i φ i : ` i F i → ` i G i is c artesian. Proof. (1): given a family ( X i : i ∈ I ) of o b jects of A , the functor ( ` ) ( X i ) is just the functor Q i ∈ I A /X i → A /  ` i ∈ I X i  which is a n equiv alence, and thus a right adjoint. (2): ` i F i is the comp osite A ∆ / / A I Q i F i / / B I ‘ / / B of a right adjoint (since A ha s copr oducts) followed b y a p.r .a (by lemma(5.1) follow ed by another p.r.a (b y (1), a nd so is p.r.a. (3): t he naturality sq uare for ` i φ i corres p onding to f : X → Y in A is the copro duct of the car tesian naturality squar e s F i X φ i,X / / F i f   G i X G i f   F i Y φ i,Y / / G i Y and so by (1) is itself a pullba c k.  3 Usually l exte nsivity is deﬁned using only ﬁnite copro ducts whereas we w ork with small ones. 18 MICHAEL BA T ANIN AND MARK WEBER Denote by PraDist( V ) and Pra End ( V ) the sub catego ries of Dist( V ) and End( V ) resp ectively , whose ob jects a re p.r.a’s and arrows are cartesian tr ansformations. Prop osition 5. 7. Le t V b e lextensive. The monoidal structu r e of Dist( V ) r estricts to Pra Dist ( V ) , and Γ r est ricts to a str ong monoidal functor PraDist( V ) → Pr aEnd( V ) (which we shal l also denote by Γ ). Proof. An y functor 1 →A out of the terminal catego ry is p.r.a, and thus one readily v eriﬁes that the functors V n →V consta n t a t t he initial ob ject 0 o f V a re p.r.a a lso. Since 1 V is p.r.a the unit of Dist( V ) is p.r.a . F or p.r.a E and F ∈ Dist( V ) we m ust verify that E ◦ F is p.r.a. By the formula ( E ◦ F ) n = a n 1 + ... + n k = n E k ( F n 1 , ..., F n k ) and lemma(5.6) it s uﬃces to show that each summand is p.r.a. But E k ( F n 1 , ..., F n k ) is the comp osite Q i V n i Q i F n i / / V k E k / / V which is p.r.a by lemma(5.1). Given ε : E → E ′ and φ : F → F ′ in P raDist( V ) w e m ust show that ε ◦ φ is car tesian. By lemma(5.6) it s uﬃces to show that E k ( F n 1 , ..., F n k ) ε k ( φ n 1 ,...,φ n k ) / / E ′ k ( F ′ n 1 , ..., F ′ n k ) is ca rtesian. But this natural tr ansformation is the comp osite Q i V n i V k V Q i F n i # # Q i F ′ n i ; ; E k " " E ′ k < < Q i φ n i   ε k   and so as a hor izon tal compo site of cartesia n transfor mations b et ween pullback preserving functors, is indeed c a rtesian. Thus the mono idal str ucture of Dist( V ) restricts to P raDist( V ), and to ﬁnish the pro of we m ust verify that Γ preserves p.r.a ob jects and cartesian transformations. Let E ∈ Dist( V ) be p.r.a. By lemma(5.6), to establish tha t Γ( E ) is p.r.a it suﬃces to sho w that for all n ∈ N , the functor X 7→ E n ( X, ..., X ) is p.r .a , but this is just the co mp osite V ∆ / / V n E n / / V which is p.r.a since E n is. Let φ : E → F in Dist ( V ) b e cartesian and let us see that Γ( φ ) is cartesia n. B y lemma(5.6) this comes down to the cartesia n naturality in X of the maps φ n,X,..., X : E n ( X, ..., X ) → F n ( X, ..., X ) which is a n instance of the cartesianness of φ n .  ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 19 Example 5.8. F rom exa mples(5.3) and example(3.2) T × is a p.r .a distr ibutive m ultitensor when T is a copro duct preserving p.r .a monad o n a lextensive category V . By prop osition(5.7), the monad M T describ ed in exa mple(4.3) is p.r.a and the distributive law λ : T M → M T is cartes ia n. Mo dulo one la st digress io n we are now rea dy to exhibit the equiv alence b etw een T - m ultitensors and M T -op erads a s promised at the b e g inning of this section. Recall that if W is a mono idal category and ( M , i, m ) a monoid therein, that the slice W / M gets a canonica l monoidal structur e. The unit is the unit i : I → M of the monoid, the tensor pro duct of a rrows α : A → M a nd β : B → M is the comp osite A ⊗ B α ⊗ β / / M ⊗ M m / / M and the c o herences are inherited from W so that the forgetful functor W / M →W is s trict monoida l. T o give α : A → M a monoid structure in W / M is the same as giving A a mono id str ucture for which α b ecomes a monoid homo mo rphism, and this is just the o b ject part of a n isomo rphism Mon( W / M ) ∼ = Mon( W ) / M commuting with the for getful functors into W . Moreover given a monoidal functor F : W →W ′ , F M is ca no nically a monoid and o ne has a commutativ e squa re W / M F M / /   W ′ /F M   W F / / W ′ of mono idal functors. Applying these observ ations to Γ : PraDist( V ) → Pr a End ( V ) one o bta ins for each p.r .a distributive multitensor E , a monoidal functor Γ E : PraDist( V ) /E → P raEnd( V ) / Γ E . An ob ject of P raDist( V ) /E amounts to a functor A : MV →V together with a cartesian transformation α : A → E . Giv en such data the distributivit y of A is a consequence of the cartesianness o f α , the distributivity of E and the stability of V ’s copro ducts. The p.r.a’nes s of A is a ls o a conseq ue nc e , because the domain of any cartesian transformation into a p.r.a functor is ag ain p.r.a. A morphism in PraDist( V ) /E from α to β : B → E is just a natural transfor ma tion φ : A → B suc h that β φ = α , beca use b y the elementary prop erties of pullbacks φ is automatically cartesian. T hus a monoid in Pr aDist( V ) /E is simply a cartesian mu ltitensor mor- phism into E . Similarly a monoid in PraEnd( V ) / Γ E is just a c a rtesian mona d morphism into Γ E , and so b y obs e rving its eﬀect on monoids in t he case E = T × where T is a copro duct pres! erving p.r.a monad o n V , one has a functor Γ T : T -Mult → M T -O p from the ca teg ory of T - multitensors to the categor y of M T -op erads. Theorem 5.9. L et V b e lextensive and T a c opr o duct pr eserving p .r.a monad on V . Then the fun ctor Γ T just describ e d is an e quivalenc e of c ate gories T -Mult ≃ M T -Op . 20 MICHAEL BA T ANIN AND MARK WEBER Proof. By the wa y we have set things up it suﬃces to show that for any p.r.a distributive multitensor E on V , the functor Γ E : PraDist( V ) /E → Pra End( V ) / Γ E is esse ntially surjective on o b jects and fully faithful. Let α : A → Γ( E ) b e a car tesian transformatio n. Choos ing pullbacks A i X i E i X i E n (1 , ..., 1) Γ E (1) A 1 α X i   E i t X i / / c n / / / / c n   for e a c h ﬁnite sequence ( X i : 1 ≤ i ≤ n ) of ob jects of V , one obtains a car tesian transformatio n α : A → E . The stability of V ’s copr oducts applied to the pullbacks A n (1 , ..., 1)   α / / E n (1 , ..., 1) c n   A 1 α 1 / / Γ E (1) for each X ∈ V and n ∈ N ensure s that Γ E ( α ) ∼ = α thu s verifying es sen tial s ur - jectivit y . Let α : A → E and β : B → E b e cartesian, and φ : Γ A → Γ B such that Γ( β ) φ =Γ α . T o ﬁnish the pr oo f we must show there is a unique φ ′ : A → B such that β φ ′ = α a nd Γ φ ′ = φ . The equation Γ φ ′ = φ implies in particular that ` n φ ′ n, 1 = φ 1 , and this determines the comp onents φ ′ n, 1 ,..., 1 uniquely be cause of A n (1 , ..., 1) B n (1 , ..., 1) E n (1 , ..., 1) Γ E (1) Γ B (1) Γ A (1) φ ′ n, 1 ,..., 1 / / β n, 1 ,..., 1 / / α n, 1 ,..., 1 * * φ 1 / / ‘ β n / / ‘ α n 4 4 c n   c n   c n   and these co mponents determine φ ′ uniquely be cause of A i X i B i X i E i X i A i 1 B i 1 E i 1 φ ′ X i / / β X i / / α X i * * φ ′ n, 1 ,..., 1 / / β n, 1 ,..., 1 / / α n, 1 ,..., 1 5 5 A i t X i   B i t X i   E i tX i   ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 21 and the equation β φ ′ = α . T o see that Γ φ ′ = φ , that is ` n φ ′ n,X,..., X = φ X for a ll X ∈ V , one deduces that the inner squar e in A n ( X, ..., X ) B n ( X, ..., X ) Γ B ( X ) Γ A ( X ) Γ A (1) A n (1 , ..., 1) B n (1 , ..., 1) Γ B (1) φ ′ n,X,...,X / / c n   c n   φ X / / φ ′ n, 1 ,..., 1 / / c n   c n   φ 1 / / A n ( t X ,...,t X ) R R R R i i R R R R B n ( t X ,...,t X ) l l l l 5 5 l l l l Γ A ( t X ) l l l l u u l l l l l Γ B ( t X ) R R R R ) ) R R R R R is a pullback since the outer s quare a nd all other regio ns in this dia gram are pull- backs, and so the result follows by lextensivity .  6. The strict ω -category monad The setting of the previous sec tio n inv olved a copro duct preser ving p.r .a mo nad T , and after this section we shall b e concer ned with the case wher e T = T the str ict ω -categor y monad on b G the categor y of glo bular sets, and its ﬁnite dimens ional analogues the strict n -catego ry monads. W e give a precise and purely inductiv e combinatorial des cription o f T in section(6.2), using some f urther theory of p.r.a monads o n presheaf categor ies which we develop in section(6.1), to facilitate our description of the details. 6.1. Sp ecifying p.r. a monads on presheaf categories. F rom [ 13 ] we know that to sp ecify a p.r.a T : b B → b C one can beg in with P ∈ b C and a functor E T : el( P ) → b B . Here we will usua lly not distinguish no ta tionally b etw een p ∈ P C and E T ( p, C ). Given k : D → C in C we shall denote by pk the element P k ( p ) and by k : pk → p the map E T ( k : ( pk , D ) → ( p, C )) . Given this data one can then deﬁne an elemen t of T X ( C ) to b e a pair ( p, h ) where p ∈ P C and h : p → X in b B . F o r a map k : D → C one deﬁnes T X ( k )( p, h ) = ( pk , hk ), and one identiﬁes P = T 1. If the E T ( p, C ) a re all connected, then T preserves copro ducts. With T so sp eciﬁed it is no t hard to characterise generic mo rphisms. T o g ive a map f : A → T X is to g iv e for a ∈ AC a n ele men t p a ∈ P C tog ether with a map f a : p a → X in b B , and this data sho uld be natural in C . The ass ignmen t ( C , a ) 7→ p a is the ob ject map o f a functor f : el( A ) → b B and the f a are the comp onents of a co cone with v ertex X . F actoring this co cone through its colimit Z gives a fac torisation A g / / T Z T h / / T X where the g a are the comp o nen ts of the universal co cone. One can easily v erify directly that such a g is generic, and since generic facto r isations are unique up to isomorphism, o ne obtains 22 MICHAEL BA T ANIN AND MARK WEBER Lemma 6.1. F or T : b B → b C sp e ciﬁe d as ab ove, f : A → T X is generic iﬀ its asso ci- ate d c o c one exhibits X as a c olimit. Examples 6.2 . (1) If in particular A is a representable C , then f : A → T X amounts to a pair ( p, h : p → X ). The associa ted co cone consists of the one ma p p and so f is generic in this case iﬀ p is an isomorphism. (2) In the case T = 1 b C , f : A → X is g eneric iﬀ it is an isomorphism. (3) Given T : b C → b C sp e ciﬁed as a bov e, a morphism f : C → T 2 X amounts to a pair ( p, h : p → T X ). This morphism is T 2 -generic iﬀ h is T -ge neric bec ause to give a commuting diagra m as depicted on the left C α / / f   T 2 Y T 2 γ   T 2 X T 2 β / / T 2 Z p α ′ / / h   T Y T γ   T X T 2 β / / T Z is the sa me as giving a commuting dia gram as depicted on the r igh t in the previous display , and s o the asser tion follows by deﬁnition o f “gener ic ” . Suppo se now tha t such a T : b C → b C comes with a ca rtesian transformation η : 1 → T . The comp onent η 1 picks out elements u C ∈ P C a nd fo r all X ∈ b C the naturality of η with resp ect to the map X → 1 shows that the comp onents of η have the explicit form x ∈ X C 7→ ( u C , x ′ : u C → X ) . Observing u C C η / / x ′   T u C ( C ) T x ′   X C η / / T X ( C ) ι  / / _   ( u C , 1 u C ) _   x  / / ( u C , x ′ ) we hav e a unique elemen t o f ι ∈ u C C which is sent by η to 1 u C . It is a general fact [ 1 2 ] that co mponents of cartesia n transfo r mations reﬂect generic morphisms, and so b y examples(6 .2)(1) and (2) the mor phism C → u C corres p onding to ι is an isomorphism. One may assume that this isomorphis m is an iden tity by r edeﬁning the yoneda em b edding if necessary to agre e with C 7→ u C and simila rly on arr o ws, so w e shall wr ite C = u C . Then the comp onent s of η may be wr itten as x 7→ ( C, x : C → X ) where the x on the right hand side co rresp onds to the x on the left hand s ide by the y oneda lemma. Deﬁnition 6. 3. Let T be a p.r.a endofunctor of b C and η : 1 → T b e a cartesian transformatio n. A pair ( P, E T ) giving the explicit description of ( T , η ) as ab ov e is called a sp e ciﬁc ation of ( T , η ). By the discussion preceeding deﬁnition(6.3) every such ( T , η ) has a speciﬁcatio n. Let us denote the assignments of an arbitary natural trans formation µ : T 2 → T by ( p ∈ P C , f : p → T X ) 7→ ( q f ∈ P C, h f : q f → X ) . Naturality of µ in C says that for k : D → C , q f k = q f k and h f k = h f k . Natura lit y of µ in X says that for h : X → Y , q T ( h ) f = q f and h T ( h ) f = hh f . Supp ose that µ ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 23 is ca rtesian. Obser ving T 2 q f µ / / T 2 h f   T q f T h f   T 2 X µ / / T X g f  / / _   ( q f , 1 q f ) _   ( p, f : p → X )  / / ( q f , h f ) one ﬁnds that ∀ p ∈ P C and f : p → X , ∃ ! g f : p → T q f such that f = T ( h f ) g f and h g f = id. By example(6.2)(3) and the fact that ca r tesian trans fo rmations reﬂect generics, such g f ’s ar e a utomatically generic. Conv ers e ly given such g f ’s one ca n readily v erify that the naturality square s of µ c o rresp onding to maps X → 1 are pullbacks and s o verify that µ is cartesian. W e record these observ ations in Lemma 6. 4. Le t ( T , η ) b e sp e ciﬁe d as in deﬁnition(6.3). T o give a c artesian natu- r al tr ansformation µ : T 2 → T is to give for e ach p ∈ P C and f : p → T X , an element q f ∈ P C and a factorisation p g f / / T q f T h f / / T X satisfying (1) F or k : D → C , q f k = q f k and h f k = h f k . (2) F or h : X → Y , q T ( h ) f = q f and h T ( h ) f = hh f . (3) F or al l p ∈ P C and f : p → T X , g f is unique such that f = T ( h f ) g f and h g f = id . and given this data, the g f ar e automatic al ly generic morphisms. Thu s a cartesian transformation µ : T 2 → T a moun ts to a nice choice of certain generic facto risations for T . Given such a characterisatio n it is straight-forward to unpack what the monad axioms for ( T , η , µ ) say in terms o f these factorisa tions. Lemma 6.5. L et ( T , η ) b e sp e ciﬁe d as in deﬁnition(6.3). T o give µ : T 2 → T making ( T , η , µ ) a p.r.a monad is to give factorisations as in lemma(6.4 ) which s atisfy the fol lowing further c onditions: (1) F or al l p ∈ P C and f : p → X , q ηf = p and h ηf = f . (2) F or al l p ∈ P C and f : p → X , q ( p,f ) = p and h ( p,f ) = f wher e ( p, f ) denotes t he map C → T X c orr esp onding to t he element ( p, f ) ∈ T X ( C ) by the yone da lemma. (3) F or al l p ∈ P C and f : p → T X , q h f = q µf and h h f = h µf . T o summarise, giv en a speciﬁcation of a p.r.a T : b C → b C , one has for e a c h C ∈ C and f : C → T X , p ∈ P C and a gener ic factorisation C g / / T p T h / / T X of f . The data of a p.r.a monad ( T , η , µ ) enables us to r egard C ∈ P C and gives us for each p ∈ P C and f : p → T X , a ch oice of q f ∈ P C and ge ner ic factoris ation p g f / / T q f T h f / / T X of f , a nd these choices satisfy certain axioms. In the case o f the str ict ω -categ ory monad below some fur ther simpliﬁcations are p ossible enabling one to disp ense with need to verify the additional conditions 24 MICHAEL BA T ANIN AND MARK WEBER of le mma(6.5) when describing it. The reason a s we shall see, is tha t this case conforms to the following deﬁnition. Deﬁnition 6. 6. A p.r.a T : b B → b C sp eciﬁed b y E T : el( P ) → b B is tight when fo r a ll p and q ∈ P C and ι : p ∼ = q in b B , o ne has p = q in P C and ι = id. Clearly tig htness is a prop erty of T , that is, is indep endent of the sp eciﬁcation. Examples 6.7 . (1) L e t T b e the free mo noid endofunctor o f Set. Then E T is a functor N → Set sending n ∈ N to a set with n elemen ts. There are of course ma ny non- trivial auto mo rphisms of a ﬁnite set, and so T is not tight. (2) Let T b e the free ca teg ory endofunctor on Gra ph which we rega rd as presheav es on 0 / / / / 1 . The n P 0 = { 0 } and P 1 = { [ n ] : n ∈ N } and the graph [ n ] has ob ject set { i : 0 ≤ i ≤ n } and a unique edge ( i − 1) → i for each 1 ≤ i ≤ n . With these details at hand one r eadily veriﬁes that this T is tight. (3) The free symmetric multicategory endofunctor on the category of multi- graphs as describ ed in example(2.1 4) o f [ 12 ] is not tigh t. In this case one actually ha s distinct p and q in P C s en t b y E T to isomorphic multigraphs. Lemma 6.8. If T : b B → b C is a t ight p.r.a then for al l A : b B → b C ther e exist s at most one c artesian t r ansformation A → T . Proof. Let α and β : A → T b e cartesian transformatio ns and a ∈ AX ( C ). F or a given sp eciﬁcation P one has p α and a g eneric factor isation C a / / g α ! ! C C C C C C C C AX α X / / T X T p α T h α < < y y y y y y y y and using the cartesia n natura lit y s quare for α corresp onding to h α , one has g ′ α : C → Ap α unique such that αg ′ α = g α and a = A ( h α ) g ′ α . Since car tesian tr ans- formations r eﬂect generics, this last equation is an A -ge ne r ic factorisation of a , and similarly one obtains ano ther one: a = A ( h β ) g ′ β by using β instead of α . Thus there is a unique is omorphism δ : p α → p β so that A ( δ ) g ′ α = g ′ β and h α δ = h β . By tightn ess δ is an identit y and s o α X a = β X a .  Thu s given a tight p.r.a T : b C → b C , car tesian transformatio ns η : 1 → T and µ : T 2 → T are unique if they exist, and when they do the mona d axioms fo r ( T , η , µ ) ar e automatic. This g iv es the following reﬁnement of lemma(6.5) in the tight ca se. Corollary 6.9. L et ( T , η ) b e sp e ciﬁe d as in deﬁnition(6.3) and let T b e tight. T o give µ : T 2 → T m aking ( T , η , µ ) a p.r.a monad is to give factorisations as in lemma(6.4). Moreov er for a tight p.r.a monad T o n b C , the multitensor T × admits the same simpliﬁcations. Lemma 6. 10. L et ( T , η , µ ) b e a p.r.a monad on b C such that T is tight. Then for al l E : M b C → b C , ther e ex ists at m ost one c artesian tr ansformation ε : E → T × . ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 25 Proof. T o give such an ε is to give fo r ea c h n ∈ N a cartesian trans formation ε n : E n → T × n , a nd s o it s uﬃces b y lemma(6.8), to show that T × n : b C n → b C is tight for all n ∈ N . The functor E T × n has ob ject map (( p 1 , ..., p n ) , C ) 7→ ( p 1 , ..., p n ). F or q 1 , ..., q n ∈ T 1 ( C ), to give a n isomor phism ι : ( p 1 , ..., p n ) ∼ = ( q 1 , ..., q n ) in b C n , is to give isomorphisms ι i : p i ∼ = q i for 1 ≤ i ≤ n , in which case the ι i are iden tities by the tightn ess of T , and so T × n is also tight.  Thu s for a tight mona d T on b C , b eing a T - oper ad is actually a pr op erty of a monad on b C , and similarly for T -multitensors. W e shall e xploit this obse rv ation notationally b elow, for instance, b y denoting a T -op erad α : A → T as we just have as a monad morphism, or just by re fer ring to the monad A , dep ending o n what is most co n venien t for the given situation. 6.2. Inductiv e descriptio n of the s trict ω -category mo nad. A g oal of this pape r to clarify the inductive na ture o f the op eradic approach to higher ca te- gory theory o f [ 1 ]. The sta rting p oin t of that approach is a precise description of the monad ( T , η , µ ) on the category b G of globular sets who s e alg ebras ar e strict ω -categor ies. Thus in this section we recall this monad, but des cribe it a little diﬀerently to the wa y it ha s b een des c ribed in the past. W e s hall give here a purely inductive de s cription of this fundamental ob ject, a nd we sha ll use the results of the previous section to exp edite our account of the details. That the algebr a s fo r the monad describ ed in this section r eally a re strict ω -catego ries deﬁned in the usual wa y by success ive enrichmen ts, is pr esen ted in s ection(8) a s a pleasant application of our gener al theory . The category G ha s as ob jects natura l n umbers and for n < m maps n σ / / τ / / m and these satis fy σ τ = τ τ and τ σ = σ σ . Thus a n ob ject of the category b G o f globula r sets is a diagr am X 0 X 1 t o o s o o X 2 t o o s o o X 3 t o o s o o ... t o o s o o of sets a nd functions such that ss = st and ts = tt . The elements o f X n are called n-cells, and f or an ( n + 1)-cell x , the n-cells sx a nd tx are ca lled the source and target of x re s pectively . In fact for each k ≤ n , we can deﬁne s ource and targ et k-cells of x a nd we denote these b y s k x a nd t k x , o nly dropping the indexing when there is little risk of confusion. Given a pair ( a, b ) of n - cells of X , one can deﬁne the globular set X ( a, b ). A k -cell of X ( a, b ) is an ( n + k )-cell x o f X such t hat s k x = a and t k x = b . Sources and targets fo r X ( a, b ) are inherited f ro m X . In particular the globular sets X ( a, b ) where a a nd b a re 0-cells a re called the homs of X . A morphism f : X → Z of globula r sets induces ma ps X ( a, b ) → Z ( f 0 a, f 0 b ) on the homs. Conv ersely , to giv e f it suﬃces to sp ecify a function f 0 : X 0 → Z 0 and for a ll a, b ∈ X 0 , morphisms X ( a, b ) → Z ( f 0 a, f 0 b ) of glo bular sets. A ﬁnite sequence ( X 1 , ..., X n ) of globular sets ma y be regar de d as a globular set, whos e set of 0 -cells is { i ∈ N : 0 ≤ i ≤ n } and whos e only non- empt y homs ar e given by ( X 1 , ..., X n )( i − 1 , i ) = X i for 1 ≤ i ≤ n . This cons truction is the ob ject map of a functor b G n → b G . 26 MICHAEL BA T ANIN AND MARK WEBER W e now b egin our description o f the e ndo functor T in the spirit of section(6.1). The role of P is play ed by the g lobular set T r of trees. The s et T r 0 contains o ne element denoted a s 0 and its ass ocia ted glo bular set co n tains one 0-c e ll, also called 0, and no thing else. By induction an element o f T r n +1 is a ﬁnite seq ue nce ( p 1 , ..., p k ) of elements of T r n and its as socia ted globula r set is just the s equence of globular sets ( p 1 , ..., p k ) re garded a s a glo bular set as in the previo us pa ragraph. So far we hav e deﬁned the elements o f T r n for all n and the ob ject ma p of E T : el(T r) → b G . W e denote by σ : 0 → p the map which selects the ob ject 0 ∈ p , and by τ : 0 → p the map which s elects the maximum vertex of p (using ≤ inherited from N ). The source a nd tar get maps s , t : T r n +1 → T r n coincide and ar e deno ted a s ∂ . F or each n we must deﬁne this map and give maps σ : ∂ p → p and τ : ∂ p → p whic h satisfy the eq ua tions σ σ = τ σ and τ τ = σ τ in (4) ∂ 2 p σ / / τ / / ∂ p σ / / τ / / p for all p ∈ T r n +2 , in order to c o mplete the des c r iption of T r and the fu nctor E T , and thus the deﬁnition of T . The maps ∂ , σ and τ ar e given by induction as follows. F or the initial step ∂ is uniquely determined since T r 0 is singleton and σ a nd τ ar e as desc ribed in the previous par agraph. F o r the inductiv e step let p = ( p 1 , ..., p k ) ∈ T r n +2 . Then ∂ p = ( ∂ p 1 , ..., ∂ p k ) a nd the maps σ , τ : ∂ p → p a re the identities on 0 -cells, and the non-empty hom maps are given by σ, τ : ∂ p i → p i resp ectively for 1 ≤ i ≤ k . The veriﬁcation o f σσ = τ σ and τ τ = σ τ as in (4) is given by inductio n as follo ws. The initial step when n = 0 is clea r since ∂ 2 p = 0 , and the 0-cell maps of σ , τ : ∂ p → p a re both the ident ity . F or the inductive step let p ! ∈ T r n +3 , then all the maps in (4) ar e identities on 0 - cells, and on the homs the desired equations follow by inductio n. By s e c tion(6.1) we hav e completed the description of a p.r.a T : b G → b G a nd we will now s ee that it is tight. Once again we arg ue b y induction on n . In the cas e n = 0 the result follows beca use T r 0 = { 0 } and the only a utomorphism of 0 ∈ b G is the iden tit y . F or the inductiv e step let p, q ∈ T r n +1 and supp ose that one has ι : p ∼ = q in b G . Since the only non-empty homs for p and q ar e b etw een consecutive elements of their vertex sets, an y f : p → q in b G is o rder prese rving in dimension 0. Thus the 0-cell map of ι is an order pr eserving bijection, a nd so m ust be the ident ity . The hom maps of ι must also b e iden tities by induction. Since the glo bular sets a sso ciated to p ∈ T r n are also co nnec ted we hav e the following res ult. Prop osition 6. 11. T : b G → b G deﬁne d as fol lows is p.r.a, tight a nd c opr o duct pr e- serving: • an n -c el l of T X is a p air ( p, f : p → X ) wher e p ∈ T r n . • for n ≥ 1 , s ( p, f ) = ( ∂ p, f σ ) and t ( p, f ) = ( ∂ p, f τ ) . • for h : X → Y , T ( h )( p, f ) = ( p, hf ) . W e will now s pecify the cartes ia n unit η : 1 →T , and from se ction(6.1) we know that this amounts to fa c to ring the yoneda embedding thro ug h E T . W e alre a dy hav e 0 ∈ T r 0 , and b y induction w e deﬁne n + 1 = ( n ) ∈ b G . Notice that the set of k - cells o f n is { 0 , 1 } when k < n and { 0 } when k = n . Mor eo ver by an easy inductive proo f the reader ma y verify that the k -cell maps of σ : n → n + 1 a nd τ : n → n + 1 a re the identities for k < n , a nd pick o ut 0 and 1 resp ectively when k = n . One ha s functions ev 0 : b G ( n, X ) → X n given by f 7→ f n (0) clearly natural in ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 27 X ∈ b G . By another e asy induction one may verify that these functions are bijective, and na tural in n in the sense that sf n +1 (0) = ( f σ ) n (0) and tf n +1 (0) = ( f τ ) n (0). Henceforth w e rega rd the identiﬁcation of n a s a globular set in this w ay as the yoneda em b edding, and the c! omp onen ts of η a re given by x ∈ X n 7→ x : n → X . Before sp ecifying the m ultiplication µ : T 2 →T some preliminar y r e marks a r e in order. F o r 0-cells a and b of X , an n -cell of the hom T X ( a, b ) consists by deﬁnition, of p = ( p 1 , ..., p k ) ∈ T r n +1 together with f : p → X such that f σ = a and f τ = b . In other words one has a sequence ( x 0 , ..., x k ) of 0-ce lls of X suc h tha t x 0 = a and x k = b , together with maps f i : p i → X ( x i − 1 , x i ) for 1 ≤ i ≤ k . Another way to say all this is that for a given s e quence ( x 0 , ..., x k ) o f 0-cells of X such that x 0 = a a nd x k = b , one ha s an inclusion c x i : Q 1 ≤ i ≤ k T ( X ( x i − 1 , x i )) → T X ( a, b ) in b G , and the following r esult. Lemma 6. 12. The maps c x i , for al l se quenc es ( x 0 , ..., x k ) of 0 - c el ls of X s uch that x 0 = a and x k = b , form a c opr o duct c o c one. Let p = ( p 1 , ..., p k ) ∈ T r n +1 . A map f : p →T X amounts to 0-cells f i of X for 0 ≤ i ≤ k , together with hom maps f i : p i →T X ( f ( i − 1) , f i ) for 1 ≤ i ≤ k . Since the p i are connected, the f i amount to 0- cells ( x i 0 , ..., x im i ) of X s uc h that x i 0 = f ( i − 1) and x im i = f i , tog e ther with maps f ij : p i →T ( X ( x ( ij ) − 1 , x ij )) for 1 ≤ i ≤ k a nd 1 ≤ j ≤ m i where ( i, j ) − 1 =    ( i, j − 1) when j > 0. ( i − 1 , m i − 1 ) when j = 0 a nd i > 0. 0 when i = j = 0. In o ther words for p = ( p 1 , ..., p k ) ∈ T r n +1 , to giv e f : p →T X is to give ob jects x 0 and x ij of X together with maps f ij : p i →T ( X ( x ( ij ) − 1 , x ij )) for 1 ≤ i ≤ k a nd 1 ≤ j ≤ m i . W e shall ca ll x 0 and the x ij the 0 -c el ls of f , and the f ij the hom map c omp onents of f . O bserve that for h : X → Y , the 0-cells of T ( h ) f are given by hx 0 and hx ij , and the hom map comp onen ts b y hf ij where 1 ≤ i ≤ k a nd 1 ≤ j ≤ m i . Now w e sp e cify the multiplication µ : T 2 →T following lemma(6.4). F or p ∈ T r n and f : p →T X the factor isation o f f tha t we m ust provide will b e g iv en by inductio n on n . When n = 0 , p = 0 and a map f : 0 → T X picks out a 0-cell (0 , x : 0 → X ) of T X . Deﬁne q f = 0 , h f = x and g f : 0 →T 0 to pick out (0 , 1 0 ). F o r the inductive step let p = ( p 1 , ..., p k ) ∈ T r n +1 and f : p →T X . Then deﬁne q f = ( q f ij : 1 ≤ i ≤ k , 1 ≤ j ≤ m i ) where the f ij are the hom map comp onents of f a s deﬁned in th e prev io us par a- graph. Deﬁne h f to have 0-cell mapping given by 0 7→ x 0 and ( i , j ) 7→ x ij , and hom maps by h f ij . Deﬁne g f to have underlying 0-cells given by 0 a nd ( i, j ), and hom map co mponents by g f ij . B y deﬁnition we hav e f = T ( h f ) g f . Prop osition 6.1 3. ( T , η , µ ) with T as sp e ciﬁe d in pr op osition(6.11) , and η and µ given by x ∈ X n 7→ ( n, x : n → X ) ( p ∈ T r n , f : p →T X ) 7→ ( q f , h f ) is a p.r.a monad. 28 MICHAEL BA T ANIN AND MARK WEBER Proof. By co rollary(6.9) it suﬃces to verify conditions (1)-(3) of lemma(6.4). Condition(1) says that fo r p = ( p 1 , ..., p k ) ∈ T r n +1 and f : p →T X : q f σ = q f τ = ∂ q f , h f σ = h f σ and h f τ = h f τ . Let us write x 0 and x ij for the 0-cells of f a nd f ij for the hom map comp onen ts where 1 ≤ i ≤ k and 1 ≤ j ≤ m i . In the cas e n = 0, w e m ust ha ve 0 = q f σ = q f τ = ∂ q f since 0 is the only element of T r 0 . Clear ly f σ picks out x 0 and f τ picks out x km k , and so h f σ : 0 → X picks out x 0 and h f τ : 0 → X picks out x km k by the initial s tep o f the description o f the factorisations. By the deﬁnition of the o b ject map of h f , h f σ and h f τ also pic k o ut the 0-cells x 0 and x km k resp ectively , th us v erifying the n = 0 case of con! dition(1). F or the indu ctive step let p = ( p 1 , ..., p k ) ∈ T r n +2 and f : p →T X . First note that σ , τ : ∂ p → p ar e identities on 0-cells a nd so f , f σ and f τ hav e the same 0-cells whic h we are denoting by x 0 and x ij . Mor eo ver by the deﬁnition of hom map compone nts, one has ( f σ ) ij = f ij σ a nd ( f τ ) ij = f ij τ . Thus by induction q f σ = ( q f ij σ : 1 ≤ i ≤ k , 1 ≤ j ≤ m i ) = ( ∂ q f ij : 1 ≤ i ≤ k , 1 ≤ j ≤ m i ) = ∂ q f and similarly q f τ = ∂ q f . Since σ , τ : ∂ q f → q f are identities on 0-cells the equations h f σ = h f σ and h f τ = h f τ are true o n 0 - cells, and on homs thes e e q uations follow by induction. Condition(2) s a ys that for p ∈ T r n , f : p →T X and h : X → Y , q T ( h ) f = q f and h T ( h ) f = hh f . When n = 0 these equations ar e immediate. F or the inductive step let p = ( p 1 , ..., p k ) ∈ T r n +1 , f : p →T X and h : X → Y . The ob jects of q T ( h ) f and q f coincide b y deﬁnition, and the homs do by induction. The ob ject maps o f h T ( h ) f and hh f coincide by deﬁnition and their ho ms maps coincide by induction. Condition(3) says that for p ∈ T r n and f : p →T X , g f is unique such that f = T ( h f ) g f and h g f = id. F or n = 0 this is clear by ins p ection. F o r the inductive step let p = ( p 1 , ..., p k ) ∈ T r n +1 and f : p →T X . By insp ection the 0-cell map o f h g f is the ide ntit y , and by induction its hom maps are also identities. As for uniqueness, the ob ject map of g f is determined uniquely by k a nd m i ∈ N for 1 ≤ i ≤ k , and the uniqueness of the hom maps follows by induction.  7. Normalised T -op erads and T -multitensors In this section we rela te T -op erads to T -multitensors a nd so express T -o perad algebras as enr ic hed catego r ies. Under a mild co ndition on an op erad α : A →T , that it b e normalise d in the sense to b e deﬁned shortly , one can constr uct a multitensor A on b G such that A -categor ie s ar e A -algebras. Moreov er A is in fact a T -multitensor, and the co nstruction ( ) is par t of an equiv alence of ca tegories b etw een T -Mult and the full sub categor y of T -Op consisting o f the normalis ed T -o pera ds . Deﬁnition 7.1. An endo functor A o f b G is normalise d when for all X ∈ b G , { AX } 0 ∼ = X 0 . A monad ( A , η , µ ) is normalised when A is normalised a s an endo- functor, a cartesian transfor mation α : A →T is called a normalise d c ol le ct ion when A is normalis ed, and a T -op erad α : A →T is normalised when A is normalised as a monad or endofunctor. W e shall denote b y T -C o ll 0 the full subcateg ory of PraE nd( b G ) / T consisting of the normalised collectio ns, and by T -O p 0 the full sub- category o f T -Op co nsisting of the normalised op erads. A 0 -cell of T X is a pair ( p ∈ T r 0 , x : p → X ), but then p = 0 and by the y oneda lemma we can regar d x as an elemen t o f X 0 . Thus T is normalised. The catego ry T -Coll 0 inherits a strict monoidal structure from Pra End ( b G ) / T , and the catego ry ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 29 of monoids therein is exa ctly T -O p 0 . W e shall allow a very c on venien t abuse of notation and language: for nor malised A wr ite { AX } 0 = X 0 rather than ac knowl- edging the bijection, a nd speak of X and AX as having the same 0-cells. This abuse is justiﬁed b ecause for any no rmalised A , one can obviously r e de ﬁne A to A ′ which is normalised in this strict s ense, and the a s signmen t A 7→ A ′ is pa rt of an equiv a le nc e of catego ries b et ween nor malised endo functors and “s trictly nor malised endofunctors”, r egarded as full sub categories o f End( b G ). W e b egin by rec a lling a nd s etting up some notation. Recall ho w a ﬁnite se- quence ( X 1 , ..., X k ) of globular sets may b e r egarded as a g lobular set: the set of 0-cells is [ k ] 0 = { 0 , ..., k } , ( X 1 , ..., X k )( i − 1 , i ) = X i and all the other homs are empty . Since we shall use these sequences often tho ughout this s ection it is necess a ry to b e ca reful with the use of round brack ets with globular sets. F or instance X and ( X ) ar e diﬀeren t, and so for an endo functor A of b G , one c a nnot identify AX and A ( X )!! Obse rv e also that the 0-cell map of a mor phism f : ( X 1 , ..., X m ) → ( Y 1 , ..., Y n ) m ust be distance preserving, that is it sends consec utive elements to consecutive elements, whenever all the X i are non-empty globular sets. W e rega rd sequence s ( x 0 , ..., x k ) of 0-cells of a globular set X as maps x : [ k ] 0 → X in b G . Given any such x we sha ll deﬁne x ∗ X := ( X ( x i − 1 , x i ) : 1 ≤ i ≤ k ) , and a map x : x ∗ X → X of globular sets. The maps x and x ag r ee on 0-cells, and x i − 1 ,i = id fo r 1 ≤ i ≤ k sp eciﬁes the ho m maps of x . F undamen tal to this sectio n is the descr iption o f the homs of T X given in lemma(6.12). W e shall now reﬁne this and see that an analog ous lemma holds for any nor malised collection. F o r X a glo bular s et and a and b ∈ X 0 , we shall now understand the hom {T X } ( a, b ). An n -cell of {T X } ( a, b ) is a pair ( p, f ) where p ∈ T r n +1 and f : p → X , such that f σ = a and f τ = b . First we consider the ca se X = ( X 1 , ..., X k ) for globular sets X i . W riting p = ( p 1 , ..., p m ) where the p i ∈ T r n , notice that f 0 m ust b e distance preser ving. There will be no such f when a > b , and in the case a ≤ b a n n -cell of {T X } ( a, b ) consists of p i ∈ T r n where a b Q a b { Ax ∗ X } (0 , b − a ) a ≤ b wher e x : [ b − a ] 0 → X is given by xi = a + i . (2) The maps { A x } 0 ,m : { Ax ∗ X } (0 , m ) → { AX } ( a, b ) for al l m ∈ N and al l se quenc es x : [ m ] 0 → X s u ch that x 0 = a and xm = b , form a c opr o duct c o c one. (3) The maps { A x } 0 ,m for al l m ∈ N and al l c onn e cte d se quenc es x : [ m ] 0 → X such t hat x 0 = a and xm = b , form a c opr o duct c o c one. Proof. In the cas e X = ( X 1 , ..., X k ) with a > b one has { α X } a,b : { AX } ( a, b ) →∅ by lemma(7.2), and since the initial ob ject of b G is strict, one has { AX } ( a, b ) = ∅ . Given any X and x : [ m ] 0 → X suc h tha t x 0 = a and xm = b , we hav e that { Ax ∗ X } (0 , m ) { A x } 0 ,m / / { α x ∗ X } 0 ,m   { AX } ( a, b ) { α X } a,b   {T x ∗ X } (0 , m ) {T x } 0 ,m / / {T X } ( a, b ) is a pullba c k by lemma(7.4) and the car tes ianness of α . In the case X = ( X 1 , ..., X k ) with a ≤ b and x : [ b − a ] 0 → X given by xi = a + i , {T x } 0 ,b − a is the identit y by lemma(7.2), thus so is { Ax } 0 ,b − a and we hav e prov ed (1). In the gener al cas e considering all m ∈ N and sequences (resp. co nnected seque nc e s ) x : [ m ] 0 → X with x 0 = a and xm = b , the {T x } 0 ,m form a c o pro duct co cone b y lemma(7.3), a nd th us so do the { Ax } 0 ,m by extensivity , whic h gives (2) and (3).  F or a normalis e d collection A , k ∈ N and X i ∈ b G wher e 1 ≤ i ≤ k , deﬁne A i X i = { AX } (0 , k ) where X = ( X 1 , ..., X k ). 4 Theorem 7.6. The assignment A 7→ A is the obje ct map of a str ong monoidal functor ( ) : T -C o ll 0 → Dist( b G ) . 4 There is an analogy b et wee n lemma(7.5) and the Lagrangian formulation of quantum me- c hanics. In this analogy one regards an y globular set X , to which one would apply a collection, as a state sp ac e the 0-cells of which are called states . A normalis ed collection A is then a t yp e of quantum me chanic al pr o c ess , with the hom { AX } ( a, b ) playing the r ole of the amplitude that the process starts in state a and ﬁnishes in state b . The b asic amplitudes are the { AX } (0 , k ) where X = ( X 1 , ... , X k ). In terms of these analogies, lemma(7.5) expresses the sense in which the general amplitude { AX } ( a, b ) ma y be regarded as the sum of the basi c amplitudes ov er all the “paths” betw een a and b , that is, as a sort of discrete F eynman in tegral. The formula just giv en expresses this passage b et ween basic and general amplitudes as a particular strong m onoidal f unc! tor, which all o ws us to view normalised op erads as multitensors, and algebras of suc h an operad as categories enriched in the corresponding multitensor. The reader s hould b e aw are that it ﬁrst b ecame apparen t to the authors that lemma(7.5) is fundamen tal to the pro of of theorems(7.6) and (7.7), and the abov e analogy was noticed afterw ards. 32 MICHAEL BA T ANIN AND MARK WEBER F or a normalise d op er ad A , one has an isomorphism A -Alg ∼ = A -Cat c ommuting with the for getful functors into Set . Proof. The a bove deﬁnition is clearly functorial in the X i so one has A : M b G → b G . A morphism of no rmalised colle c tions φ : A → B is a cartesian trans- formation b etw een A and B , and such a φ then induces a natural transfor mation φ : A → B b y the formula φ X i = { φ X } 0 ,k . The cartes ia nness of φ and lemma(7.4) ensures that φ is c artesian. In pa rticular T = T × by lemma(7.2) a nd so for a given normalised collection α : A →T , one obtains a cartesia n α : A →T × . Now by example(3.2) T × is distributive (ie pr e serves co products in ea c h v ariable) and so A is also be c a use of the ca rtesianness of α and the stability of copro ducts in b G . The assignment ! φ 7→ φ describ ed a bove is c learly functorial, and so ( ) is indeed w ell-deﬁned as a functor into Dist( b G ). Since X (0 , k ) is empty when k 6 = 1 a nd just X 1 when k = 1, we have 1 = I the unit of Dist( b G ). Let A a nd B b e norma lis ed collections and X = ( X 1 , ..., X m ). By lemma(7.5) the mo r phisms { A x } 0 ,k : { Ax ∗ B X } (0 , k ) → { AB X } (0 , m ) where k ∈ N and x : [ k ] 0 → B X such that x 0 = 0 and xk = m , f orm a c o pro duct co cone. By the deﬁnition of the tensor pr oduct in Dist( b G ), this induces a n iso mo r- phism AB ∼ = A ◦ B . W e now argue that these isomo rphisms satisfy the coherence conditions of a strong mono idal functor . Recall that the tensor pro duct in Dist( b G ) is deﬁned using copro ducts. A diﬀerent choices of copr oducts give ris e to diﬀerent monoidal structur e s on Dist( b G ), th ough for tw o such c hoices the iden tit y functor on Dist( b G ) inherits unique coherence isomor phisms that make it str o ng monoidal and thus an iso morphism of monoidal categories. Because of this o ne may ea s ily chec k that if a given str o ng mo no idal coherence d iag r am co mm utes for a par tic- ular choice of deﬁning copro ducts of the mono ida l structure of Dist( b G ), then this diagram comm utes for a n y s uc h choice. Thus to verify a given strong monoida! l coherence diagra m, it s uﬃces to see that it commutes for some choice of copro ducts. But for any s uc h diagram one can simply c ho ose the copro ducts so that all the coher ence iso morphisms inv olved in just tha t dia g ram ar e identities. Note that this is not the sa me as s pecifying Dist( b G )’s monoida l str ucture s o as to make ( ) strict mono ida l. This ﬁnishes the pro of that ( ) is strong monoida l. Let A b e a no r malised op erad a nd Z b e a set. T o give a globular set X with X 0 = Z and x : AX → X which is the identit y on 0-cells, is to give globular sets X ( y , z ) for a ll y , z ∈ Z and maps x y ,z : { AX } ( y , z ) → X ( y , z ). By lemma(7.5) the x y ,z amount to giving for each k ∈ N and f : [ k ] 0 → X s uc h that f 0 = y and f k = z , a ma p x f : A i X ( f i − 1 , f i ) → X ( y , z ) since A i X ( f i − 1 , f i ) = { Af ∗ X } (0 , k ), that is x f = x y ,z { Af } 0 ,k . F or y , z ∈ Z , o ne has a unique f : [1] 0 → X giv en by f 0 = y and f 1 = z . The naturality square for η at f implies that { η X } y ,z = { Af } 0 , 1 { η ( X ( y ,z )) } 0 , 1 and the deﬁnition of ( ) says that { η ( X ( y ,z )) } 0 , 1 = η X ( y ,z ) . Th us to s a y that a map x : AX → X satisﬁes the unit law of an A -algebra is to say that x is the identit y on 0- cells a nd that the x f describ ed ab o ve satisfy the unit axioms of an A -categor y . ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 33 T o say that x satisﬁes the as socia tiv e law is to say that for all y , z ∈ Z , (5) { A 2 X } ( y , z ) { µ X } y,z / / { Ax } y,z   { AX } ( y , z ) x y,z   { AX } ( y , z ) x y,z / / X ( y , z ) commutes. Given f : [ m ] 0 → X with f 0 = y and f m = z , and g : [ k ] 0 → Af ∗ X with g 0 = 0 and g k = m , preco mposing (5) with the co mposite map (6) { Ag ∗ Af ∗ X } (0 , k ) { A g } 0 ,k / / { A 2 f ∗ X } (0 , m ) { Af } 0 ,m / / { A 2 X } ( y , z ) and using lemma(7.5) one can see that o ne obtains the commut ativity of (7) A i A j X ( f (( i, j ) − 1) , f ( i, j )) µ / / A i x { A f } g   A ij X ( f (( i, j ) − 1) , f ( i, j )) x f   A i X ( g ( i − 1) , g i ) x g / / X ( y , z ) where 1 ≤ i ≤ k , 1 ≤ j ≤ m i , with the m i determined in the obvious wa y b y g . That is, the asso ciative law for x , na mely (5), implies the A -categ o ry asso ciative laws (7 ). Conv ersely s ince the composites (6) o ver all c hoices of f and g form a c o pro duct co cone by lemma (7 .5), (7) also implies (5). This completes the description of the ob ject part of A -Alg ∼ = A -Cat. Let ( X , x ) and ( X ′ , x ′ ) b e A -alg ebras and F 0 : X 0 → X ′ 0 be a function. T o give F : X → X ′ with 0 -cell map F 0 is to give fo r all y , z ∈ X 0 , maps F y ,z : X ( y , z ) → X ′ ( F 0 y , F 0 z ). By lemma(7 .5) to s a y that F is an a lgebra map is eq uiv alent to saying tha t F 0 and the F y ,z form an A -functor. The isomor phism A -Alg ∼ = A -Cat just de s cribed commutes with the fo rgetful functor s into Set b y deﬁnition.  Early in the ab ov e pro of we saw that ( ) sends mor phisms in T - Coll 0 to ca rte- sian tr ansformations. Since T × is tight by pro position(6 .11) and lemma(6.10), this implies by theorem(7.6) that ( ) may in fact be rega rded a s a stro ng mono idal functor ( ) : T -C o ll 0 → PraDist( b G ) / T × . F or this manifestation of ( ) we hav e the following r esult. Theorem 7.7. The functor ( ) just describ e d is an e quivalenc e of c ate gories T - C o ll 0 ≃ PraDist( b G ) / T × . Proof. W e will verify that ( ) is e ssen tially surjective on ob jects a nd fully faithful. F or a cartesian ε : E →T × we now deﬁne α : A →T s o that α ∼ = ε . F or X ∈ b G deﬁne { AX } 0 = X 0 , and for x, y ∈ X 0 , deﬁne { AX } ( x, y ) a s a copro duct with copro duct injections c f : E i X ( f ( i − 1) , f i ) → { AX } ( x, y ) for e ac h f : [ k ] 0 → X with f 0 = x and f k = y . This deﬁnition is functoria l in X in the obvious way . The comp onents of α ar e identities on 0-cells with the hom ma ps 34 MICHAEL BA T ANIN AND MARK WEBER determined by the co mm utativity of (8) E i X ( f ( i − 1) , f i ) c f / / ε   { AX } ( x, y ) { α X } x,y   {T f ∗ X } (0 , k ) {T f } 0 ,k / / {T X } ( x, y ) for all f as a b ov e. Since b G is e x tensiv e these squares a re pullbacks, and so by lemma(7.4) α de ﬁned in this way is indeed car tesian. In the ca se where X = ( X 1 , ..., X k ) and f is the identit y o n 0- cells, one has {T f } 0 ,k = id and so (8) gives α ∼ = ε as required. T o verify fully faithfulness let α : A →T and β : B →T b e normalised colle c tions, and φ : A → B b e a cartesian trans formation. T o ﬁnish the pro of it suﬃces, by the tightness of T and T × and lemma(6.8), to deﬁne a cartesian transformatio n ψ : A → B unique such that ψ = φ . F or X ∈ b G a nd f : [ k ] 0 → X t his last eq uation says that such a ψ must satisfy { ψ f ∗ X } (0 , k ) = φ X ( f ( i − 1) ,f i ) : { Af ∗ X } (0 , k ) → { B f ∗ X } (0 , k ) . The cartesianness of ψ and the tightness of T implies ψ β = α by lemma(6.8), a nd so { ψ X } 0 is the identit y . F or x, y ∈ X 0 the map { ψ X } x,y is determined b y the commutativit y of { Af ∗ X } (0 , k ) { A f } 0 ,k / / { ψ f ∗ X } (0 ,k )   { AX } ( x, y ) { ψ X } x,y   { B f ∗ X } (0 , k ) { B f } 0 ,k / / { B X } ( x, y ) for all f , since the { A f } 0 ,k form a copro duct co cone by lemma(7.5). Note also that this squar e is a pullbac k b y the extensiv it y of b G . This co mpletes the deﬁnition o f the components o f ψ and the pr oo f that they are deter mined uniq uely b y φ and the equation ψ = φ , and so to ﬁnish the pro of one m ust v erify that the ψ X are cartesian natural in X . T o this end le t F : X → Y . Since the compo nen ts of α are ident ities in dimension 0 it suﬃces b y lemma(7 .4) to sho w that for all x, y ∈ X 0 the squares (9) { AX } ( x, y ) { AF } x,y / / { ψ X } x,y   { AY } ( F 0 x, F 0 y ) { ψ Y } F 0 x,F 0 y   { B X } ( x, y ) { B F } x,y / / { B Y } ( F 0 x, F 0 y ) are pullba c ks. F or all f : [ k ] 0 → X o ne has F f = F f by deﬁnition, and so the comp osite squa re { Af ∗ X } (0 , k ) { A f } 0 ,k / / { ψ f ∗ X } (0 ,k )   { AX } ( x, y ) { AF } x,y / / { ψ X } x,y   { AY } ( F 0 x, F 0 y ) { ψ Y } F 0 x,F 0 y   { B f ∗ X } (0 , k ) { B f } 0 ,k / / { B X } ( x, y ) { B F } x,y / / { B Y } ( F 0 x, F 0 y ) ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 35 is a pullback, and so b y the e x tensivit y of b G (9) is indeed a pullback since the { A f } 0 ,k for a ll f fo r m a copro duct co cone.  Remark 7 .8. The equiv alence of theo rem(7.7) could ha ve b een described diﬀer- ent ly . This alterna tiv e view inv olves the a djo in t endo functor s D and Σ of b G . F or X ∈ b G , D X is obtained by discarding the 0- cells and putting { D X } n = X n +1 , Σ X has o ne 0-c e ll and Σ X n +1 = X n and one has D ⊣ Σ. The eﬀect of D and Σ on arrows provides an adjunction (10) b G / T 1 D T 1 / / b G /D T 1 Σ D T 1 o o ⊥ , and the rig ht a djoin t Σ D T 1 is f ully faithful since Σ is. Th us (10) r e stricts to an equiv a le nc e betw een the full sub category N o f b G / T 1 co nsisting of those f : X →T 1 such tha t X 0 is singleton. Ev aluating a t 1 giv es a n equiv ale nc e be tw een T -Coll 0 and the full sub catego ry of b G / T 1 just des cribed. By ev aluating at 1 and by the deﬁnitions of D and T 1 one obta ins PraDist( b G ) / T × ≃ b G /D T 1. Finally thes e equiv a le nc e s ﬁt together into a squa re T -Coll 0 ( ) / / ev 1   PraDist( b G ) / T × ev 1   N D T 1 / / b G /D T 1 which o ne may eas ily verify commutes up to isomor phism. These equiv a lences ev 1 really just expres s the e q uiv alence of two diﬀere nt ways o f viewing collections and their multit ensor ial ana logues, and so mo dulo this, the equiv alence from (10) expresses in p erhaps more concrete terms what ( ) do es. How ever we hav e chosen to work with ( ) b ecause this p oint of view makes clear er the relationship betw een algebras a nd enriched categ ories that w e hav e expr e s sed in theorem(7.6). Putting together theorem(7.7) a nd theorem(5.9) one obtains the equiv a lence betw een normalised T -o p erads , T -multitensors and M T -op erads. Corollary 7.9. T -Op 0 ≃ T -Mult ≃ M T -Op . 8. Finite dim ensions and the alg ebras of T W e shall now explain how the r esults of this pap e r specia lise to ﬁnite dimen- sions, and show how one ca n see that the alg ebras of T really a re s tr ict ω -ca tegories deﬁned in the usual wa y by successive enr ic hment. The ca tegory G ≤ n is de ﬁned to b e the full sub categor y o f G c o nsisting of the k ∈ N such that 0 ≤ k ≤ n . The ob jects o f b G ≤ n are called n - globular s ets. By deﬁnition the monad T on b G restricts to n -globular s ets: the description of T X n depe nds only on the k -cells of X for k ≤ n . Thus one has a monad T ≤ n on b G ≤ n . Our descriptio n o f T fro m sectio n(6 ) r estricts also, and so the monads T ≤ n are p.r.a, copro duct pr eserving and tight. In fac t, by dir ect insp ection, ev erything we have done in this pap er that has anything to do with T re s tricts to ﬁnite dimensions. 36 MICHAEL BA T ANIN AND MARK WEBER In pa rticular for n ∈ N , d enoting b y T ≤ 1+ n -Coll 0 the ca tegory o f norma lised (1 + n )-collections, whos e ob jects are car tesian transfo r mations α : A →T ≤ 1+ n whose comp onen ts are identities in dimension 0, one has a functor ( ) : T ≤ 1+ n -Coll 0 → Dist( b G ≤ n ) whose ob ject map is g iv en b y the fo r m ula A i X i = { AX } (0 , k ) where A is a nor malised (1 + n )-collection, k ∈ N and X i ∈ b G ≤ n where 1 ≤ i ≤ k , and X ∈ b G ≤ 1+ n is deﬁned as X = ( X 1 , ..., X k ). The ﬁnite dimensional analogue of theorem(7.6) is Theorem 8.1. The functor ( ) just describ e d is a stro ng monoidal functor, and for a n ormali se d (1 + n ) -op er ad A , one has an isomorphi sm A -Alg ∼ = A -Cat c ommut ing with the for getful functors into Set . As b efore one may also r egard ( ) a s a strong monoidal functor ( ) : T ≤ 1+ n -Coll 0 → PraDist( b G ≤ n ) / T × ≤ n . and the ana logue of theo rem(7.7) is Theorem 8.2. The functor ( ) just describ e d is an e qu ival enc e of c ate gories T ≤ 1+ n -Coll 0 ≃ PraDist( b G ≤ n ) / T × ≤ n . and so we hav e Corollary 8.3. T ≤ 1+ n -Op 0 ≃ T ≤ n -Mult ≃ M T ≤ n -Op . One can think of n as an o rdinal instead o f a natural n umber, and then the original results from sec tion(7) corr espond to the ca se n = ω . All a lo ng we hav e b een working with the monads T ≤ n as for mally deﬁned combinatorial ob jects. Given the r esults of this pa per how ever, it is now easy to see that their alg e bras are indeed strict n -c ategories. The usual deﬁnition of strict n -categor ies is by success iv e enrichment . One deﬁnes 0-C a t = Set and (1 + n )-Cat = ( n -Cat)-Cat for n ∈ N where n -Ca t is reg a rded as monoidal via cartes ian pro duct. Reca sting this a little mor e forma lly , ( − )-Cat is an endofunctor of t he full subca tegory of CA T consisting of categor ies with ﬁnite pro ducts. W riting 0 for the terminal ob ject of this catego ry , that is the ter minal categ ory , one ha s b y functoriality a sequence 0 0-Cat / / 1-Cat / / 2-Cat / / 3-Cat / / ... / / Explicitly the ma ps in this diagr am ar e the obvious for g etful functors. The limit of this diagram is formed a s in CA T, and pro vides th e deﬁnition of the categ ory ω -Cat. Then by theorem(7 .6) and prop osition(2.8) we hav e isomorphisms φ n : T ≤ 1+ n -Alg → ( T ≤ n -Alg)-Cat Let us write Enr for the endofunctor V 7→ V -Cat that we hav e just b een co nsidering. The isomo rphisms φ n are natura l in the sense of the following lemma, which ena bles us to then forma lly ident ify the alg ebras of T in theo rem(8.5). ALGEBRAS OF HIGHER OPERADS AS EN RICHED CA TEGORIES 37 Lemma 8. 4. F or n ∈ N let tr n : T ≤ 1+ n -Alg →T ≤ n -Alg b e the for getful fun ctor given by tru nc ation. The squar e T ≤ 2+ n -Alg tr 1+ n / / φ 1+ n   T ≤ 1+ n -Alg φ n   ( T ≤ 1+ n -Alg)-Cat Enr(tr n ) / / ( T ≤ n -Alg)-Cat c ommu tes for al l n ∈ N . Proof. One o btains φ n explicitly as the co mposite of tw o iso morphisms T ≤ 1+ n -Alg → T × ≤ n -Cat → ( T ≤ n -Alg)-Cat the ﬁr st of whic h is descr ibed explicitly in the pro of of theorem(7.6), and the second in the pro of of prop osition(2.8), a nd using these desc r iptions one may easily verify directly the desir ed naturality .  Theorem 8.5. F or 0 ≤ n ≤ ω , T ≤ n -Alg ∼ = n -Cat . Proof. W rite t : 0-Cat → 0 for the unique functor . By the deﬁnition o f ω -Ca t it suﬃces to provide isomorphisms ψ n : T ≤ n -Alg → n -Cat for n ∈ N natural in the sense that T ≤ 1+ n -Alg tr n / / ψ 1+ n   T ≤ n -Alg ψ n   (1 + n )-Ca t Enr 1+ n ( t ) / / n -Cat commutes for all n . T ake ψ 0 = 1 Set and b y induction deﬁne ψ 1+ n as the comp osite T ≤ 1+ n -Alg φ n / / ( T ≤ n -Alg)-Cat Enr( ψ n ) / / (1 + n )-Ca t . The case n = 0 for ψ ’s naturality comes fr o m the fact that the iso morphisms that comprise φ 1 (see lemma(8.4 )) are deﬁned ov er Set. The inductive s tep follows easily from lemma (8.4).  9. Ac kno wledgements The ﬁrs t author gra tefully a c knowledges for the ﬁnancia l suppor t of Scott Rus- sell Johnso n Memorial F oundation. Both authors are also grateful for the ﬁna ncial suppo rt of Max P lanc k Institut f ¨ ur Mathematik and the Australian Research Coun- cil gr an t No. DP05583 72. This pape r was completed while the second a utho r was a po stdoc at Macqua rie Univ ersity in Sydney Australia and a t the PPS la b in Paris, and he would like to tha nk these institutions for their hospitality and pleasant working conditions. W e w ould b oth like to ackno wledge the hospitality of the Ma x Planck Institute where so me of this w ork was ca rried out. Moreover we a re also indebted to Clemens Ber ger, Denis-Cha rles Cis inski and Paul-Andr ´ e Melli` es for int eresting discussions on the substa nce of this pap er. 38 MICHAEL BA T ANIN AND MARK WEBER References [1] M. Batanin. M onoidal globular categories as a natural en vironment for the theory of weak n -categories. A dvanc es in Mathematics , 136:39–103, 1998. [2] M. Batanin. The Eck mann-Hil ton argument and higher op erads. A dvanc es in Mathematics , 217:334–385 , 2008. [3] M. Batanin and M. W eb er. Al gebras of higher op erads as enriche d categories I I. In prepara- tion. [4] C. Ber ger. Iterated wreath product of the simplex category and iterated lo op spaces. A dvanc es in Mathematics , 213:230–270, 2007. [5] A. C ar boni, S. Lack, and R . F.C.W alters. In tro duction to extensiv e and distributiv e categories. J. Pur e Appl. A lgebr a , 84:145–158, 1993. [6] B. Da y and R. Street. Lax monoids, pseudo-op erads, and con v olution. In Diagr ammatic Morphisms and Applic ations , volume 318 of Contemp or ary Mathematics , pages 75–96, 2003. [7] G.M. Kelly . Basic co nc epts of e nriche d c ate gory t he ory, LMS le ctur e note se ries , volume 64. Camb ridge University Pr ess, 1982. [8] G.M. Kelly . On the oper ads of J.P . Ma y. T A C r eprints series , 13, 2005. [9] S. Lack. Codescent ob jects and coherence. J. Pur e Appl. Algebr a , 175:223–241 , 2002. [10] T. Leinster. Higher op er ads, higher ca te gories . lecture note series. London Mathe matical Society , 2003. [11] J. McCl ure and J. Smith. Cosimpli cial ob jects and little n-cubes I. Am er. J. Math , 126:1109– 1153, 2004. [12] M. W eber. Generic m orphisms, parametric represent ations, and weakly cartesian m onads. The ory and applic ations of c ategories , 13:191–234, 2004. [13] M. W eb er. F amilial 2-functo rs and parametric r igh t adjoin ts. The ory and applic ations of c ate gories , 18:665–732 , 2007. Dep a r tment of Ma thema tics, Macquarie University E-mail addr ess : mb atanin@ics.m q.edu.au Labora toire PPS, Universit ´ e P aris Diderot – P aris 7 E-mail addr ess : we ber@pps.juss ieu.fr

Algebras of higher operads as enriched categories

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