Algebras of higher operads as enriched categories
aa r X i v : . [ m a t h . C T ] M a r Algebras of higher operads as enriched categories
Michael Batanin and Mark Weber
Abstract.
We decribe the correspondence between normalised ω -operads inthe sense of [ ] and certain lax monoidal structures on the category of globularsets. As with ordinary monoidal categories, one has a notion of category en-riched in a lax monoidal category. Within the aforementioned correspondence,we provide also an equivalence between the algebras of a given normalised ω -operad, and categories enriched in globular sets for the induced lax monoidalstructure. This is an important step in reconciling the globular and simplicialapproaches to higher category theory, because in the simplicial approaches oneproceeds inductively following the idea that a weak ( n + 1)-category is some-thing like a category enriched in weak n -categories, and in this paper we beginto reveal how such an intuition may be formulated in terms of the machineryof globular operads.
1. Introduction
The subject of enriched category theory [ ] was brought to maturity by theefforts of Max Kelly and his collaborators. Max also had a hand in the genesis ofthe study of operads, and in [ ] which for a long time went unpublished, he layedthe categorical basis for their further analysis. It is with great pleasure that we areable to present the following paper, which relates enriched category theory and thestudy of higher operads, in dedication to a great mathematician.In the combinatorial approach to defining and working with higher categoricalstructures, one uses globular operads to say what the structures of interest are in onego. However in the simplicial approaches to higher category theory, one proceedsinductively following the idea that a weak ( n + 1)-category is something like acategory enriched in weak n -categories. This is the first in a series of papers whosepurpose is to reveal and study the inductive aspects hidden within the globularoperadic approach.An ω -operad in the sense of [ ] can be succinctly described as a cartesianmonad morphism α : A →T , where T is the monad on the category b G of globularsets whose algebras are strict ω -categories. The algebras of the given operad arejust the algebras of the monad A . Among the ω -operads, one can distinguish the normalised ones, which don’t provide any structure at the object level, so that onemay regard a globular set X and the globular set AX as having the same objects.For example, the operad constructed in [ ] to define weak- ω -categories, and indeedany ω -operad that has been constructed to give a definition of weak- ω -category, is normalised. One of the main results of this paper, corollary(7.9), provides twoalternative views of normalised operads: as M T -operads and as T -multitensors.The notion of T -operad, and more generally of T -multicategory, makes sensefor any cartesian monad T on a finitely complete category V (see [ ]). A T -operadcan be defined as a cartesian monad morphism into T , in the same way as we havealready outlined in the case T = T above. Under certain conditions on V and T ,one has a monad M on V which is also cartesian and whose algebras are monoids in V , this monad distributes with T , and the composite monad M T is also cartesian,so one can consider
M T -operads. All of this is so in the case T = T .On the other hand a multitensor structure on a category V is just anothername for the structure of a lax monoidal category on V . This general notion hasbeen discussed both in [ ] within the framework of lax monoids, and in [ ] whereit is expressed in the language of internal operads. A multitensor is like a monoidalstructure, except that the coherences are not necessarily invertible, and one worksin an “unbiased” setting defining an n -ary tensor product for all n ∈ N . Just aswith monoidal categories one can consider categories enriched in a lax monoidalcategory. In particular if V has cartesian products and T is a monad on V , one candefine a canonical multitensor T × on V , with the property that categories enrichedin ( V , T × ) are exactly categories enriched in T -Alg for the cartesian tensor product.When V is lextensive and T is a p.r.a monad in the sense of [ ], one can definea T -multitensor in an analogous way to the definition of T -operad: as a cartesianmultitensor morphism into T × . These assumptions on V and T are a little strongerthan asking that T be a cartesian monad, and are clearly satisfied for all examplesof interest for us such as T = T .The correspondence between normalised T -operads and T -multitensors alreadydiscussed also includes an important feature at the level of algebras. Namely the al-gebras of a given normalised T -operad α : A →T correspond to categories enrichedin the associated T -multitensor. In this way, any higher categorical structure de-finable by a normalised T -operad is expressed as a category enriched in b G for acanonically defined lax monoidal structure on b G .This paper is organised as follows. In section(2) we recall the definition of a laxmonoidal category and of categories enriched therein, and give the example of T × .Multitensors, that is lax monoidal structures, generalise non-symmetric operads,and sections(3) and (4) explain how basic operad theory generalises to multitensors.In section(3) we see how under certain conditions, one may regard multitensors asmonoids for a certain monoidal structure, which generalises the substitution tensorproduct of collections familiar from the theory of operads. Proposition(3.3) is infact a special case of proposition(2.1) of [ ]. Nevertheless we give a self-containedaccount of proposition(3.3) and related notions, to keep the exposition relativelyself-contained and as elementary as possible for our purposes. In section(4) weexplain how one can induc!e a monad from a multitensor. The theory of T -multitensors, which is themultitensorial analogue of the theory of T -operads described in [ ], is given insection(5), and it is at this level of generality that one sees the equivalence between T -multitensors and M T -operads.From this point in the paper we begin working directly with the case T = T .In section(6) we give a self-contained inductive description of the monad T . Thisis a very beautiful mathematical object. It is a p.r.a monad and its functor part LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 3 preserves coproducts. It has another crucial property, called tightness , which impliesthat for any endofunctor A , if a cartesian transformation α : A →T exists then it isunique. This property is very useful, for instance when building up a description of T one need not check the monad axioms because these come for free once one hasgiven cartesian transformations η : 1 →T and µ : T →T . The inductive descriptionof T given here is closely related to the wreath product of Clemens Berger [ ].In section(7) we give the correspondence between normalised T -operads and T -multitensors, as well as the identification between the algebras of a given normalisedoperad and categories enriched in the associated multitensor. In the final sectionwe explain how our results may be adapted to normalised n -operads, that is tofinite dimensions, and then explain how the algebras of T , which we defined asa combinatorial object, really are strict ω -categories. This last fact is of coursewell-known, but the simplicity and canonicity of our proof is a pleasant illustrationof the theory developed in this paper.The work discussed here is in a sense purely formal. Everything works at ahigh level of generality. Things become more interesting and subtle when we wishto lift the lax monoidal structures we obtain on b G , or one of its finite dimensionalanalogues, to the category of algebras of another operad. For example alreadyin this paper, one can see that the lax monoidal structure T × on b G correspondsto cartesian product of T -algebras, in the sense that they give the same enrichedcategories. It is from the general theory of such lifted lax monoidal structures thatthe Gray tensor product and its many variants, and many other examples, will becaptured within our framework. These issues will be the subject of [ ].
2. Lax Monoidal Categories
In this section we recall the notion of lax monoidal category , which is a gen-eralisation of the well-known concept of monoidal category. As with monoidalcategories, one can consider categories enriched in a lax monoidal category. Anymonad T on a finitely complete category V defines a canonical lax monoidal struc-ture T × on V , and for this structure enriched categories correspond to categoriesenriched in T -Alg regarded as monoidal via cartesian product.Given a 2-monad T on a 2-category K one may consider lax algebras for T .A lax T -algebra structure on an object A ∈ K is a triple ( a, u, σ ) consisting of anaction a : T A → A together with 2-cells A η A / / A (cid:31) (cid:31) @@@@@@@ T A a } } |||||||| A u + T A µ A / / T a (cid:15) (cid:15)
T A a (cid:15) (cid:15) T A a / / A σ + satisfying some well-known axioms. See [ ] for a complete description of theseaxioms, and of the 2-category Lax- T -Alg. When T is the identity, lax algebras arejust monads in K . The example most important for us however is when T is themonoid monad M on CAT. Definition 2.1. A multitensor on a category V is a lax M -algebra structure( E, u, σ ) on V . A category V equipped with a multitensor structure is called a lax monoidal category . When u is the identity the multitensor and lax monoidalstructure are said to be normal . MICHAEL BATANIN AND MARK WEBER
We shall now unpack this definition. Since MV = ` n ≥ V n a functor E : MV→V amounts to functors E n : V n →V for n ∈ N .Before proceeding further we digress a little on notation. For functors of manyvariables we shall use some space saving notation: we deem that the followingexpressions E n ( X , ..., X n ) E ≤ i ≤ n X i E i X i are synonymous, and we will frequently use the latter, often leaving the “ n ” un-mentioned when no confusion would result. In particular for X ∈V and 1 ≤ i ≤ n , E i X denotes E n ( X, ..., X ). We identify the number n with the ordered set { , ..., n } andwe refer to elements of the ordinal sum n • := n + ... + n k as pairs ( i, j ) where 1 ≤ i ≤ k and 1 ≤ j ≤ n i . Following these conventions E i E j X ij and E ij X ij are synonymous with E k ( E n ( X , ..., X n ) , ..., E n k ( X k , ..., X kn k ))and E n • ( X , ..., X n , ......, X k , ..., X kn k )respectively. We will use multiply indexed expressions (like E i E jk E l X ijkl ) to moreefficiently convey expressions that have multiple layers of brackets and applicationsof E ’s.The remaining data for a multitensor on V amounts to maps u X : X → E X σ X ij : E i E j X ij → E ij X ij that are natural in the arguments and satisfyE i X i u E i / / (cid:15) (cid:15) E E i X iσ (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) E i X i = E i E j E k X ijk σ E k / / E i σ (cid:15) (cid:15) E ij E k X ijkσ (cid:15) (cid:15) E i E jk X ijk σ / / E ijk X ijk = E i E X iσ (cid:30) (cid:30) ======= E i X i (cid:15) (cid:15) E i u o o E i X i =Thus a multitensor is very much like a functor-operad in the sense of [ ], exceptthat there are no symmetric group actions with respect to which the substitutionsare equivariant . An equivalent formulation of definition(2.1), in the language of [ ],is that a multitensor on V is a non-symmetric operad internal to the endomorphismoperad of V . Example 2.2.
A normal multitensor on V such that σ is invertible is just amonoidal structure on V , with E n playing the role of the n -fold tensor product.In the case where V is finitely complete and E n is n -fold cartesian product and forthe sake of the next example, we denote the isomorphism “ σ ” as ι : Q i Q j X ij → Q ij X ij More precisely, functor-operads in the sense of [ ] are normal lax algebras for the symmetric monoidal category 2-monad on CAT. LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 5
Example 2.3.
Let T be a monad on a finitely complete category V . Denote by k X i : T Q i X i → Q i T X i the canonical maps which measure the extent to which T preserves products. Onedefines a multitensor ( T × , u, σ ) as follows: T × k ( X , ..., X k ) = Q ≤ i ≤ n T ( X i ) u is the unit η X : X → T X of the monad, and σ is defined as the composite Q i T Q j T X ij Q i kT / / Q i Q j T X ij ιµ / / Q ij T X ij For the remainder of this section let ( V , E ) be a lax monoidal category. Definition 2.4. An E -category ( X, κ ), or in other words a category enriched in( V , E ), consists of • a set X of objects. • for all pairs ( x , x ) of elements of X , an object X ( x , x ) of V . Theseobjects are called the homs of X . • for all n ∈ N and ( n +1)-tuples ( x , ..., x n ) of elements of X , maps κ x i : E ≤ i ≤ n X ( x i − , x i ) → X ( x , x n )called the compositions of X .satisfying unit and associative laws, which say that X ( x , x ) E X ( x , x ) u / / X ( x , x ) κ (cid:15) (cid:15) id % % LLLLLLLLLLL E i E j X ( x ( ij ) − , x ij ) E ij X ( x ( ij ) − , x ij ) σ / / X ( x , x mn m ) κ (cid:15) (cid:15) E i X ( x ( i − , x in i ) E i κ (cid:15) (cid:15) κ / / commute, where 1 ≤ i ≤ m , 1 ≤ j ≤ n i and x (11) − = x . Since a choice of i and j ref-erences an element of the ordinal n • , the predecessor ( ij ) − ij ) iswell-defined when i and j are not both 1. An E -monoid is an E -category with oneobject. Definition 2.5.
Let (
X, κ ) and (
Y, λ ) be E -categories. An E -functor f : ( X, κ ) → ( Y, λ )consists of a function f : X → Y , and for all pairs ( x , x ) from X , arrows f x ,x : X ( x , x ) → Y ( f x , f x )satisfying a functoriality axiom, which says thatE i X ( x i − , x i ) E i f / / κ (cid:15) (cid:15) E i Y ( f x i − , f x i ) λ (cid:15) (cid:15) X ( x , x n ) f / / Y ( f x , f x n )commutes. We denote by E -Cat the category of E -categories and E -functors, andby Mon( E ) the full subcategory of E -Cat consisting of the E -monoids. MICHAEL BATANIN AND MARK WEBER
Example 2.6.
A non-symmetric operad( A n : n ∈ N ) u : I → A σ : A k ⊗ A n ⊗ ... ⊗ A n k → A n • in a braided monoidal category V defines a multitensor E on V via the formulaE ≤ i ≤ n X i = A n ⊗ X ⊗ ... ⊗ X n with u and σ providing the structure maps in the obvious way. The categoryMon( E ) of E -monoids is the usual category of algebras of A , and thus E -categoriesare a natural notion of “many object algebra” for an operad A .Our notation for multitensors makes evident the analogy with monads and algebras:a multitensor E is analogous to a monad and an E -category is the analogue of analgebra for E . In particular observe that the following basic facts are instances ofthe axioms for the lax monoidal category ( V , E ) and categories enriched therein. Lemma 2.7. (1) ( E , u, σ ) is a monad on V . (2) The monad E acts on E n for all n ∈ N , that is σ : E E i X i → E i X i is an E -algebra structure on E i X i . (3) With respect to the E -algebra structures of (2) all of the components of σ are E -algebra morphisms. (4) Each hom of an E -category ( X, κ ) is an E -algebra, with the algebra struc-ture on X ( x , x ) given by κ : E X ( x , x ) → X ( x , x ) . (5) With respect to the E -algebras of (2) and (4), all the components of κ are morphisms of E -algebras. Proposition 2.8.
Let T be a monad on a finitely complete category V . Regarding T -Alg as a monoidal category via cartesian product one has T × -Cat ∼ = ( T -Alg)-Cat commuting with the forgetful functors into Set . Proof.
Let X be a set and for a, b ∈ X let X ( a, b ) ∈ V . Suppose that κ x i : Q i T X ( x i − , x i ) → X ( x , x n )for each n ∈ N and x , ..., x n in X , are the structure maps for a T × -categorystructure. Then by lemma(2.7) the κ a,b : T X ( a, b ) → X ( a, b ) are algebra structuresfor the homs, and for x ij ∈ X with 1 ≤ i ≤ k and 1 ≤ j ≤ n i one has the inner regions LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 7 of T Q i X ( x i − , x i ) Q i T X ( x i − , x i ) Q i X ( x i − , x i ) T Q i T X ( x i − , x i ) T X ( x , x n ) X ( x , x n ) Q i T X ( x i − , x i ) Q i T X ( x i − , x i ) Q i T X ( x i − , x i ) T Q i η / / T κ / / κ (cid:15) (cid:15) k (cid:15) (cid:15) Q i κ (cid:15) (cid:15) Q i η / / κ / / k (cid:5) (cid:5) (cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11) Q i T κ (cid:21) (cid:21) +++++++ Q i ηT / / Q i µ , , XXXXX κ ' ' PPPPPP commutative, and the commutative outer region is the associativity axiom for thecomposites κ ′ x i : Q i X ( x i − , x i ) Q i η / / Q i T X ( x i − , x i ) κ xi / / X ( x , x n )for each x , ..., x n ∈ X . Taking the product structure on T -Alg as normal, theunit axiom for the κ ′ is clearly satisfied, and so they are the structure maps for a( T -Alg)-category structure. Conversely given algebra structures κ a,b and structuremaps κ ′ x i one can define κ x i as the composite Q i T X ( x i − , x i ) Q i κ xi − ,xi / / Q i X ( x i − , x i ) κ ′ xi / / X ( x , x n )and since the regions of Q i T Q j T X ( x ( ij ) − , x ij ) Q ij T X ( x ( ij ) − , x ij ) Q ij T X ( x ( ij ) − , x ij ) Q ij X ( x ( ij ) − , x ij ) X ( x , x n ) Q i X ( x i − , x i ) Q i T X ( x i − , x i ) Q i T Q j X ( x ( ij ) − , x ij ) Q ij T ( x ( ij ) − , x ij ) k / / Q ij µ / / Q ij κ (cid:15) (cid:15) κ ′ (cid:15) (cid:15) Q i T Q j κ (cid:15) (cid:15) Q i T κ ′ (cid:15) (cid:15) Q i κ / / κ ′ / / k / / Q ij κ / / Q i κ ′ u u lllllllllllll Q ij T κ (cid:15) (cid:15) commute, the commutativity of the outside of this diagram shows that the κ x i satisfy the associativity condition of a T × -category structure, and the unit axiomfollows from the unit T -algebra axiom on the homs. The correspondence just de-scribed is clearly a bijection, and completes the description of the isomorphism onobjects over Set.Let f : X → Y be a function, κ x i : Q i T X ( x i − , x i ) → X ( x , x n ) λ y i : Q i T Y ( y i − , y i ) → Y ( y , y n ) MICHAEL BATANIN AND MARK WEBER be the structure maps for T × -categories X and Y , κ ′ x i and λ ′ y i be the associated( T -Alg)-category structures, and f a,b : X ( a, b ) → Y ( f a, f b )for a, b ∈ X be maps in V . In the following display the diagram on the left Q i X ( x i − , x i ) Q i f / / Q i η (cid:15) (cid:15) Q i Y ( y i − , y i ) Q i η (cid:15) (cid:15) Q i T X ( x i − , x i ) Q i T f / / κ (cid:15) (cid:15) Q i T Y ( y i − , y i ) λ (cid:15) (cid:15) X ( x , x n ) f x ,xn / / Y ( y , y n ) Q i T X ( x i − , x i ) Q i T f / / Q i κ (cid:15) (cid:15) Q i T Y ( y i − , y i ) Q i λ (cid:15) (cid:15) Q i X ( x i − , x i ) Q i f / / κ ′ (cid:15) (cid:15) Q i Y ( y i − , y i ) λ ′ (cid:15) (cid:15) X ( x , x n ) f x ,xn / / Y ( y , y n )explains how the T × -functor axiom for the f a,b implies the ( T -Alg)-functor axiom,and the diagram on the right shows the converse. (cid:3)
3. Distributive multitensors as monoids
It is well-known that monads on a category V are monoids in the strict monoidalcategory End( V ) of endofunctors of V whose tensor product is given by composition.Given the analogy between monads and multitensors, one is led to ask under whatcircumstances are multitensors monoids in a certain monoidal category. One naturalanswer to this question, that we shall present now, requires that we restrict attentionto distributive multitensors to be defined below. Throughout this section V isassumed to have coproducts. Definition 3.1.
A functor E : MV→V is distributive when for all n ∈ N , E n preserves coproducts in each variable. We denote by Dist( V ) the category whoseobjects are such functors MV→V , and whose morphisms are natural transforma-tions between them. A multitensor (
E, u, σ ) (resp. lax monoidal category ( V , E ))is said to be distributive when E is distributive. Examples 3.2.
In the case where ( V , ⊗ , I ) is a genuine monoidal category, V isdistributive in the above sense iff ( X ⊗− ) and ( −⊗ X ) preserve coproducts for each X ∈ V . If in addition ⊗ is just cartesian product and T is a monad on V whosefunctor part preserves coproducts, then the multitensor T × of example(2.3) is alsodistributive.When E is distributive we haveE ≤ i ≤ n a j ∈ J i X ij ∼ = a j ∈ J ... a j n ∈ J n E i X ij i for any doubly indexed family X ij of objects of V . To characterise distributivityvia this formula we must be more precise and say that a certain canonical mapbetween these objects is an isomorphism. It is however more convenient to expressall this in terms of coproduct cocones. To state such an equation we must have foreach 1 ≤ i ≤ n a family of maps( c ij : X ij → X i • : j ∈ J i ) LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 9 which forms a coproduct cocone in V . Given a choice for each i of j ∈ J i , oneobtains a map E i c ij : E i X ij → E i X i • , and distributivity says that all such maps together form a coproduct cocone. Themorphisms that comprise this cocone are indexed by elements of Q i J i in agreementwith the right hand side of the above formula. For what will soon follow it isworth recalling that the (obviously true) statement “a coproduct of coproducts is acoproduct” can be described in a similar way. That is, given c ij as above togetherwith another coproduct cocone( c i : X i • → X •• : 1 ≤ i ≤ n ) , for each choice of i and j one obtains a composite arrow X ij c ij / / X i • c i / / X •• , and the collection of all such composites is a coproduct cocone.Define the unit I of Dist( V ) by I =1 V and for n =1, I n is constant at ∅ . Thetensor product E ◦ F of E and F in Dist( V ) is defined as:( E ◦ F ) n = a k ≥ a n + ... + n k = n E i F n i and so for all k and n i ∈ N where 1 ≤ i ≤ k we have mapsE i F j c ij / / E ◦ F ij which we shall also denote by c ( n ,...,n k ) as convenience dictates. For all n ∈ N theset of all such maps such that n • = n form a coproduct cocone. In the case where E = I one has I i F j ∼ = ∅ when k =1, and so c ( n ) : F n → ( I ◦ F ) n is invertible, the inverse of which we denote by λ . In the case where F = I one hasE i I j ∼ = ∅ when not all the n i ’s are 1, and so c (1 ,..., : E n → ( E ◦ n is invertible, the inverse of which we denote by ρ . Given E , F and G in Dist( V ),one has for all r ∈ N , m i ∈ N such that 1 ≤ i ≤ r , and n ij ∈ N for all i and 1 ≤ j ≤ m i ,a composite 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 composites obtained from such choices with n •• = n forms a coproduct cocone (the coproduct of coproducts is a coproduct). Fora given choice of r , m i and n ij as above one can also form a compositeE i F j G k E i c jk / / E i (F ◦ G jk ) c i ( jk ) / / ( E ◦ ( F ◦ G )) n •• and for all n ∈ N , the set of all such composites obtained from such choices with n •• = n forms a coproduct cocone because E is distributive. Thus for each n , E , F and G one has a unique isomorphism α , such that for all choices of r , m i and n ij with n •• = n , the diagram(1) E i F j G k c ij G k / / E i c jk FFFFFFFFF (E ◦ F ij ) G k c ( ij ) k / / (( E ◦ F ) ◦ G ) nα (cid:15) (cid:15) E i (F ◦ G jk ) c i ( jk ) / / ( E ◦ ( F ◦ G )) n commutes. Proposition 3.3.
The data ( I, ◦ , α, λ, ρ ) just described is a monoidal structure for Dist( V ) . The category Mon(Dist( V )) is isomorphic to the category of distributivemultitensors and morphisms thereof. Proof.
The case of (1) for which m i =1 amounts to the commutativity of theoutside of (E ◦ I i ) F k c ik / / (( E ◦ I ) ◦ F ) nα (cid:15) (cid:15) E i F kρ − F k > > ||||| c ik / / E i λ − BBBBB E ◦ F ik ρ − ◦ F uu : : uuu E ◦ λ − III $ $ III E i (I ◦ F k ) c ik / / ( E ◦ ( I ◦ F )) n ==and the inner commutativities indicated here are obtained from the definition ofthe arrow map of “ ◦ ”. But the c ik : E i F k → E ◦ F ik for all choices with n •• = n form a coproduct cocone, and so the triangle in the abovediagram, which is the unit coherence for Dist( V ), must commute also. For E , F , G and H in Dist( V ) we will now see that the corresponding associativity pentagoncommutes. For each n and choice of r , p i for all 1 ≤ i ≤ r , m ij for all i and 1 ≤ j ≤ p i , LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 11 and n ijk for all i , j and 1 ≤ k ≤ m ij , such that n ••• = n , we get a diagram of the form: • •• •• / / /// (cid:23) (cid:23) ///yyyy | | yyyy(cid:15)(cid:15)(cid:15) (cid:7) (cid:7) (cid:15)(cid:15)(cid:15) EEEE " " EEEE ••• (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) • • • (cid:127) (cid:127) (cid:127) (cid:127) (cid:127) (cid:127) •• •• (cid:31) (cid:31) / / * * ? ? • • • o o o o o o ••• O O O O O O id / / id (cid:28) (cid:28) id { { vvvvvvvvvvvvvvvvv(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5) id (cid:2) (cid:2) (cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5) HHHHHHHHHHHHHHHHH id HHHHHHHHHHHHHHHHH(cid:9)(cid:9)(cid:9)(cid:9)(cid:9) id (cid:4) (cid:4) (cid:9)(cid:9)(cid:9)(cid:9)(cid:9) IIIIIIIIII id $ $ IIIIIIIIII / / id (cid:27) (cid:27) { { ◦ ααt α tαt ◦ αn where the inner-most pentagon what we are trying to prove the commutativity of.The outer pentagon has all vertices equal to E i F j G k H l . The composites of the dottedpaths of length 3, when taken over all choices, form coproduct cocones of each ofthe vertices of the inner pentagon. For instance for the top left vertex we haveE i F j G k H l (E ◦ F ij ) G k H lcGH / / ((E ◦ F) ◦ G ijk ) H lcH / / ((( E ◦ F ) ◦ G ) ◦ H ) nc / / and the two indicated paths involving the left most vertex areE i F j G k H l (E ◦ F ij ) G k H lcGH / / (E ◦ F ij )(G ◦ H kl ) E ◦ F c / / ((( E ◦ F ) ◦ G ) ◦ H ) nc / / and E i F j G k H l E i F j (G ◦ H kl ) EF c / / (E ◦ F ij )(G ◦ H kl ) cG ◦ H / / ((( E ◦ F ) ◦ G ) ◦ H ) nc / / and in a similar vein the reader will easily supply the details of the other dottedpaths. The labels of the regions of the diagram indicate why the correspondingregion commutes: “ α ” means the region commutes by the definition of α , “n”indicates commutativity because of naturality, “ ◦ ” indicates commutativity becauseof the definition of the arrow map of ◦ , and “t” indicates that the region commutestrivially. The outer pentagon of course also commutes trivially. Since all this is truefor all choices of the r , p i , m ij and n ijk , we obtain the commutativity of the innerpentagon since the top left dotted composites together exhibit ((( E ◦ F ) ◦ G ) ◦ H ) n asa coproduct. The statement about Mon(Dist( V )) follows immediately by unpackingthe definitions involved. (cid:3)
4. Monads from multitensors
Multitensors generalise non-symmetric operads by example(2.6). Given certainhypotheses on the ambient braided monoidal category V , a non-symmetric operadtherein gives rise to a monad on V whose algebras are those of the original operad. Thus one is led to ask whether one can define a monad from a multitensor ina similar way. Such a construction is described in the present section, and wecontinue to assume throughout this section that V has coproducts.Define the functor Γ : Dist( V ) → End( V ) asΓ( E )( X ) = a n ≥ E ≤ i ≤ n X and so for each X in V we get c n : E ≤ i ≤ n X → Γ( E )( X )for n ∈ N forming a coproduct cocone. By the definition of I the map c : X → Γ( I )( X ) is an isomorphism, and we define that the inverses of these mapsare the components of an isomorphism γ : 1 V → Γ( I ). For X in V and m and n i in N where 1 ≤ i ≤ m , we can consider compositesE i F j X E i Γ F X E i c j / / Γ( E )Γ( F ) X c m / / and since E is distributive all such composites exhibit Γ( E )Γ( F ) X as a coproduct.For X , m and n i as above one also has compositesE i F j X (E ◦ F ij ) X c ij / / Γ( E ◦ F ) X c n • / / and all such composites exhibit Γ( E ◦ F ) X as a coproduct. Thus there is a uniqueisomorphism γ makingE 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 * * TTTTTTT c ij / / c n • jjjjjjjj commute, and γ is clearly natural in X . Proposition 4.1.
The data ( γ , γ ) make Γ into a monoidal functor. For anydistributive multitensor E , one has an isomorphism Mon( E ) ∼ = Γ E -Alg commutingwith the forgetful functors into V . Proof.
The definition of γ in the case where E = I and the m =1 says thatthe 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 Γ F / / γ * * TTTTTTTT λ − / / c n jjjjjjjjj Γ λ − =commutes for all m ∈ N , and the region labelled with “=” commutes because ofthe definition of the arrow maps of ◦ . Thus the inner triangle, which is the left unit LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 13 monoidal functor coherence axiom, commutes also. The definition of γ in the casewhere F = I and the n i ’s are all 1 says that the outside ofE 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 O O c m / / γ * * TTTTTTTTTTTTT ρ − / / c n • jjjjjjjjjjjjjj Γ( E ) X c m mmmmmmmm Γ ρ − aaaaaa aaaaaaaaaaaaaaaaa Γ( E ) γ ? ? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ==commutes for all m ∈ N , and the regions labelled with “=” commute because of thedefinition of the arrow maps of ◦ . Thus the inner triangle, which is the right unitmonoidal functor coherence axiom, 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 Γ( G ) / / γ (cid:28) (cid:28) :::::::: Γ α w w nnnnnnnnnnnnn Γ( E ) γ (cid:2) (cid:2) (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) γ ' ' PPPPPPPPPPPPP commutes. Now given X in V and r , m i and n ij in N where 1 ≤ i ≤ r and 1 ≤ j ≤ m i ,one obtains a diagram of the form • •• •• / / /// (cid:23) (cid:23) ///yyyy | | yyyy(cid:15)(cid:15)(cid:15) (cid:7) (cid:7) (cid:15)(cid:15)(cid:15) EEEE " " EEEE ••• (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) ••• / / / / / / • •• • w w (cid:15) (cid:15) (cid:127) (cid:127) (cid:11) (cid:11) o o ••• / / / / / / • • • o o o o o o ••• O O O O O O id / / id (cid:28) (cid:28) id vvvvvvvvvvvvvvvvv { { vvvvvvvvvvvvvvvvv id (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) id (cid:15)(cid:15)(cid:15) (cid:7) (cid:7) (cid:15)(cid:15)(cid:15) id HHHHHHHHHHHHHHHHH
HHHHHHHHHHHHHHHHH id / / id ;;;;; (cid:29) (cid:29) ;;;;; id (cid:127)(cid:127)(cid:127)(cid:127) (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127) id HHHHHHHHHHHH { { (cid:7) (cid:7) (cid:15)(cid:15)(cid:15) tγ n tγ α Γ t nγ Γ Et γ where the inner-most pentagon is (2) instantiated at X , and all the outer verticesare E i F j G k X . The two 3-fold paths into Γ( E )Γ( F )Γ( G )( X ) are the top-leftmost 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 because of naturality. The composites so formed by taking allchoices of r , m i and n ij exhibit Γ( E )Γ( F )Γ( G )( X ) as a coproduct because E and F are distributive. The left-most dotted path into Γ( E ◦ F )Γ( G )( X ) isE 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 path into Γ( E ◦ F )Γ( G )( X ) isE 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 reader will easily supply the definitions of the other dotted pathsin the above diagram. The labelled regions of that diagram commute for the reasonsindicated by the labels as with the proof of proposition(3.3), the region labelled by“Γ” commutes by the definition of the arrow map of Γ, and the region labelledby “Γ E ” commutes by the definition of the arrow map of Γ E . The outer diagramcommutes trivially and since this is all true for all choices of the r , m i and n ij ,the inner pentagon commutes as required. The statement about Mon( E ) followsimmediately by unpacking the definitions involved. (cid:3) Example 4.2.
One can apply proposition(4.1) to the case of example(2.6) when( V , ⊗ , I ) is a distributive braided monoidal category, because then the multitensoron V determined by a non-symmetric operad will also be distributive. In thisway one obtains the usual construction of the monad induced by a non-symmetricoperad. Example 4.3.
Applying proposition(4.1) to the case of a distributive monoidal cat-egory ( V , ⊗ , I ) as in example(3.2), one recovers the usual monoid monad M :=Γ( ⊗ ).In the case where ⊗ is cartesian product and T preserves coproducts, in view ofΓ( T × )= M T one obtains a monad structure on
M T , and thus a monad distributivelaw λ : T M → M T , and the algebras of
M T are monoids in T -Alg by proposi-tion(2.8). In terms of Γ and T × one can describe λ explicitly. The substitution for T × , described in example(2.3), is a map µ × : T × ◦ T × → T × in Dist( V ), and λ is thecomposite T M ηT Mη / / M T M T Γ µ × / / M T in End( V ). LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 15
5. Multitensors as operads
Given a cartesian monad T on a finitely complete category V one has the well-known notion of T -operad as described for example in [ ]. There is an analogousnotion of T -multitensor and we shall describe this in the present section. Undercertain conditions the given monad T distributes with the monoid monad M on V and the composite monad M T is again cartesian, in which case one has anequivalence of categories between T -multitensors and M T -operads. The theorydescribed in this section requires that T is a little more than cartesian, namely thatit is p.r.a in the sense of [ ], and that V is lextensive. Both notions will be recalledhere for the readers’ convenience.We recall some aspects of the theory parametric right adjoints from [ ]. Afunctor T : A→B is a parametric right adjoint (p.r.a) when for all A ∈ A , theinduced functors T A : A /A → B /T A given by applying T to arrows have left adjoints, and when A has a terminalobject 1, this is equivalent to asking that T has a left adjoint. Right adjoints areclearly p.r.a and p.r.a functors are closed under composition. Moreover one has thefollowing simple observation which we shall use often in this work. Lemma 5.1.
Let I be a set and F i : A i →B i for i ∈ I be 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 , we have ( Q i F i ) ( X i ) = Q i (( F i ) X i ), which as aproduct of right adjoints is a right adjoint. (cid:3) There is a more explicit characterisation of p.r.a functors which is sometimes useful.A map f : B → T A is T -generic when for any α , β , and γ making the outside of B α / / f (cid:15) (cid:15) T X
T γ (cid:15) (cid:15)
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 alternativecharacterisation says that T is p.r.a iff every map f : B → T A factors as B g / / T C
T h / / T A where g is generic, and such generic factorisations are unique up to isomorphism ifthey exist (see [ ] for more details). One defines a monad ( T, η, µ ) on a category V to be p.r.a when T is p.r.a as a functor, and η and µ are cartesian transformations.One has the following corresponding definition for multitensors. Definition 5.2.
A multitensor (
E, u, σ ) on V is p.r.a when E : MV→V is p.r.aand u and σ are cartesian transformations. We reserve the right to use this abbreviation also as an adjective, as in “ T is p arametrically r epresent a ble”. It is straight-forward to observe that 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 products.First note that T × n is the composite V n T n / / V n Q / / V and so is p.r.a. by lemma(5.1) and the composability of p.r.a’s. From [ ] lemma(2.14)the canonical maps k X i : T Q i X i → Q i T X i which measure the extent to which T preserves products are cartesian natural inthe X i . Thus T × is a p.r.a multitensor.For a p.r.a monad ( T, η, µ ) on a category V recall that a T -operad is cartesian monadmorphism α : A → T . That is, A is a monad on V , α is a natural transformation A → T which is compatible with the monad structures, and the naturality squares of α are pullbacks. The cartesianness of α and p.r.a’ness of T implies that A is itselfa p.r.a monad. For instance when T = T the monad on the category b G of globularsets whose algebras are strict ω -categories, to be recalled in detail in section(6), T -operads are the ω -operads of Batanin [ ]. By analogy one has the followingdefinition for multitensors. Definition 5.4.
Let (
T, η, µ ) be a p.r.a monad on V a category with finite products.A T -multitensor is a cartesian multitensor morphism ε : E → T × . Example 5.5.
We will now unpack this notion in the case where V = Set and T is the identity monad. Because of the pullback squaresE i X i Q i T X i ( T n E n ε Xi / / Q i T t Xi (cid:15) (cid:15) E i t Xi (cid:15) (cid:15) ε / / the data for ε amounts to a sequence of objects E n := E n ∈ V for n ∈ N , togetherwith maps ε n,i : E n → T ≤ i ≤ n . In this case T ε amounts to a sequence( E n : n ∈ N ) of sets. In terms of this data one has(3) E ≤ i ≤ n X i = E n × Q i X i The unit of the multitensor amounts to an element u : 1 → E , and the substitution σ amounts to functions σ n ,...,n k : E k × E n × ... × E n k → E n • for each finite sequence ( n , ..., n k ) of natural numbers. The multitensor axioms for( E, u, σ ) correspond to axioms that make (
E, u, σ ) a non-symmetric operad in Set.
LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 17
We assume throughout this section that V is lextensive. Let us now recall thisnotion. A category V is lextensive [ ] when it has finite limits, coproducts and foreach family of objects ( X i : i ∈ I ) of V the functor Q i ∈ I V /X i → V / (cid:18) ` i ∈ I X i (cid:19) which sends a family of maps ( h i : Z i → X i ) to their coproduct is an equivalence.This last property is equivalent to saying that V has a strict initial object and thatcoproducts in V are disjoint and stable. There are many examples of lextensivecategories: for instance every Grothendieck topos is lextensive, as is CAT. Moreoverif T is a coproduct preserving monad on a lextensive category V then T -Alg is alsolextensive: for such a T the forgetful functor T -Alg →V creates finite limits andcoproducts, and so these exist in T -Alg and interact as nicely as they did in V .Thus in particular the category of algebras of any higher operad is lextensive. Notein particular that lextensivity implies distributivity (see [ ]) and so the resultsof the previous two sections apply in this one. The next result summarises howlextensivity interacts well with p.r.a’ness. Lemma 5.6.
Let A and B be lextensive and I be a set. (1) The functor ` : A I →A , which takes an I -indexed family of objects of A to its coproduct, is p.r.a. (2) If F i : A→B for i ∈ I are p.r.a functors, then ` i F i : A→B is p.r.a. (3) If F i : A→B for i ∈ I are functors and φ i : F i → G i are cartesian trans-formations, then ` i φ i : ` i F i → ` i G i is cartesian. Proof. (1): given a family ( X i : i ∈ I ) of objects of A , the functor ( ` ) ( X i ) isjust the functor Q i ∈ I A /X i → A / (cid:18) ` i ∈ I X i (cid:19) which is an equivalence, and thus a right adjoint.(2): ` i F i is the composite A ∆ / / A I Q i F i / / B I ‘ / / B of a right adjoint (since A has coproducts) followed by a p.r.a (by lemma(5.1)followed by another p.r.a (by (1), and so is p.r.a.(3): the naturality square for ` i φ i corresponding to f : X → Y in A is the coproductof the cartesian naturality squares F i X φ i,X / / F i f (cid:15) (cid:15) G i X G i f (cid:15) (cid:15) F i Y φ i,Y / / G i Y and so by (1) is itself a pullback. (cid:3) Usually lextensivity is defined using only finite coproducts whereas we work with small ones.
Denote by PraDist( V ) and PraEnd( V ) the subcategories of Dist( V ) and End( V )respectively, whose objects are p.r.a’s and arrows are cartesian transformations. Proposition 5.7.
Let V be lextensive. The monoidal structure of Dist( V ) restrictsto PraDist( V ) , and Γ restricts to a strong monoidal functor PraDist( V ) → PraEnd( V ) (which we shall also denote by Γ ). Proof.
Any functor 1 →A out of the terminal category is p.r.a, and thus onereadily verifies that the functors V n →V constant at the initial object 0 of V arep.r.a also. Since 1 V is p.r.a the unit of Dist( V ) is p.r.a. For p.r.a E and F ∈ Dist( V )we must verify that E ◦ F is p.r.a. By the formula( E ◦ F ) n = a n + ... + n k = n E k ( F n , ..., F n k )and lemma(5.6) it suffices to show that each summand is p.r.a. But E k ( F n , ..., F n k )is the composite Q i V n i Q i F ni / / V k E k / / V which is p.r.a by lemma(5.1). Given ε : E → E ′ and φ : F → F ′ in PraDist( V ) wemust show that ε ◦ φ is cartesian. By lemma(5.6) it suffices to show that E k ( F n , ..., F n k ) ε k ( φ n ,...,φ nk ) / / E ′ k ( F ′ n , ..., F ′ n k )is cartesian. But this natural transformation is the composite Q i V n i V k V Q i F ni Q i F ′ ni ; ; E k " " E ′ k < < Q i φ ni (cid:11) (cid:19) ε k (cid:11) (cid:19) and so as a horizontal composite of cartesian transformations between pullbackpreserving functors, is indeed cartesian. Thus the monoidal structure of Dist( V )restricts to PraDist( V ), and to finish the proof we must verify that Γ preserves p.r.aobjects and cartesian transformations. Let E ∈ Dist( V ) be p.r.a. By lemma(5.6),to establish that Γ( E ) is p.r.a it suffices to show that for all n ∈ N , the functor X E n ( X, ..., X ) is p.r.a, but this is just the composite V ∆ / / V n E n / / V which is p.r.a since E n is. Let φ : E → F in Dist( V ) be cartesian and let us see thatΓ( φ ) is cartesian. By lemma(5.6) this comes down to the cartesian naturality in X of the maps φ n,X,...,X : E n ( X, ..., X ) → F n ( X, ..., X )which is an instance of the cartesianness of φ n . (cid:3) LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 19
Example 5.8.
From examples(5.3) and example(3.2) T × is a p.r.a distributivemultitensor when T is a coproduct preserving p.r.a monad on a lextensive category V . By proposition(5.7), the monad M T described in example(4.3) is p.r.a and thedistributive law λ : T M → M T is cartesian.Modulo one last digression we are now ready to exhibit the equivalence between T -multitensors and M T -operads as promised at the beginning of this section. Recallthat if W is a monoidal category and ( M, i, m ) a monoid therein, that the slice W /M gets a canonical monoidal structure. The unit is the unit i : I → M of the monoid,the tensor product of arrows α : A → M and β : B → M is the composite A ⊗ B α ⊗ β / / M ⊗ M m / / M and the coherences are inherited from W so that the forgetful functor W /M →W is strict monoidal. To give α : A → M a monoid structure in W /M is the same asgiving A a monoid structure for which α becomes a monoid homomorphism, andthis is just the object part of an isomorphism Mon( W /M ) ∼ =Mon( W ) /M commutingwith the forgetful functors into W . Moreover given a monoidal functor F : W→W ′ , F M is canonically a monoid and one has a commutative square W /M F M / / (cid:15) (cid:15) W ′ /F M (cid:15) (cid:15) W F / / W ′ of monoidal functors.Applying these observations to Γ : PraDist( V ) → PraEnd( V ) one obtains foreach p.r.a distributive multitensor E , a monoidal functorΓ E : PraDist( V ) /E → PraEnd( V ) / Γ E. An object of PraDist( V ) /E amounts to a functor A : MV→V together with acartesian transformation α : A → E . Given such data the distributivity of A is aconsequence of the cartesianness of α , the distributivity of E and the stability of V ’s coproducts. The p.r.a’ness of A is also a consequence, because the domain ofany cartesian transformation into a p.r.a functor is again p.r.a. A morphism inPraDist( V ) /E from α to β : B → E is just a natural transformation φ : A → B suchthat βφ = α , because by the elementary properties of pullbacks φ is automaticallycartesian. Thus a monoid in PraDist( V ) /E is simply a cartesian multitensor mor-phism into E . Similarly a monoid in PraEnd( V ) / Γ E is just a cartesian monadmorphism into Γ E , and so by observing its effect on monoids in the case E = T × where T is a coproduct pres!erving p.r.a monad on V , one has a functorΓ T : T -Mult → M T -Opfrom the category of T -multitensors to the category of M T -operads.
Theorem 5.9.
Let V be lextensive and T a coproduct preserving p.r.a monad on V . Then the functor Γ T just described is an equivalence of categories T -Mult ≃ M T -Op . Proof.
By the way we have set things up it suffices to show that for any p.r.adistributive multitensor E on V , the functor Γ E : PraDist( V ) /E → PraEnd( V ) / Γ E is essentially surjective on objects and fully faithful. Let α : A → Γ( E ) be a cartesiantransformation. Choosing pullbacksA i X i E i X i E n (1 , ...,
1) Γ E (1) A α Xi (cid:15) (cid:15) E i t Xi / / c n / / / / c n (cid:15) (cid:15) for each finite sequence ( X i : 1 ≤ i ≤ n ) of objects of V , one obtains a cartesiantransformation α : A → E . The stability of V ’s coproducts applied to the pullbacks A n (1 , ..., (cid:15) (cid:15) α / / E n (1 , ..., c n (cid:15) (cid:15) A α / / Γ E (1)for each X ∈ V and n ∈ N ensures that Γ E ( α ) ∼ = α thus verifying essential sur-jectivity. Let α : A → E and β : B → E be cartesian, and φ : Γ A → Γ B such thatΓ( β ) φ =Γ α . To finish the proof we must show there is a unique φ ′ : A → B such that βφ ′ = α and Γ φ ′ = φ . The equation Γ φ ′ = φ implies in particular that ` n φ ′ n, = φ , andthis determines the components φ ′ n, ,..., uniquely because of A n (1 , ..., B n (1 , ..., E n (1 , ..., E (1)Γ B (1)Γ A (1) φ ′ n, ,..., / / β n, ,..., / / α n, ,..., * * φ / / ‘ β n / / ‘ α n c n (cid:15) (cid:15) c n (cid:15) (cid:15) c n (cid:15) (cid:15) and these components determine φ ′ uniquely because ofA i X i B i X i E i X i A i i i φ ′ Xi / / β Xi / / α Xi * * φ ′ n, ,..., / / β n, ,..., / / α n, ,..., A i t Xi (cid:15) (cid:15) B i t Xi (cid:15) (cid:15) E i tX i (cid:15) (cid:15) LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 21 and the equation βφ ′ = α . To see that Γ φ ′ = φ , that is ` n φ ′ n,X,...,X = φ X for all X ∈ V ,one deduces that the inner square in A n ( X, ..., X ) B n ( X, ..., X )Γ B ( X )Γ A ( X )Γ A (1) A n (1 , ..., B n (1 , ..., B (1) φ ′ n,X,...,X / / c n (cid:15) (cid:15) c n (cid:15) (cid:15) φ X / / φ ′ n, ,..., / / c n (cid:15) (cid:15) c n (cid:15) (cid:15) φ / / A n ( t X ,...,t X ) RRRR i i RRRR B n ( t X ,...,t X ) llll llll Γ A ( t X ) llll u u lllll Γ B ( t X ) RRRR ) ) RRRRR is a pullback since the outer square and all other regions in this diagram are pull-backs, and so the result follows by lextensivity. (cid:3)
6. The strict ω -category monad The setting of the previous section involved a coproduct preserving p.r.a monad T , and after this section we shall be concerned with the case where T = T the strict ω -category monad on b G the category of globular sets, and its finite dimensionalanalogues the strict n -category monads. We give a precise and purely inductivecombinatorial description of T in section(6.2), using some further theory of p.r.amonads on presheaf categories which we develop in section(6.1), to facilitate ourdescription of the details. From [ ] we knowthat to specify a p.r.a T : b B → b C one can begin with P ∈ b C and a functor E T :el( P ) → b B . Here we will usually not distinguish notationally between 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 element of
T X ( C ) to be a pair ( p, h ) where p ∈ P C and h : p → X in b B . For 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 ) are all connected, then T preservescoproducts.With T so specified it is not hard to characterise generic morphisms. To give amap f : A → T X is to give for a ∈ AC an element p a ∈ P C together with a map f a : p a → X in b B , and this data should be natural in C . The assignment ( C, a ) p a isthe object map of a functor f : el( A ) → b B and the f a are the components of a coconewith vertex X . Factoring this cocone through its colimit Z gives a factorisation A g / / T Z
T h / / T X where the g a are the components of the universal cocone. One can easily verifydirectly that such a g is generic, and since generic factorisations are unique up toisomorphism, one obtains Lemma 6.1.
For T : b B → b C specified as above, f : A → T X is generic iff its associ-ated cocone exhibits X as a colimit. 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 associated cocone consists of theone map 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 generic iff it is an isomorphism.(3) Given T : b C → b C specified as above, a morphism f : C → T X amountsto a pair ( p, h : p → T X ). This morphism is T -generic iff h is T -genericbecause to give a commuting diagram as depicted on the left C α / / f (cid:15) (cid:15) T Y T γ (cid:15) (cid:15) T X T β / / T Z p α ′ / / h (cid:15) (cid:15) T Y
T γ (cid:15) (cid:15)
T X T β / / T Z is the same as giving a commuting diagram as depicted on the right in theprevious display, and so the assertion follows by definition of “generic”.Suppose now that such a T : b C → b C comes with a cartesian transformation η : 1 → T . The component η picks out elements u C ∈ P C and for all X ∈ b C thenaturality of η with respect to the map X → η havethe explicit form x ∈ XC ( u C , x ′ : u C → X ) . Observing u C C η / / x ′ (cid:15) (cid:15) T u C ( C ) T x ′ (cid:15) (cid:15) XC η / / T X ( C ) ι (cid:31) / / _ (cid:15) (cid:15) ( u C , u C ) _ (cid:15) (cid:15) x (cid:31) / / ( u C , x ′ )we have a unique element of ι ∈ u C C which is sent by η to 1 u C . It is a generalfact [ ] that components of cartesian transformations reflect generic morphisms,and so by examples(6.2)(1) and (2) the morphism C → u C corresponding to ι is anisomorphism. One may assume that this isomorphism is an identity by redefiningthe yoneda embedding if necessary to agree with C u C and similarly on arrows,so we shall write C = u C . Then the components of η may be written as x ( C, x : C → X )where the x on the right hand side corresponds to the x on the left hand side bythe yoneda lemma. Definition 6.3.
Let T be a p.r.a endofunctor of b C and η : 1 → T be a cartesiantransformation. A pair ( P, E T ) giving the explicit description of ( T, η ) as above iscalled a specification of (
T, η ).By the discussion preceeding definition(6.3) every such (
T, η ) has a specification.Let us denote the assignments of an arbitary natural transformation µ : T → T by( p ∈ P C, f : p → T X ) ( q f ∈ P C, h f : q f → X ) . Naturality of µ in C says that for k : D → C , q fk = q f k and h fk = h f k . Naturalityof µ in X says that for h : X → Y , q T ( h ) f = q f and h T ( h ) f = hh f . Suppose that µ LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 23 is cartesian. Observing T q f µ / / T h f (cid:15) (cid:15) T q fT h f (cid:15) (cid:15) T X µ / / T X g f (cid:31) / / _ (cid:15) (cid:15) ( q f , q f ) _ (cid:15) (cid:15) ( p, f : p → X ) (cid:31) / / ( 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 cartesian transformations reflectgenerics, such g f ’s are automatically generic. Conversely given such g f ’s one canreadily verify that the naturality squares of µ corresponding to maps X → µ is cartesian. We record these observations in Lemma 6.4.
Let ( T, η ) be specified as in definition(6.3). To give a cartesian natu-ral transformation µ : T → T is to give for each 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)
For k : D → C , q fk = q f k and h fk = h f k . (2) For h : X → Y , q T ( h ) f = q f and h T ( h ) f = hh f . (3) For all 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 are automatically generic morphisms. Thus a cartesian transformation µ : T → T amounts to a nice choice of certaingeneric factorisations for T . Given such a characterisation it is straight-forward tounpack what the monad axioms for ( T, η, µ ) say in terms of these factorisations.
Lemma 6.5.
Let ( T, η ) be specified as in definition(6.3). To give µ : T → T making ( T, η, µ ) a p.r.a monad is to give factorisations as in lemma(6.4) which satisfy thefollowing further conditions: (1) For all p ∈ P C and f : p → X , q ηf = p and h ηf = f . (2) For all p ∈ P C and f : p → X , q ( p,f ) = p and h ( p,f ) = f where ( p, f ) denotes the map C → T X corresponding to the element ( p, f ) ∈ T X ( C ) bythe yoneda lemma. (3) For all p ∈ P C and f : p → T X , q h f = q µf and h h f = h µf . To summarise, given a specification of a p.r.a T : b C → b C , one has for each C ∈ C and f : C → T X , p ∈ P C and a generic 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 regard C ∈ P C and givesus for each p ∈ P C and f : p → T X , a choice of q f ∈ P C and generic factorisation p g f / / T q f T h f / / T X of f , and these choices satisfy certain axioms.In the case of the strict ω -category monad below some further simplificationsare possible enabling one to dispense with need to verify the additional conditions of lemma(6.5) when describing it. The reason as we shall see, is that this caseconforms to the following definition. Definition 6.6.
A p.r.a T : b B → b C specified by E T : el( P ) → b B is tight when for all p and q ∈ P C and ι : p ∼ = q in b B , one has p = q in P C and ι = id.Clearly tightness is a property of T , that is, is independent of the specification. Examples 6.7. (1) Let T be the free monoid endofunctor of Set. Then E T is a functor N → Set sending n ∈ N to a set with n elements. There areof course many non-trivial automorphisms of a finite set, and so T is nottight.(2) Let T be the free category endofunctor on Graph which we regard aspresheaves on 0 / / / / P { } and P { [ n ] : n ∈ N } andthe graph [ n ] has object set { i : 0 ≤ i ≤ n } and a unique edge ( i − → i foreach 1 ≤ i ≤ n . With these details at hand one readily verifies that this T is tight.(3) The free symmetric multicategory endofunctor on the category of multi-graphs as described in example(2.14) of [ ] is not tight. In this case oneactually has distinct p and q in P C sent by E T to isomorphic multigraphs. Lemma 6.8. If T : b B → b C is a tight p.r.a then for all A : b B → b C there exists at mostone cartesian transformation A → T . Proof.
Let α and β : A → T be cartesian transformations and a ∈ AX ( C ).For a given specification P one has p α and a generic factorisation C a / / g α ! ! CCCCCCCC AX α X / / T XT p α T h α < < yyyyyyyy and using the cartesian naturality square for α corresponding 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 cartesian trans-formations reflect generics, this last equation is an A -generic factorisation of a , andsimilarly one obtains another one: a = A ( h β ) g ′ β by using β instead of α . Thusthere is a unique isomorphism δ : p α → p β so that A ( δ ) g ′ α = g ′ β and h α δ = h β . Bytightness δ is an identity and so α X a = β X a . (cid:3) Thus given a tight p.r.a T : b C → b C , cartesian transformations η : 1 → T and µ : T → T are unique if they exist, and when they do the monad axioms for ( T, η, µ ) areautomatic. This gives the following refinement of lemma(6.5) in the tight case.
Corollary 6.9.
Let ( T, η ) be specified as in definition(6.3) and let T be tight.To give µ : T → T making ( T, η, µ ) a p.r.a monad is to give factorisations as inlemma(6.4). Moreover for a tight p.r.a monad T on b C , the multitensor T × admits the samesimplifications. Lemma 6.10.
Let ( T, η, µ ) be a p.r.a monad on b C such that T is tight. Then forall E : M b C → b C , there exists at most one cartesian transformation ε : E → T × . LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 25
Proof.
To give such an ε is to give for each n ∈ N a cartesian transformation ε n : E n → T × n , and so it suffices by lemma(6.8), to show that T × n : b C n → b C is tightfor all n ∈ N . The functor E T × n has object map (( p , ..., p n ) , C ) ( p , ..., p n ). For q , ..., q n ∈ T C ), to give an isomorphism ι : ( p , ..., p n ) ∼ = ( q , ..., q n ) in b C n , is togive isomorphisms ι i : p i ∼ = q i for 1 ≤ i ≤ n , in which case the ι i are identities by thetightness of T , and so T × n is also tight. (cid:3) Thus for a tight monad T on b C , being a T -operad is actually a property of amonad on b C , and similarly for T -multitensors. We shall exploit this observationnotationally below, for instance, by denoting a T -operad α : A → T as we just haveas a monad morphism, or just by referring to the monad A , depending on what ismost convenient for the given situation. ω -category monad. A goal ofthis paper to clarify the inductive nature of the operadic approach to higher cate-gory theory of [ ]. The starting point of that approach is a precise description ofthe monad ( T , η, µ ) on the category b G of globular sets whose algebras are strict ω -categories. Thus in this section we recall this monad, but describe it a littledifferently to the way it has been described in the past. We shall give here a purelyinductive description of this fundamental object, and we shall use the results of theprevious section to expedite our account of the details. That the algebras for themonad described in this section really are strict ω -categories defined in the usualway by successive enrichments, is presented in section(8) as a pleasant applicationof our general theory.The category G has as objects natural numbers and for n < m maps n σ / / τ / / m and these satisfy στ = τ τ and τ σ = σσ . Thus an object of the category b G of globularsets is a diagram X X t o o s o o X t o o s o o X t o o s o o ... t o o s o o of sets and functions such that ss = st and ts = tt . The elements of X n are calledn-cells, and for an ( n + 1)-cell x , the n-cells sx and tx are called the source andtarget of x respectively. In fact for each k ≤ n , we can define source and targetk-cells of x and we denote these by s k x and t k x , only dropping the indexing whenthere is little risk of confusion. Given a pair ( a, b ) of n -cells of X , one can definethe globular set X ( a, b ). A k -cell of X ( a, b ) is an ( n + k )-cell x of X such that s k x = a and t k x = b . Sources and targets for X ( a, b ) are inherited from X . Inparticular the globular sets X ( a, b ) where a and b are 0-cells are called the homs of X . A morphism f : X → Z of globular sets induces maps X ( a, b ) → Z ( f a, f b ) onthe homs. Conversely, to give f it suffices to specify a function f : X → Z andfor all a, b ∈ X , morphisms X ( a, b ) → Z ( f a, f b ) of globular sets.A finite sequence ( X , ..., X n ) of globular sets may be regarded as a globularset, whose set of 0-cells is { i ∈ N : 0 ≤ i ≤ n } and whose only non-empty homs aregiven by ( X , ..., X n )( i − , i ) = X i for 1 ≤ i ≤ n . This construction is the object mapof a functor b G n → b G . We now begin our description of the endofunctor T in the spirit of section(6.1).The role of P is played by the globular set Tr of trees. The set Tr contains oneelement denoted as 0 and its associated globular set contains one 0-cell, also called0, and nothing else. By induction an element of Tr n +1 is a finite sequence ( p , ..., p k )of elements of Tr n and its associated globular set is just the sequence of globularsets ( p , ..., p k ) regarded as a globular set as in the previous paragraph. So far wehave defined the elements of Tr n for all n and the object map of E T : el(Tr) → b G .We denote by σ : 0 → p the map which selects the object 0 ∈ p , and by τ : 0 → p themap which selects the maximum vertex of p (using ≤ inherited from N ).The source and target maps s, t : Tr n +1 → Tr n coincide and are denoted as ∂ .For each n we must define this map and give maps σ : ∂p → p and τ : ∂p → p whichsatisfy the equations σσ = τ σ and τ τ = στ in(4) ∂ p σ / / τ / / ∂p σ / / τ / / p for all p ∈ Tr n +2 , in order to complete the description of Tr and the functor E T ,and thus the definition of T . The maps ∂ , σ and τ are given by induction asfollows. For the initial step ∂ is uniquely determined since Tr is singleton and σ and τ are as described in the previous paragraph. For the inductive step let p = ( p , ..., p k ) ∈ Tr n +2 . Then ∂p = ( ∂p , ..., ∂p k ) and the maps σ, τ : ∂p → p arethe identities on 0-cells, and the non-empty hom maps are given by σ, τ : ∂p i → p i respectively for 1 ≤ i ≤ k . The verification of σσ = τ σ and τ τ = στ as in (4) is givenby induction as follows. The initial step when n = 0 is clear since ∂ p = 0, andthe 0-cell maps of σ, τ : ∂p → p are both the identity. For the inductive step let p ! ∈ Tr n +3 , then all the maps in (4) are identities on 0-cells, and on the homs thedesired equations follow by induction.By section(6.1) we have completed the description of a p.r.a T : b G → b G and wewill now see that it is tight. Once again we argue by induction on n . In the case n = 0 the result follows because Tr = { } and the only automorphism of 0 ∈ b G is the identity. For the inductive step let p, q ∈ Tr n +1 and suppose that one has ι : p ∼ = q in b G . Since the only non-empty homs for p and q are between consecutiveelements of their vertex sets, any f : p → q in b G is order preserving in dimension0. Thus the 0-cell map of ι is an order preserving bijection, and so must be theidentity. The hom maps of ι must also be identities by induction. Since the globularsets associated to p ∈ Tr n are also connected we have the following result. Proposition 6.11. T : b G → b G defined as follows is p.r.a, tight and coproduct pre-serving: • an n -cell of T X is a pair ( p, f : p → X ) where p ∈ Tr n . • for n ≥ , s ( p, f ) = ( ∂p, f σ ) and t ( p, f ) = ( ∂p, f τ ) . • for h : X → Y , T ( h )( p, f ) = ( p, hf ) . We will now specify the cartesian unit η : 1 →T , and from section(6.1) we knowthat this amounts to factoring the yoneda embedding through E T . We alreadyhave 0 ∈ Tr , and by induction we define n + 1 = ( n ) ∈ b G . Notice that the setof k -cells of n is { , } when k < n and { } when k = n . Moreover by an easyinductive proof the reader may verify that the k -cell maps of σ : n → n + 1 and τ : n → n + 1 are the identities for k < n , and pick out 0 and 1 respectively when k = n . One has functions ev : b G ( n, X ) → X n given by f f n (0) clearly natural in LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 27 X ∈ b G . By another easy induction one may verify that these functions are bijective,and natural in n in the sense that sf n +1 (0) = ( f σ ) n (0) and tf n +1 (0) = ( f τ ) n (0).Henceforth we regard the identification of n as a globular set in this way as the yoneda embedding, and the c!omponents of η are given by x ∈ X n x : n → X .Before specifying the multiplication µ : T →T some preliminary remarks are inorder. For 0-cells a and b of X , an n -cell of the hom T X ( a, b ) consists by definition,of p = ( p , ..., p k ) ∈ Tr n +1 together with f : p → X such that f σ = a and f τ = b .In other words one has a sequence ( x , ..., x k ) of 0-cells of X such that x = a and x k = b , together with maps f i : p i → X ( x i − , x i ) for 1 ≤ i ≤ k . Another way to say allthis is that for a given sequence ( x , ..., x k ) of 0-cells of X such that x = a and x k = b , one has an inclusion c x i : Q ≤ i ≤ k T ( X ( x i − , x i )) → T X ( a, b )in b G , and the following result. Lemma 6.12.
The maps c x i , for all sequences ( x , ..., x k ) of -cells of X such that x = a and x k = b , form a coproduct cocone. Let p = ( p , ..., p k ) ∈ Tr n +1 . A map f : p →T X amounts to 0-cells f i of X for0 ≤ i ≤ k , together with hom maps f i : p i →T X ( f ( i − , f i ) for 1 ≤ i ≤ k . Since the p i are connected, the f i amount to 0-cells ( x i , ..., x im i ) of X such that x i = f ( i − x im i = f i , together with maps f ij : p i →T ( X ( x ( ij ) − , x ij )) for 1 ≤ i ≤ k and1 ≤ j ≤ m i where( i, j ) − ( i, j −
1) when j > i − , m i − ) when j = 0 and i > i = j = 0.In other words for p = ( p , ..., p k ) ∈ Tr n +1 , to give f : p →T X is to give objects x and x ij of X together with maps f ij : p i →T ( X ( x ( ij ) − , x ij )) for 1 ≤ i ≤ k and1 ≤ j ≤ m i . We shall call x and the x ij the -cells of f , and the f ij the hom mapcomponents of f . Observe that for h : X → Y , the 0-cells of T ( h ) f are given by hx and hx ij , and the hom map components by hf ij where 1 ≤ i ≤ k and 1 ≤ j ≤ m i .Now we specify the multiplication µ : T →T following lemma(6.4). For p ∈ Tr n and f : p →T X the factorisation of f that we must provide will be given by inductionon 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 , ). For the inductivestep let p = ( p , ..., p k ) ∈ Tr n +1 and f : p →T X . Then define q f = ( q f ij : 1 ≤ i ≤ k, ≤ j ≤ m i )where the f ij are the hom map components of f as defined in the previous para-graph. Define h f to have 0-cell mapping given by 0 x and ( i, j ) x ij , and hommaps by h f ij . Define g f to have underlying 0-cells given by 0 and ( i, j ), and hommap components by g f ij . By definition we have f = T ( h f ) g f . Proposition 6.13. ( T , η, µ ) with T as specified in proposition(6.11), and η and µ given by x ∈ X n ( n, x : n → X ) ( p ∈ Tr n , f : p →T X ) ( q f , h f ) is a p.r.a monad. Proof.
By corollary(6.9) it suffices to verify conditions (1)-(3) of lemma(6.4).Condition(1) says that for p = ( p , ..., p k ) ∈ Tr n +1 and f : p →T X : q fσ = q fτ = ∂q f , h fσ = h f σ and h fτ = h f τ . Let us write x and x ij for the 0-cells of f and f ij for the hom map components where 1 ≤ i ≤ k and 1 ≤ j ≤ m i . In the case n = 0, wemust have 0 = q fσ = q fτ = ∂q f since 0 is the only element of Tr . Clearly f σ picksout x and f τ picks out x km k , and so h fσ : 0 → X picks out x and h fτ : 0 → X picks out x km k by the initial step of the description of the factorisations. By thedefinition of the object map of h f , h f σ and h f τ also pick out the 0-cells x and x km k respectively, thus verifying the n = 0 case of con!dition(1). For the inductive step let p = ( p , ..., p k ) ∈ Tr n +2 and f : p →T X .First note that σ, τ : ∂p → p are identities on 0-cells and so f , f σ and f τ have thesame 0-cells which we are denoting by x and x ij . Moreover by the definition ofhom map components, one has ( f σ ) ij = f ij σ and ( f τ ) ij = f ij τ . Thus by induction q fσ = ( q f ij σ : 1 ≤ i ≤ k, ≤ j ≤ m i ) = ( ∂q f ij : 1 ≤ i ≤ k, ≤ j ≤ m i ) = ∂q f and similarly q fτ = ∂q f . Since σ, τ : ∂q f → q f are identities on 0-cells the equations h fσ = h f σ and h fτ = h f τ are true on 0-cells, and on homs these equations followby induction.Condition(2) says that for p ∈ Tr 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 are immediate. For the inductive steplet p = ( p , ..., p k ) ∈ Tr n +1 , f : p →T X and h : X → Y . The objects of q T ( h ) f and q f coincide by definition, and the homs do by induction. The object maps of h T ( h ) f and hh f coincide by definition and their homs maps coincide by induction.Condition(3) says that for p ∈ Tr n and f : p →T X , g f is unique such that f = T ( h f ) g f and h g f = id. For n = 0 this is clear by inspection. For the inductivestep let p = ( p , ..., p k ) ∈ Tr n +1 and f : p →T X . By inspection the 0-cell map of h g f is the identity, and by induction its hom maps are also identities. As for uniqueness,the object map of g f is determined uniquely by k and m i ∈ N for 1 ≤ i ≤ k , and theuniqueness of the hom maps follows by induction. (cid:3)
7. Normalised T -operads and T -multitensors In this section we relate T -operads to T -multitensors and so express T -operadalgebras as enriched categories. Under a mild condition on an operad α : A →T , thatit be normalised in the sense to be defined shortly, one can construct a multitensor A on b G such that A -categories are A -algebras. Moreover A is in fact a T -multitensor,and the construction ( ) is part of an equivalence of categories between T -Mult andthe full subcategory of T -Op consisting of the normalised T -operads. Definition 7.1.
An endofunctor A of b G is normalised when for all X ∈ b G , { AX } ∼ = X . A monad ( A, η, µ ) is normalised when A is normalised as an endo-functor, a cartesian transformation α : A →T is called a normalised collection when A is normalised, and a T -operad α : A →T is normalised when A is normalisedas a monad or endofunctor. We shall denote by T -Coll the full subcategory ofPraEnd( b G ) / T consisting of the normalised collections, and by T -Op the full sub-category of T -Op consisting of the normalised operads.A 0-cell of T X is a pair ( p ∈ Tr , x : p → X ), but then p = 0 and by the yonedalemma we can regard x as an element of X . Thus T is normalised. The category T -Coll inherits a strict monoidal structure from PraEnd( b G ) / T , and the category LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 29 of monoids therein is exactly T -Op . We shall allow a very convenient abuse ofnotation and language: for normalised A write { AX } = X rather than acknowl-edging the bijection, and speak of X and AX as having the same A , one can obviously redefine A to A ′ which is normalised in this strict sense, and the assignment A A ′ is part of anequivalence of categories between normalised endofunctors and “strictly normalisedendofunctors”, regarded as full subcategories of End( b G ).We begin by recalling and setting up some notation. Recall how a finite se-quence ( X , ..., X k ) of globular sets may be regarded as a globular set: the set of0-cells is [ k ] = { , ..., k } , ( X , ..., X k )( i − , i ) = X i and all the other homs are empty. Since we shall usethese sequences often thoughout this section it is necessary to be careful with theuse of round brackets with globular sets. For instance X and ( X ) are different, andso for an endofunctor A of b G , one cannot identify AX and A ( X )!! Observe alsothat the 0-cell map of a morphism f : ( X , ..., X m ) → ( Y , ..., Y n )must be distance preserving, that is it sends consecutive elements to consecutiveelements, whenever all the X i are non-empty globular sets. We regard sequences( x , ..., x k ) of 0-cells of a globular set X as maps x : [ k ] → X in b G . Given any such x we shall define x ∗ X := ( X ( x i − , x i ) : 1 ≤ i ≤ k ) , and a map x : x ∗ X → X of globular sets. The maps x and x agree on 0-cells, and x i − ,i = id for 1 ≤ i ≤ k specifies the hom maps of x .Fundamental to this section is the description of the homs of T X given inlemma(6.12). We shall now refine this and see that an analogous lemma holds forany normalised collection. For X a globular set and a and b ∈ X , we shall nowunderstand the hom {T X } ( a, b ). An n -cell of {T X } ( a, b ) is a pair ( p, f ) where p ∈ Tr n +1 and f : p → X , such that f σ = a and f τ = b . First we consider the case X = ( X , ..., X k ) for globular sets X i . Writing p = ( p , ..., p m ) where the p i ∈ Tr n ,notice that f must be distance preserving. There will be no such f when a > b ,and in the case a ≤ b an n -cell of {T X } ( a, b ) consists of p i ∈ Tr n where a
Let X = ( X , ..., X k ) in b G . Then for ≤ a, b ≤ k we have {T X } ( a, b ) = ( ∅ a > b Q a
Let X be a globular set and a and b ∈ X . (1) The maps {T x } ,m : {T x ∗ X } (0 , m ) → {T X } ( a, b ) for all m ∈ N and all sequences x : [ m ] → X such that x a and xm = b ,form a coproduct cocone. (2) The maps {T x } ,m for all m ∈ N and all connected sequences x : [ m ] → X such that x a and xm = b , form a coproduct cocone. Now for a normalised collection α : A →T , the extensivity of b G and the cartesiannessof α enables us to lift our understanding of the homs of T X expressed in theprevious two lemmas, to an understanding of the homs of AX . In order to do thisin lemma(7.5) below, we require a basic lemma regarding pullbacks and homs in b G . Lemma 7.4.
Given a commutative square (I) W f (cid:15) (cid:15) h / / X g (cid:15) (cid:15) Y k / / Z I W ( a, b ) f a,b (cid:15) (cid:15) h a,b / / X ( ha, hb ) g ha,hb (cid:15) (cid:15) Y ( a, b ) k a,b / / Z ( ha, hb )II in b G such that f and g are identities, one has for each a, b ∈ W commutingsquares (II) as in the previous display. The square (I) is a pullback iff for all a, b ∈ W , the square (II) is a pullback. Proof.
Suppose that (I) is a pullback and a, b ∈ W . Let y ∈ Y ( a, b ) n and x ∈ X ( ha, hb ) n such that ky = gx . Then there is a unique w ∈ W n +1 such that f w = y and hw = x , and since f = id and its components commute with sourcesand targets, one has w ∈ W ( a, b ) n whence (II) is a pullback. Conversely supposethat (II) is a pullback for all a, b ∈ W . In dimension 0 (I) is a pullback since f and g are identities. For n ∈ N let y ∈ Y n +1 and x ∈ X n +1 such that ky = gx .Put a = s y and b = t b so that y ∈ Y ( a, b ) n . Since the components of maps in b G commute with sources and targets we have x ∈ X ( ha, hb ) n , and since (II) isa pullback there is a unique w ∈ W ( a, b ) n such that f w = y and hw = x . Any w ′ ∈ W n +1 such that f w ′ = y and hw ′ = x is in W ( a, b ) n since the components of f commute with sources and targets, and so w ′ = w . (cid:3) Lemma 7.5.
Fix a choice of initial object ∅ and pullbacks in b G , such that thepullback of an identity arrow is an identity. Let α : A →T be a normalised collection. LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 31 (1)
Let X = ( X , ..., X k ) in b G . Then for ≤ a, b ≤ k we have { AX } ( a, b ) = (cid:26) ∅ a > b { Ax ∗ X } (0 , b − a ) a ≤ b where x : [ b − a ] → X is given by xi = a + i . (2) The maps { Ax } ,m : { Ax ∗ X } (0 , m ) → { AX } ( a, b ) for all m ∈ N and all sequences x : [ m ] → X such that x a and xm = b ,form a coproduct cocone. (3) The maps { Ax } ,m for all m ∈ N and all connected sequences x : [ m ] → X such that x a and xm = b , form a coproduct cocone. Proof.
In the case X = ( X , ..., X k ) with a > b one has { α X } a,b : { AX } ( a, b ) →∅ by lemma(7.2), and since the initial object of b G is strict, one has { AX } ( a, b ) = ∅ .Given any X and x : [ m ] → X such that x a and xm = b , we have that { Ax ∗ X } (0 , m ) { Ax } ,m / / { α x ∗ X } ,m (cid:15) (cid:15) { AX } ( a, b ) { α X } a,b (cid:15) (cid:15) {T x ∗ X } (0 , m ) {T x } ,m / / {T X } ( a, b )is a pullback by lemma(7.4) and the cartesianness of α . In the case X = ( X , ..., X k )with a ≤ b and x : [ b − a ] → X given by xi = a + i , {T x } ,b − a is the identity bylemma(7.2), thus so is { Ax } ,b − a and we have proved (1). In the general caseconsidering all m ∈ N and sequences (resp. connected sequences) x : [ m ] → X with x a and xm = b , the {T x } ,m form a coproduct cocone by lemma(7.3), andthus so do the { Ax } ,m by extensivity, which gives (2) and (3). (cid:3) For a normalised collection A , k ∈ N and X i ∈ b G where 1 ≤ i ≤ k , defineA i X i = { AX } (0 , k )where X = ( X , ..., X k ). Theorem 7.6.
The assignment A A is the object map of a strong monoidalfunctor ( ) : T -Coll → Dist( b G ) . There is an analogy between lemma(7.5) and the Lagrangian formulation of quantum me-chanics. In this analogy one regards any globular set X , to which one would apply a collection,as a state space the 0-cells of which are called states . A normalised collection A is then a type ofquantum mechanical process , with the hom { AX } ( a, b ) playing the role of the amplitude that theprocess starts in state a and finishes in state b . The basic amplitudes are the { AX } (0 , k ) where X = ( X , ..., X k ). In terms of these analogies, lemma(7.5) expresses the sense in which the generalamplitude { AX } ( a, b ) may be regarded as the sum of the basic amplitudes over all the “paths”between a and b , that is, as a sort of discrete Feynman integral. The formula just given expressesthis passage between basic and general amplitudes as a particular strong monoidal func!tor, which allows us to view normalised operads as multitensors, and algebras of such anoperad as categories enriched in the corresponding multitensor.The reader should be aware that it first became apparent to the authors that lemma(7.5) isfundamental to the proof of theorems(7.6) and (7.7), and the above analogy was noticed afterwards. For a normalised operad A , one has an isomorphism A -Alg ∼ = A -Cat commutingwith the forgetful functors into Set . Proof.
The above definition is clearly functorial in the X i so one has A : M b G → b G . A morphism of normalised collections φ : A → B is a cartesian trans-formation between A and B , and such a φ then induces a natural transformation φ : A → B by the formula φ X i = { φ X } ,k . The cartesianness of φ and lemma(7.4)ensures that φ is cartesian. In particular T = T × by lemma(7.2) and so for agiven normalised collection α : A →T , one obtains a cartesian α : A →T × . Now byexample(3.2) T × is distributive (ie preserves coproducts in each variable) and so A is also because of the cartesianness of α and the stability of coproducts in b G . Theassignment ! φ φ described above is clearly functorial, and so ( ) is indeed well-defined asa functor into Dist( b G ).Since X (0 , k ) is empty when k = 1 and just X when k = 1, we have 1 = I theunit of Dist( b G ). Let A and B be normalised collections and X = ( X , ..., X m ). Bylemma(7.5) the morphisms { Ax } ,k : { Ax ∗ BX } (0 , k ) → { ABX } (0 , m )where k ∈ N and x : [ k ] → BX such that x xk = m , form a coproductcocone. By the definition of the tensor product in Dist( b G ), this induces an isomor-phism AB ∼ = A ◦ B . We now argue that these isomorphisms satisfy the coherenceconditions of a strong monoidal functor. Recall that the tensor product in Dist( b G )is defined using coproducts. A different choices of coproducts give rise to differentmonoidal structures on Dist( b G ), though for two such choices the identity functoron Dist( b G ) inherits unique coherence isomorphisms that make it strong monoidaland thus an isomorphism of monoidal categories. Because of this one may easilycheck that if a given strong monoidal coherence diagram commutes for a partic-ular choice of defining coproducts of the monoidal structure of Dist( b G ), then thisdiagram commutes for any such choice. Thus to verify a given strong monoida!l coherence diagram, it suffices to see that it commutes for some choice ofcoproducts. But for any such diagram one can simply choose the coproducts sothat all the coherence isomorphisms involved in just that diagram are identities.Note that this is not the same as specifying Dist( b G )’s monoidal structure so as tomake ( ) strict monoidal. This finishes the proof that ( ) is strong monoidal.Let A be a normalised operad and Z be a set. To give a globular set X with X = Z and x : AX → X which is the identity on 0-cells, is to give globular sets X ( y, z ) for all 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 ] → X such that f y and f k = z ,a map x f : A i X ( f i − , f i ) → X ( y, z )since A i X ( f i − , f i ) = { Af ∗ X } (0 , k ), that is x f = x y,z { Af } ,k . For y, z ∈ Z , onehas a unique f : [1] → X given by f y and f z . The naturality square for η at f implies that { η X } y,z = { Af } , { η ( X ( y,z )) } , and the definition of ( ) says that { η ( X ( y,z )) } , = η X ( y,z ) . Thus to say that a map x : AX → X satisfies the unit lawof an A -algebra is to say that x is the identity on 0-cells and that the x f describedabove satisfy the unit axioms of an A -category. LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 33
To say that x satisfies the associative law is to say that for all y, z ∈ Z ,(5) { A X } ( y, z ) { µ X } y,z / / { Ax } y,z (cid:15) (cid:15) { AX } ( y, z ) x y,z (cid:15) (cid:15) { AX } ( y, z ) x y,z / / X ( y, z )commutes. Given f : [ m ] → X with f y and f m = z , and g : [ k ] → Af ∗ X with g gk = m , precomposing (5) with the composite map(6) { Ag ∗ Af ∗ X } (0 , k ) { Ag } ,k / / { A f ∗ X } (0 , m ) { Af } ,m / / { A X } ( y, z )and using lemma(7.5) one can see that one obtains the commutativity of(7) A i A j X ( f (( i, j ) − , f ( i, j )) µ / / A i x { Af } g (cid:15) (cid:15) A ij X ( f (( i, j ) − , f ( i, j )) x f (cid:15) (cid:15) A i X ( g ( i − , gi ) x g / / X ( y, z )where 1 ≤ i ≤ k , 1 ≤ j ≤ m i , with the m i determined in the obvious way by g . That is,the associative law for x , namely (5), implies the A -category associative laws (7).Conversely since the composites (6) over all choices of f and g form a coproductcocone by lemma(7.5), (7) also implies (5). This completes the description of theobject part of A -Alg ∼ = A -Cat.Let ( X, x ) and ( X ′ , x ′ ) be A -algebras and F : X → X ′ be a function. Togive F : X → X ′ with 0-cell map F is to give for all y, z ∈ X , maps F y,z : X ( y, z ) → X ′ ( F y, F z ). By lemma(7.5) to say that F is an algebra map is equivalentto saying that F and the F y,z form an A -functor. The isomorphism A -Alg ∼ = A -Catjust described commutes with the forgetful functors into Set by definition. (cid:3) Early in the above proof we saw that ( ) sends morphisms in T -Coll to carte-sian transformations. Since T × is tight by proposition(6.11) and lemma(6.10), thisimplies by theorem(7.6) that ( ) may in fact be regarded as a strong monoidalfunctor ( ) : T -Coll → PraDist( b G ) / T × . For this manifestation of ( ) we have the following result.
Theorem 7.7.
The functor ( ) just described is an equivalence of categories T -Coll ≃ PraDist( b G ) / T × . Proof.
We will verify that ( ) is essentially surjective on objects and fullyfaithful. For a cartesian ε : E →T × we now define α : A →T so that α ∼ = ε . For X ∈ b G define { AX } = X , and for x, y ∈ X , define { AX } ( x, y ) as a coproductwith coproduct injections c f : E i X ( f ( i − , f i ) → { AX } ( x, y )for each f : [ k ] → X with f x and f k = y . This definition is functorial in X inthe obvious way. The components of α are identities on 0-cells with the hom maps determined by the commutativity of(8) E i X ( f ( i − , f i ) c f / / ε (cid:15) (cid:15) { AX } ( x, y ) { α X } x,y (cid:15) (cid:15) {T f ∗ X } (0 , k ) {T f } ,k / / {T X } ( x, y )for all f as above. Since b G is extensive these squares are pullbacks, and so bylemma(7.4) α defined in this way is indeed cartesian. In the case where X =( X , ..., X k ) and f is the identity on 0-cells, one has {T f } ,k = id and so (8) gives α ∼ = ε as required. To verify fully faithfulness let α : A →T and β : B →T benormalised collections, and φ : A → B be a cartesian transformation. To finish theproof it suffices, by the tightness of T and T × and lemma(6.8), to define a cartesiantransformation ψ : A → B unique such that ψ = φ . For X ∈ b G and f : [ k ] → X thislast equation says that such a ψ must satisfy { ψ f ∗ X } (0 , k ) = φ X ( f ( i − ,fi ) : { Af ∗ X } (0 , k ) → { Bf ∗ X } (0 , k ) . The cartesianness of ψ and the tightness of T implies ψβ = α by lemma(6.8), andso { ψ X } is the identity. For x, y ∈ X the map { ψ X } x,y is determined by thecommutativity of { Af ∗ X } (0 , k ) { Af } ,k / / { ψ f ∗ X } (0 ,k ) (cid:15) (cid:15) { AX } ( x, y ) { ψ X } x,y (cid:15) (cid:15) { Bf ∗ X } (0 , k ) { Bf } ,k / / { BX } ( x, y )for all f , since the { Af } ,k form a coproduct cocone by lemma(7.5). Note also thatthis square is a pullback by the extensivity of b G . This completes the definition ofthe components of ψ and the proof that they are determined uniquely by φ andthe equation ψ = φ , and so to finish the proof one must verify that the ψ X arecartesian natural in X . To this end let F : X → Y . Since the components of α areidentities in dimension 0 it suffices by lemma(7.4) to show that for all x, y ∈ X the squares(9) { AX } ( x, y ) { AF } x,y / / { ψ X } x,y (cid:15) (cid:15) { AY } ( F x, F y ) { ψ Y } F x,F y (cid:15) (cid:15) { BX } ( x, y ) { BF } x,y / / { BY } ( F x, F y )are pullbacks. For all f : [ k ] → X one has F f = F f by definition, and so thecomposite square { Af ∗ X } (0 , k ) { Af } ,k / / { ψ f ∗ X } (0 ,k ) (cid:15) (cid:15) { AX } ( x, y ) { AF } x,y / / { ψ X } x,y (cid:15) (cid:15) { AY } ( F x, F y ) { ψ Y } F x,F y (cid:15) (cid:15) { Bf ∗ X } (0 , k ) { Bf } ,k / / { BX } ( x, y ) { BF } x,y / / { BY } ( F x, F y ) LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 35 is a pullback, and so by the extensivity of b G (9) is indeed a pullback since the { Af } ,k for all f form a coproduct cocone. (cid:3) Remark 7.8.
The equivalence of theorem(7.7) could have been described differ-ently. This alternative view involves the adjoint endofunctors D and Σ of b G . For X ∈ b G , DX is obtained by discarding the 0-cells and putting { DX } n = X n +1 , Σ X has one 0-cell and Σ X n +1 = X n and one has D ⊣ Σ. The effect of D and Σ onarrows provides an adjunction(10) b G / T D T / / b G /D T Σ D T o o ⊥ , and the right adjoint Σ D T is fully faithful since Σ is. Thus (10) restricts to anequivalence between the full subcategory N of b G / T f : X →T X is singleton. Evaluating at 1 gives an equivalence between T -Coll and the full subcategory of b G / T D and T b G ) / T × ≃ b G /D T
1. Finally theseequivalences fit together into a square T -Coll / / ev (cid:15) (cid:15) PraDist( b G ) / T × ev (cid:15) (cid:15) N D T / / b G /D T really just express the equivalence of two different ways of viewing collectionsand their multitensorial analogues, and so modulo this, the equivalence from (10)expresses in perhaps more concrete terms what ( ) does. However we have chosento work with ( ) because this point of view makes clearer the relationship betweenalgebras and enriched categories that we have expressed in theorem(7.6).Putting together theorem(7.7) and theorem(5.9) one obtains the equivalencebetween normalised T -operads, T -multitensors and M T -operads. Corollary 7.9. T -Op ≃ T -Mult ≃ M T -Op .
8. Finite dimensions and the algebras of T We shall now explain how the results of this paper specialise to finite dimen-sions, and show how one can see that the algebras of T really are strict ω -categoriesdefined in the usual way by successive enrichment.The category G ≤ n is defined to be the full subcategory of G consisting of the k ∈ N such that 0 ≤ k ≤ n . The objects of b G ≤ n are called n -globular sets. Bydefinition the monad T on b G restricts to n -globular sets: the description of T X n depends only on the k -cells of X for k ≤ n . Thus one has a monad T ≤ n on b G ≤ n .Our description of T from section(6) restricts also, and so the monads T ≤ n are p.r.a,coproduct preserving and tight. In fact, by direct inspection, everything we havedone in this paper that has anything to do with T restricts to finite dimensions. In particular for n ∈ N , denoting by T ≤ n -Coll the category of normalised(1+ n )-collections, whose objects are cartesian transformations α : A →T ≤ n whosecomponents are identities in dimension 0, one has a functor( ) : T ≤ n -Coll → Dist( b G ≤ n )whose object map is given by the formulaA i X i = { AX } (0 , k )where A is a normalised (1 + n )-collection, k ∈ N and X i ∈ b G ≤ n where 1 ≤ i ≤ k ,and X ∈ b G ≤ n is defined as X = ( X , ..., X k ). The finite dimensional analogue oftheorem(7.6) is Theorem 8.1.
The functor ( ) just described is a strong monoidal functor, and fora normalised (1 + n ) -operad A , one has an isomorphism A -Alg ∼ = A -Cat commutingwith the forgetful functors into Set . As before one may also regard ( ) as a strong monoidal functor( ) : T ≤ n -Coll → PraDist( b G ≤ n ) / T ×≤ n . and the analogue of theorem(7.7) is Theorem 8.2.
The functor ( ) just described is an equivalence of categories T ≤ n -Coll ≃ PraDist( b G ≤ n ) / T ×≤ n . and so we have Corollary 8.3. T ≤ n -Op ≃ T ≤ n -Mult ≃ M T ≤ n -Op . One can think of n as an ordinal instead of a natural number, and then the originalresults from section(7) correspond to the case n = ω .All along we have been working with the monads T ≤ n as formally definedcombinatorial objects. Given the results of this paper however, it is now easyto see that their algebras are indeed strict n -categories. The usual definition ofstrict n -categories is by successive enrichment. One defines 0-Cat = Set and (1 + n )-Cat = ( n -Cat)-Cat for n ∈ N where n -Cat is regarded as monoidal via cartesianproduct. Recasting this a little more formally, ( − )-Cat is an endofunctor of thefull subcategory of CAT consisting of categories with finite products. Writing 0for the terminal object of this category, that is the terminal category, one has byfunctoriality a sequence 00-Cat / / / / / / / / ... / / Explicitly the maps in this diagram are the obvious forgetful functors. The limitof this diagram is formed as in CAT, and provides the definition of the category ω -Cat. Then by theorem(7.6) and proposition(2.8) we have isomorphisms φ n : T ≤ n -Alg → ( T ≤ n -Alg)-CatLet us write Enr for the endofunctor V 7→ V -Cat that we have just been considering.The isomorphisms φ n are natural in the sense of the following lemma, which enablesus to then formally identify the algebras of T in theorem(8.5). LGEBRAS OF HIGHER OPERADS AS ENRICHED CATEGORIES 37
Lemma 8.4.
For n ∈ N let tr n : T ≤ n -Alg →T ≤ n -Alg be the forgetful functor givenby truncation. The square T ≤ n -Alg tr n / / φ n (cid:15) (cid:15) T ≤ n -Alg φ n (cid:15) (cid:15) ( T ≤ n -Alg)-Cat Enr(tr n ) / / ( T ≤ n -Alg)-Cat commutes for all n ∈ N . Proof.
One obtains φ n explicitly as the composite of two isomorphisms T ≤ n -Alg → T ×≤ n -Cat → ( T ≤ n -Alg)-Catthe first of which is described explicitly in the proof of theorem(7.6), and the secondin the proof of proposition(2.8), and using these descriptions one may easily verifydirectly the desired naturality. (cid:3) Theorem 8.5.
For ≤ n ≤ ω , T ≤ n -Alg ∼ = n -Cat . Proof.
Write t : 0-Cat → ω -Catit suffices to provide isomorphisms ψ n : T ≤ n -Alg → n -Cat for n ∈ N natural in thesense that T ≤ n -Alg tr n / / ψ n (cid:15) (cid:15) T ≤ n -Alg ψ n (cid:15) (cid:15) (1 + n )-Cat Enr n ( t ) / / n -Catcommutes for all n . Take ψ = 1 Set and by induction define ψ n as the composite T ≤ n -Alg φ n / / ( T ≤ n -Alg)-Cat Enr( ψ n ) / / (1 + n )-Cat . The case n = 0 for ψ ’s naturality comes from the fact that the isomorphisms thatcomprise φ (see lemma(8.4)) are defined over Set. The inductive step follows easilyfrom lemma(8.4). (cid:3)
9. Acknowledgements
The first author gratefully acknowledges for the financial support of Scott Rus-sell Johnson Memorial Foundation. Both authors are also grateful for the financialsupport of Max Planck Institut f¨ur Mathematik and the Australian Research Coun-cil grant No. DP0558372. This paper was completed while the second author was apostdoc at Macquarie University in Sydney Australia and at the PPS lab in Paris,and he would like to thank these institutions for their hospitality and pleasantworking conditions. We would both like to acknowledge the hospitality of the MaxPlanck Institute where some of this work was carried out. Moreover we are alsoindebted to Clemens Berger, Denis-Charles Cisinski and Paul-Andr´e Melli`es forinteresting discussions on the substance of this paper.
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Department of Mathematics, Macquarie University
E-mail address : [email protected] Laboratoire PPS, Universit´e Paris Diderot – Paris 7
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