The Lifting Theorem for Multitensors
aa r X i v : . [ m a t h . C T ] J un The Lifting Theorem for Multitensors
Michael Batanin, Denis-Charles Cisinski, and Mark Weber
Abstract.
We continue to develop the theory of [ ] and [ ] on monadsand multitensors. The central result of this paper – the lifting theorem formultitensors – enables us to see the Gray tensor product of 2-categories andthe Crans tensor product of Gray categories as part of our emerging framework.Moreover we explain how our lifting theorem gives an alternative descriptionof Day convolution [ ] in the unenriched context. Contents
1. Multitensors and functor operads 12. The lifting theorem 33. Multitensor lifting made explicit 74. Gray and Crans tensor products 135. Contractibility 146. Standard convolution 16Appendix A. Transfinite constructions in monad theory 18Acknowledgements 25References 25
1. Multitensors and functor operads
This paper continues the developments of [ ] and [ ] on the interplay betweenmonads and multitensors in the globular approach to higher category theory. Totake an important example, according to [ ] there are two related combinatorialobjects which can be used to describe the notion of Gray category. One has themonad A on the category G ( Set ) of 3-globular sets whose algebras are Gray cat-egories, which was first described in [ ]. On the other hand there is a multitensor(ie a lax monoidal structure) on the category G ( Set ) of 2-globular sets, such thatcategories enriched in E are exactly Gray categories. The theory described in [ ]explains how A and E are related as part of a general theory which applies to alloperads of the sort defined originally in [ ].However there is a third object which is missing from this picture, namely,the Gray tensor product of 2-categories. It is a simpler object than A and E ,and categories enriched in 2-Cat for the Gray tensor product are exactly Graycategories. The purpose of this paper is to exhibit the Gray tensor product as part of our emerging framework. This is done by means of the lifting theorem formultitensors – theorem(7) of this article.Recall [
2, 16 ] that a multitensor ( E, u, σ ) on a category V consists of n-arytensor product functors E n : V n → V , whose values on objects are denoted in anyof the following ways E ( X , ..., X n ) E n ( X , ..., X n ) E ≤ i ≤ n X i E i X i depending on what is most convenient, together with unit and substitution maps u X : Z → E X σ X ij : E i E j X ij → E ij X ij for all X , X ij from V which are natural in their arguments and satisfy the obviousunit and associativity axioms. It is also useful to think of ( E, u, σ ) more abstractlyas a lax algebra structure on V for the monoid monad M on CAT , and so todenote E as a functor E : M V → V . The basic example to keep in mind is that ofa monoidal structure on V , for in this case E is given by the n -ary tensor products, u is the identity and the components of σ are given by coherence isomorphisms forthe monoidal structure.A category enriched in E consists of a V -enriched graph X together with com-position maps κ x i : E i X ( x i − , x i ) → X ( x , x n )for all n ∈ N and sequences ( x , ..., x n ) of objects of X , satisfying the evident unitand associativity axioms. With the evident notion of E -functor (see [ ]), one hasa category E -Cat of E -categories and E -functors together with a forgetful functor U E : E -Cat → G V. When E is a distributive multitensor, that is when E n commutes with coproductsin each variable, one can construct a monad Γ E on G V over Set . The object mapof the underlying endofunctor is given by the formulaΓ EX ( a, b ) = a a = x ,...,x n = b E i X ( x i − , x i ) , the unit u is used to provide the unit of the monad and σ is used to provide themultiplication. The identification of the algebras of Γ E and categories enriched in E is witnessed by a canonical isomorphism E -Cat ∼ = G ( V ) Γ E over G V . This con-struction, the senses in which it is 2-functorial, and its respect of various categoricalproperties, is explained fully in [ ]. We use the notation and terminology of [ ]freely.If one restricts attention to unary operations, then E , u and the components σ X : E X → E X provide the underlying endofunctor, unit, and multiplicationfor a monad on V . This monad is called the unary part of E . When the unarypart of E is the identity monad, the multitensor is a functor operad . This coincideswith existing terminology, see [ ] for instance, except that we don’t in this paperconsider any symmetric group actions. Since units for functor operads are identities,we denote any such as a pair ( E, σ ), where as for general multitensors E denotesthe functor part and σ the substitution.By definition then, a functor operad is a multitensor. On the other hand, asobserved in [ ] lemma(2.7), the unary part of a multitensor E acts on E , in the HE LIFTING THEOREM FOR MULTITENSORS 3 sense that as a functor E factors as M V V E V / / U E / / and in addition, the substitution maps are morphisms of E -algebras. Moreover an E -category structure on a V -enriched graph X includes in particular an E -algebrastructure on each hom X ( a, b ) of X with respect to which the composition mapsare morphisms of E -algebras. These observations lead to Question 1.
Given a multitensor (
E, u, σ ) on a category V can one find a functoroperad ( E ′ , σ ′ ) on V E such that E ′ -categories are exactly E -categories?The main result of this paper, theorem(7), says that question(1) has a niceanswer: when E is distributive and accessible and V is cocomplete, one can indeedfind a unique distributive accessible such E ′ . Moreover as we will see in section(6),this construction generalises Day convolution [ ] and some of its lax analogues [ ].Perhaps the first appearance of a case of our lifting theorem in the literature,that does not involve convolution, is in the work of Ginzburg and Kapranov onKoszul duality [ ]. Formula (1.2.13) of that paper, in the case of a K -collection E coming from an operad, implicitly involves the lifting of the multitensor corre-sponding (as in [ ] example(2.6)) to the given operad. For instance, our liftingtheorem gives a satisfying general explanation for why one must tensor over K inthat formula.This paper is organised in the following way. The lifting theorem is provedin section(2), using some transfinite constructions from monad theory which arerecalled in appendix(A). The lifted functor operad is unpacked explicitly in sec-tion(3). In section(4) we explain how the Gray tensor product of 2-categories andCrans tensor product of Gray categories is obtained as a lifting via theorem(7).Part of the interplay between monads and multitensors described in [ ] coverscontractible multitensors and their relation to the contractible operads of [ ]. Insection(5) we extend this analysis to the lifted multitensors, and in example(25)explain how this gives a different proof of the contractibility of the operad for Graycategories. In section(6) we explain how Day convolution, in the unenriched setting,can also be obtained via our theorem(7).
2. The lifting theorem
The idea which enables us to answer question(1) is the following. Given adistributive multitensor E on V one can consider also the multitensor f E whoseunary part is also E , but whose non-unary parts are all constant at ∅ . This isclearly a sub-multitensor of E , also distributive, and moreover as we shall see onehas f E -Cat ∼ = G ( V E ) over G V . Thus from the inclusion f E ֒ → E one induces theforgetful functor U fitting in the commutative triangle G ( V E ) E -Cat G V o o U U E (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) (cid:31) (cid:31) G ( U E ) ?????? For sufficiently nice V and E this forgetful functor has a left adjoint. The categoryof algebras of the induced monad T will be E -Cat since U is monadic. Thus problemis reduced to that of establishing that this monad T arises from a multitensor on MICHAEL BATANIN, DENIS-CHARLES CISINSKI, AND MARK WEBER V E . Theorem(42) of [ ] gives the properties that T must satisfy in order thatthere is such a multitensor, and gives an explicit formula for it in terms of T .In the interplay between multitensors and monads described in [ ] the con-struction E Γ E of a monad on G V over Set from a distributive multitensorprovides the object map of 2-functorsΓ : DISTMULT → MND(
CAT / Set )Γ ′ : OpDISTMULT → OpMND(
CAT / Set ) . That the monads (
S, η, µ ) on G V that arise from this construction are “over Set ”means that for all X ∈ G V , the V -graph SX has the same object set as X , andthe components of the unit η and multiplication µ are identities on objects. The-orem(42) of [ ] alluded to above characterises the monads on G V over Set ofthe form Γ E as those which are distributive and path-like in the sense of defini-tions(41) and (38) of [ ] respectively. Note that the properties of distributivityand path-likeness concern only the functor part of a given monad on G V over Set .On the way to the proof of theorem(7) below, it is necessary to have available thesedefinitions for functors over
Set between categories of enriched graphs.Suppose that categories V and W have coproducts. Recall that a finite sequence( Z , ..., Z n ) of objects of V may be regarded as a V -graph whose object set is { , ..., n } , hom from ( i −
1) to i is Z i for 1 ≤ i ≤ n , and other homs are initial.Then a functor T : G V → G W over Set determines a functor T : M V → W whoseobject map is given by T ( Z , ..., Z n ) = T ( Z , ..., Z n )(0 , n ) . By definition T amounts to functors T n : V n → W for each n ∈ N , and onemay consider the various categorical properties that such T may enjoy, as in thediscussion of [ ] section(4.3). Definition 2.
Let V and W be categories with coproducts. A functor T : G V →G W over Set is distributive when for each n ∈ N , T n preserves coproducts in eachvariable.Given a V -graph X and sequence x = ( x , ..., x n ) of objects of X , one can definethe morphism x : ( X ( x , x ) , X ( x , x ) , ..., X ( x n − , x n )) → X whose object map is i x i , and whose hom map between ( i −
1) and i is theidentity. For all such sequences x one has T ( x ) ,n : T i X ( x i − , x i ) → T X ( x , x n )and so taking all sequences x starting at a and finishing at b one induces thecanonical map π T,X,a,b : a a = x ,...,x n = b T i X ( x i − , x i ) → T X ( a, b )in W . Definition 3.
Let V and W be categories with coproducts. A functor T : G V →G W over Set is path-like when for all X ∈ G V and a, b ∈ X , the maps π T,X,a,b areisomorphisms.
HE LIFTING THEOREM FOR MULTITENSORS 5
Clearly a monad (
T, η, µ ) on G V over Set is distributive (resp. path-like) in thesense of [ ] iff the underlying endofunctor T is so in the sense just defined. Lemma 4.
Let V , W and Y be categories with coproducts and R : V → W , T : G V → G W and S : G W → G Y be functors.(1) If R preserves coproducts then G R is distributive and path-like.(2) If S and T are distributive and path-like, then so is ST . Proof. (1): Since R preserves the initial object one has G R ( Z , ..., Z n ) =( RZ , ..., RZ n ) and so G R : M V → W sends sequences of length n = 1 to ∅ , andits unary part is just R . Thus G R is distributive since R preserves coproducts, andcoproducts of copies of ∅ are initial. The summands of the domain of π G R,X,a,b areinitial unless ( x , ..., x n ) is the sequence ( a, b ), thus π G R,X,a,b is clearly an isomor-phism, and so G R is path-like.(2): Since S and T are path-like and distributive one has ST ( Z , ..., Z n )(0 , n ) ∼ = a r ≤ ... ≤ r m = n S ≤ i ≤ m T r i − Let V be a category with coproducts, W be a cocomplete category, J bea small connected category and F : J → [ G V, G W ] be a functor. Suppose that F sends objects and arrows of J functors and naturaltransformations over Set . (1) Then the colimit K : G V →G W of F may be chosen to be over Set .Given such a choice of K : (2) If F j is path-like for all j ∈ J , then K is also path-like. (3) If F j is distributive for all j ∈ J , then K is also distributive. Proof. Colimits in [ G V, G W ] are computed componentwise from colimits in G W and so for X ∈ G V we must describe a universal cocone with components κ X,j : F j ( X ) → KX. By remark(21) of [ ] we may demand that the κ X,j are identities on objects, andthen compute the hom of the colimit between a, b ∈ X by taking a colimit cocone { κ X,j } a,b : F j ( X )( a, b ) → KX ( a, b ) MICHAEL BATANIN, DENIS-CHARLES CISINSKI, AND MARK WEBER in W . This establishes (1). Since the properties of path-likeness and distributiv-ity involve only colimits at the level of the homs as does the construction of K just given, (2) and (3) follow immediately since colimits commute with colimits ingeneral. (cid:3) Recall the structure-semantics result of Lawvere, which says that for any cate-gory E , the canonical functorMnd( E ) op → CAT / E T U : E T → E with object map indicated is fully faithful (see [ ] for a proof). An importantconsequence of this is that for monads S and T on E , an isomorphism E T ∼ = E S over E is induced by a unique isomorphism S ∼ = T of monads. We now have allthe pieces we need to implement our strategy. First, in the following lemma, wegive the result we need to recognise the induced monad on G ( V E ) as arising froma multitensor. Lemma 6. Let λ be a regular cardinal. Suppose that V is a cocomplete category, R is a coproduct preserving monad on V , S is a λ -accessible monad on G V over Set , and φ : G R → S is a monad morphism over Set . Denote by T the monad on G ( V R ) induced by φ ! ⊣ φ ∗ . (1) One may choose φ ! so that T is over Set .Given such a choice of φ ! : (2) If S is distributive and path-like then so is T . (3) If R is λ -accessible then so is T . Proof. Let us denote by ρ : RU R → U R the 2-cell datum of the Eilenberg-Moore object for R , and note that by [ ] lemma(16) one may identify U G R = G ( U R ) and G ρ as the 2-cell datum for G R ’s Eilenberg-Moore object. Now T is over Set iff G ( U R ) T is. Moreover since R preserves coproducts U R creates them, andso T is path-like and distributive iff G ( U R ) T is. Since G ( U R ) T = U S φ ! , it followsthat T is over Set , path-like and distributive iff U S φ ! is. From corollary(32) thebeginning of the transfinite construction in [ G ( V R ) , G V ] giving U S φ ! is depicted in S G ( RU R ) S G ( U R ) S Q S Q ...S G ( RU R ) S G ( U R ) SQ SQ ...S G ( RU R ) S G ( U R ) Q Q ... S G ( ρ ) / / S µ S S ( φ ) G ( U R ) / / / / / / / / S G ( ρ ) / / Sµ S S ( φ ) G ( U R ) / / / / / / / / S G ( ρ ) / / µ S S ( φ ) G ( U R ) / / / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) T a (cid:15) (cid:15) O O (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) O O (cid:15) (cid:15) O O (cid:15) (cid:15) O O $ $ JJJJJJJJ $ $ JJJJJJJJ $ $ JJJJJJJJ $ $ JJJJJJJJJ $ $ JJJJJJJJJ $ $ JJJJJJJJJ Since the monads S and G ( R ) are over Set , as are ρ and φ , it follows by a transfiniteinduction using lemma(5) that all successive stages of this construction give functorsand natural transformations over Set , whence U S φ ! is itself over Set . Lemma(4)ensures that the functors G ( R ) and G ( RU R ) are distributive and path-like, since R preserves coproducts and U R creates them. When S is also distributive andpath-like, then by the same sort of transfinite induction using lemmas(4) and (5),all successive stages of this construction give functors that are distributive andpath-like, whence U S φ ! is itself distributive and path-like. HE LIFTING THEOREM FOR MULTITENSORS 7 Supposing R to be λ -accessible, note that G ( R ) is also λ -accessible. One way tosee this is to consider the distributive multitensor ˜ R on V whose unary part is R andnon-unary parts are constant at ∅ . Thus ˜ R will be λ -accessible since R is. To givean ˜ R -category structure on X ∈ G V amounts to giving R -algebra structures to thehoms of X , and similarly on morphisms, whence one has a canonical isomorphism˜ R -Cat ∼ = G ( V R ) over G V . By proposition(26) of [ ] together with structure-semantics one obtains Γ ˜ R ∼ = G R . Thus by [ ] theorem(29), G R is indeed λ -accessible. But then it follows that U G R = G ( U R ) creates λ -filtered colimits, andso T is λ -accessible iff G ( U R ) T = U S φ ! is. In the transfinite construction of U S φ ! ,it is now clear that the functors involved at every stage are λ -accessible by yetanother transfinite induction, and so U S φ ! is λ -accessible as required.To finish the proof we must check that T ’s monad structure is over Set . Since µ T is a retraction of η T T it suffices to verify that η T is over Set , which is equivalentto asking that the components of G ( U R ) η T are identities on objects. Writing q : S → U S φ ! for the transfinite composite constructed as part of the definition of φ ! recall from the end of section(A.3) that one has a commutative square G ( RU R ) G ρ / / φ G ( U R ) (cid:15) (cid:15) G U R G ( U R ) η T (cid:15) (cid:15) S G ( U R ) q / / U S φ ! Now G ρ and φ G ( U R ) are over Set by definition, and q is by construction, so theresult follows. (cid:3) Theorem 7. ( Multitensor lifting theorem ) Let λ be a regular cardinal and let E be a λ -accessible distributive multitensor on a cocomplete category V . Then thereis, to within isomorphism, a unique functor operad ( E ′ , σ ′ ) on V E such that(1) ( E ′ , σ ′ ) is distributive.(2) E ′ -Cat ∼ = E -Cat over G V .Moreover E ′ is also λ -accessible. Proof. Write ψ : f E ֒ → E for the multitensor inclusion of the unary partof E , and then apply lemma(6) with S = Γ E , R = E and φ = Γ ψ to producea λ -accessible distributive and path-like monad T on G ( V E ) over Set . Thus by[ ] proposition(40) and theorem(42), T is a distributive multitensor on V E with T -Cat ∼ = E -Cat. Moreover since T ∼ = Γ T it follows by [ ] theorem(29) that T is λ -accessible. As for uniqueness suppose that ( E ′ , σ ′ ) is given as in the statement.Then by [ ] theorem(42) Γ( E ′ ) is a distributive monad on G ( V E ) and one has G ( V E ) Γ( E ′ ) ∼ = E -Catover G ( V E ). By structure-semantics one has an isomorphism Γ( E ′ ) ∼ = T of monads,and thus by [ ] proposition(43), an isomorphism E ′ ∼ = T of multitensors. (cid:3) 3. Multitensor lifting made explicit Let us now instantiate the constructions of section(A.3) to produce a more ex-plicit description of the functor operad E ′ produced by theorem(7). Beyond mereinstantiation this task amounts to reformulating everything in terms of hom mapswhich live in V , because in our case the colimits being formed in G V at each stage MICHAEL BATANIN, DENIS-CHARLES CISINSKI, AND MARK WEBER of the construction are connected colimits diagrams whose morphisms are all iden-tity on objects. Moreover these fixed object sets are of the form { , ..., n } for n ∈ N . Notation . We shall be manipulating sequences of data and so we describe heresome notation that will be convenient. A sequence ( a , ..., a n ) from some set will bedenoted more tersely as ( a i ) leaving the length unmentioned. Similarly a sequenceof sequences (( a , ..., a n ) , ..., ( a k , ..., a kn k ))of elements from some set will be denoted ( a ij ) – the variable i ranges over 1 ≤ i ≤ k and the variable j ranges over 1 ≤ j ≤ n i . Triply-nested sequences look like this ( a ijk ),and so on. These conventions are more or less implicit already in the notation wehave been using all along for multitensors. See especially section( ?? ) and [ ]. Wedenote by con( a i ,...,i k )the ordinary sequence obtained from the k -tuply nested sequence ( a i ,...,i k ) byconcatenation. In particular given a sequence ( a i ), the set of ( a ij ) such thatcon( a ij ) = ( a i ) is just the set of partitions of the original sequence into doubly-nested sequences, and will play an important role below. This is because to givethe substitution maps for a multitensor E on V , is to give maps σ : E i E j X ij → E i X i for all ( X ij ) and ( X i ) from V such that con( X ij ) = ( X i ).The monad map φ : M → S is taken as Γ( ψ ) : G E → Γ( E ) where ψ : f E ֒ → E is the inclusion of the unary part of the multitensor E . The role of ( X, x ) in V M is played by sequences ( X i , x i ) of E -algebras regarded as objects of G ( V E ). Thetransfinite induction produces for each ordinal m and each sequence of E -algebrasas above of length n , morphisms v ( m )( X i ,x i ) : SQ m ( X i , x i ) → Q m +1 ( X i , x i ) q ( m )( X i ,x i ) : Q m ( X i , x i ) → Q m +1 ( X i , x i ) q ( To summarise, the output of the transfinite process we are going to describe is,for each ordinal m , the following data. For each sequence ( X i , x i ) of E -algebras,one has an object E (m) i ( X i , x i )and morphisms v ( m )( X ij ,x ij ) : E i E (m) j ( X ij , x ij ) → E (m+1) i ( X i , x i ) q ( m )( X i ,x i ) : E (m) i ( X i , x i ) → E (m+1) i ( X i , x i ) q ( Let V be a cocomplete category, λ a regular cardinal, and E a dis-tributive λ -accessible multitensor on V . Then for any ordinal m with | m | ≥ λ onemay take (E (m) i ( X i , x i ) , a ( X i , x i )) where the action a ( X i , x i ) is given as the composite E E (m) i ( X i , x i ) E (m+1) i ( X i , x i ) E (m) i ( X i , x i ) v ( m ) / / ( q ( m ) ) − / / as an explicit description of the object map of the lifted multitensor E ′ on V E . In corollaries (33) and (36), in which the initial data is a monad map φ : M → S between monads on a category V together with an algebra ( X, x ) for M , we notedthe simplification of our constructions when S and S preserve the coequaliser(2) SM X SX Q X Sx / / µ S S ( φ ) / / q / / in V , which is part of the first step of the inductive construction of φ ! . In thepresent situation the role of V is played by the category G V , the role of S is playedby Γ E , and the role of ( X, x ) played by a given sequence ( X i , x i ) of E -algebras,and so the role of the coequaliser (2) is now played by the coequaliser(3) Γ E ( E X i ) Γ E ( X i ) Q Sx / / µ S S ( φ ) / / q (0) / / HE LIFTING THEOREM FOR MULTITENSORS 11 in G V . Here we have denoted by Q the V -graph with objects { , ..., n } and homsgiven by Q ( i, j ) = ( ∅ if i > j E (1) i Let F : A × ... ×A n →B be a functor. If F preserves reflexive co-equalisers in each variable separately then F preserves reflexive coequalisers. and this can be proved by induction on n using the 3 × ] lemma(1). The most well-known instance of this is Corollary 11. [ ] Let V be a biclosed monoidal category. Then the n -fold tensorproduct of reflexive coequalisers in V is again a reflexive coequaliser. In particular note that by corollary(10) a multitensor E preserves (some class of)reflexive coequalisers iff it preserves them in each variable separately.Returning to our basic coequalisers an immediate consequence of the explicitdescription of Γ E and corollary(10) is Lemma 12. Let E be a distributive multitensor on V a cocomplete category, and ( X i , x i ) a sequence of E -algebras. If E preserves the basic coequalisers associated toall the subsequences of ( X i , x i ) , then for all r ∈ N , (Γ E ) r preserves the coequaliser(3). and applying this lemma and corollary(33) gives Corollary 13. Let V be a cocomplete category, λ a regular cardinal, E a distributive λ -accessible multitensor on V and ( X i , x i ) a sequence of E -algebras. If E preservesthe basic coequalisers associated to all the subsequences of ( X i , x i ) , then one maytake E ′ i ( X i , x i ) = (E (1) i ( X i , x i ) , a ) where the action a is defined as the unique map such that aE ( q (0) ) = q (0) σ . Note in particular that when the sequence ( X i , x i ) of E -algebras is of length n = 0or n = 1, the associated basic coequaliser is absolute. In the n = 0 case the basiccoequaliser is constant at E , and when n = 1 the basic coequaliser may be takento be the canonical presentation of the given E -algebra. Thus in these cases itfollows from corollary(13) that E ′ = ( E , σ ) and E ′ ( X, x ) = ( X, x ). Reformulatingthe explicit description of the unit in corollary(36) one recovers the fact from ourexplicit descriptions, that the unit of E ′ is the identity, which was of course trueby construction.To complete the task of giving a completely explicit description of the multi-tensor E ′ we now turn to unpacking its substitution. So we assume that E is adistributive λ -accessible multitensor on V a cocomplete category, and fix an ordinal m so that | m | ≥ λ , so that E ′ may be constructed as E ( m ) as in corollary(8). Bytransfinite induction on r we shall generate the following data: σ ( r ) X ij ,x ij : E (r) i (E (m) j ( X ij ) , x ij ) → E (m) i ( X i , x i )and σ ( r +1) X ij ,x ij whenever con( X ij , x ij ) = ( X i , x i ), such thatE i E (r) j E (m) k E (r+1) ij E (m) k E (m) ijk E i E (m) jk v ( r ) E ( m ) / / σ ( r +1) (cid:15) (cid:15) / / ( q ( m ) ) − v ( m ) (cid:15) (cid:15) E i σ ( r ) commutes. Initial step . Define σ (0) to be the identity and σ (1) as the unique map suchthat σ (1) q (0) = ( q ( m ) ) − v ( m ) by the universal property of the coequaliser q (0) . Inductive step . Define σ ( r +2) as the unique map such that σ ( r +2) ( v ( r +1) E ( m ) ) = ( q ( m ) ) − v ( m ) (E i σ ( r +1) )using the universal property of v ( r +1) as a coequaliser. Limit step . When r is a limit ordinal define σ ( r ) as induced by the µ ( s ) for HE LIFTING THEOREM FOR MULTITENSORS 13 s < r and the universal property of E ( r ) as the colimit of the sequence of the E ( s ) for s < r . Then define σ ( r +1) as the unique map such that σ ( r +1) ( v ( r ) E ( m ) ) = ( q ( m ) ) − v ( m ) (E i σ ( r ) )using the universal property of v ( r ) as a coequaliser.The fact that the transfinite construction just specified was obtained from thatfor corollary(35), by taking S = Γ E and looking at the homs, means that by corol-laries (35) and (36) one has Corollary 14. Let V be a cocomplete category, λ a regular cardinal, E a distributive λ -accessible multitensor on V and ( X i , x i ) a sequence of E -algebras. Then one has σ ′ ( X i ,x i ) = σ ( m )( X i ,x i ) as an explicit description of the substitution of E ′ . If moreover E preserves thebasic coequalisers of all the subsequences of ( X i , x i ) , then one may take σ (1)( X i ,x i ) asthe explicit description of the substitution. 4. Gray and Crans tensor products Let A be a T ≤ n +1 -operad over Set and let E be the associated T ×≤ n -multitensor.Thus by definition one has A = Γ E E = A and G n +1 ( Set ) A ∼ = E -Cat over G n +1 ( Set ). The monad E on G n Set has as alge-bras, the structure borne by the homs of an A -algebra. Theorem(7) produces thefunctor operad E ′ on G n ( Set ) E such that G n +1 ( Set ) A ∼ = E ′ -Cat ∼ = E -Catover G n ( Set ) E . Moreover E ′ is the unique such functor operad which is distribu-tive. Example 15. When A is the terminal T ≤ n +1 -operad, E is the terminal T ×≤ n -multitensor, and so E = T ≤ n . Since strict ( n +1)-categories are categories enrichedin n -Cat using cartesian products, and these commute with coproducts (in fact allcolimits), it follows by the uniqueness part of theorem(7) that E ′ is just the cartesianproduct of n -categories. Example 16. Suppose that E is a multitensor on V and T is an opmonoidal monadon ( V, E ). Then one has by theorem(49) of [ ] a lifted multitensor E ′ on V T . Onthe other hand if moreover V is cocomplete, E is a distributive and accessiblefunctor operad, and T is coproduct preserving and accessible, then E ′ may also beobtained by applying theorem(7) to the composite multitensor EM ( T ). When E is given by cartesian product and T = T ≤ n , we recover example(15). Example 17. Take A to be the T ≤ -operad for Gray categories constructed in[ ] (example(4) after corollary(8.1.1)). Since E is the monad on G ( Set ) for 2-categories, in this case E ′ is a functor operad for 2-categories. However the Graytensor product of 2-categories [ ] is part of a symmetric monoidal closed structure.Thus it is distributive as a functor operad, and since Gray categories are categoriesenriched in the Gray tensor product by definition, it follows that E ′ is the Gray tensor product. In other words, the general methods of this paper have succeededin producing the Gray tensor product of 2-categories from the operad A . Example 18. In [ ] Sjoerd Crans explicitly constructed a tensor product on thecategory of Gray-categories. This explicit construction was extremely complicated.It is possible to exhibit the Crans tensor product as an instance of our generaltheory, by rewriting his explicit constructions as the construction of the T ≤ -operad A whose algebras are teisi in his sense. The associated multitensor E has E equalto the T ≤ -operad for Gray categories. Thus theorem(7) constructs a functor operad E ′ of Gray categories whose enriched categories are teisi. Since the tensor productexplicitly constructed by Crans is distributive, the uniqueness of part of theorem(7)ensures that it is indeed E ′ , since teisi are categories enriched in the Crans tensorproduct by definition.Honestly writing the details of the T ≤ -operad of example(18) is a formidabletask and we have omitted this here. In the end though, such details will not beimportant, because such a tensor product (or more properly a biclosed versionthereof) will only be really useful once it is constructed in a conceptual way as partof a general inductive machine. 5. Contractibility5.1. Functoriality and comparison. Recall [ ] [ ] that when a 2-category K has Eilenberg-Moore objects, the one and 2-cells of the 2-category MND( K ) ad-mit another description. Given monads ( V, T ) and ( W, S ) in K , to give a monadfunctor ( H, ψ ) : ( V, T ) → ( W, S ), is to give ˜ H : V T → W S such that U S ˜ H = HU T .This follows immediately from the universal property of Eilenberg-Moore objects.Similarly to give a monad 2-cell φ : ( H , ψ ) → ( H , ψ ) is to give φ : H → H and˜ φ : ˜ H → ˜ H commuting with U T and U S . Note that Eilenberg-Moore objects in CAT / Set are computed as in CAT , and we shall soon apply these observationsto the case K = CAT / Set . Remark 19. Suppose we have a lax monoidal functor ( H, ψ ) : ( V, E ) → ( W, F ).Then we obtain a commutative diagram E -Cat G ( V E ) G V G W G ( W F ) F -Cat / / / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) of forgetful functors in CAT / Set . If moreover V and W are cocomplete and E and F are distributive and accessible, then by theorem(7) we have distributivemultitensors E ′ and F ′ on V E and V F respectively, and from the left-most squareabove we have a monad morphism ( G V, Γ E ′ ) → ( G W, Γ F ′ ) with underlying functor G ( ψ ∗ ). By [ ] proposition(44) this monad functor is the result of applying Γ to aunique lax monoidal functor( ψ ∗ , ψ ′ ) : ( V E , E ′ ) → ( W F , F ′ ) . Arguing similarly for monoidal transformations and monad 2-cells, one finds thatthe assignment ( V, E ) ( V E , E ′ ), for cocomplete V and accessible E , is 2-functorial. HE LIFTING THEOREM FOR MULTITENSORS 15 Remark 20. Suppose that ε : E → T ×≤ n is a T ×≤ n -multitensor. Applying remark(19)in the case V = W = G n Set , H = id, ψ = ε and example(15) one obtains a map ε ′ ( X i ,x i ) : E i ′ ε ∗ ( X i , x i ) → Y i ε ∗ ( X i , x i )of E -algebras for each sequence (( X , x ) , ..., ( X n , x n )) of strict n -categories, since ε ∗ : n -Cat → E -Alg as a right adjoint preserves products. This gives a generalcomparison map between the functor operad E ′ produced by theorem(7) and carte-sian products, defined for sequences of E -algebras that underlie strict n -categories. Example 21. When E is the multitensor of example(17) for Gray categories, E is itself T ≤ and ε =id, and ε ′ gives the well-known comparison map from the Graytensor product of 2-categories to the cartesian product, which we recall is actuallya componentwise biequivalence.Returning to the situation of remark(19), it is routine to unpack the assignment( H, ψ ) ( ψ ∗ , ψ ′ ) as in section(3) and so obtain the following 1-cell counterpart ofcorollary(13). Corollary 22. Let ( H, ψ ) : ( V, E ) → ( W, F ) be a lax monoidal functor such that V and W are cocomplete, and E and F are accessible. Let ( X , ..., X n ) be a sequenceof objects of V . Then the component of ψ ′ at the sequence ( E X , ..., E X n ) of free E -algebras is just ψ X i . In section(7.3) of [ ] we sawthat a T ≤ n +1 -operad A over Set is contractible iff its associated T ×≤ n -multitensor E is contractible. We now extend this result to the associated functor operad E ′ on G n ( Set ) E . Proposition 23. Let ( H, ψ ) : ( V, E ) → ( W, F ) be a lax monoidal functor betweendistributive lax monoidal categories, and I a class of maps in W . Suppose that W is extensive, H preserves coproducts and the codomains of maps in I are connected.Then the following statements are equivalent (1) ψ is a trivial I -fibration. (2) Γ ψ is a trivial I + -fibration.and moreover when in addition V and W are cocomplete and E and F are accessible,these conditions are also equivalent to (3) The components of U F ψ ′ at sequences ( E X , ..., E X n ) of free E -algebrasare trivial I -fibrations. Proof. (1) ⇔ (2) is proposition(58) of [ ] and (2) ⇔ (3) follows immediatelyfrom corollary(22). (cid:3) Recall from [ ] section(7.2) that I ≤ n is the set of boundary inclusions of m -globes for m ≤ n . Then proposition(23) has the immediate Corollary 24. Let ≤ n ≤ ∞ , α : A → T ≤ n +1 be an n +1 -operad over Set and ε : E → T ×≤ n be the corresponding n -multitensor. TFSAE:(1) α : A → T ≤ n +1 is contractible.(2) ε : E → T ×≤ n is contractible. (3) The components of ε ′ ( X i ,x i ) of remark(20) are trivial I ≤ n -fibrations of n -globular sets, when the ( X i , x i ) are free strict n -categories. The equivalence of (1) and (2) appeared already as [ ] corollary(59). Example 25. When α is the T ≤ -operad for Gray-categories, the contractibilityof α is a consequence of the fact that the canonical 2-functors from the Gray to thecartesian tensor product are identity-on-object biequivalences. 6. Standard convolution6.1. Recalling convolution for multicategories. The set of multimaps( X , ..., X n ) → Y in a given multicategory C shall be denoted as C ( X , ..., X n ; Y ).Recall that a linear map in C is a multimap whose domain is a sequence of length 1.The objects of C and linear maps between them form a category, which we denoteas C l , and we call this the linear part of C . The set of objects of C is denoted as C . Given objects A , ..., A n , ......, A k , ..., A kn k B , ..., B k C of C , we denote by σ A,B,C : C ( B , ..., B k ; C ) × Y i C ( A i , ..., A in i ; B i ) → C ( A , ..., A kn k ; C )the substitution functions of the multicategory C . One thus induces a function σ A,C : Z B ,...,B k C ( B , ..., B k ; C ) × Y i C ( A i , ..., A in i ; B i ) → C ( A , ..., A kn k ; C )in which for the purposes of making sense of this coend, the objects B , ..., B k areregarded as objects of the category C l . A promonoidal category in the sense of Day[ ], in the unenriched context, can be defined as a multicategory C such that theseinduced functions σ A,C are all bijective. A promonoidal structure on a category D is a promonoidal category C such that C l = D op .A lax monoidal category ( V, E ) is cocomplete when V is cocomplete as a cat-egory and E n : V n → V preserves colimits in each variable for all n ∈ N . In thissituation the multitensor E is also said to be cocomplete. When C is small it definesa functor operad on the functor category [ C l , Set ] whose tensor product F is givenby the coend F i X i = Z C ,...,C n C ( C , ..., C n ; − ) × Y i X i C i and substitution is defined in the evident way from that of C . By proposition(2.1)of [ ] F is a cocomplete functor operad and is called the standard convolution structure of C on [ C l , Set ]. By proposition(2.2) of [ ], for each fixed category D ,standard convolution gives an equivalence between multicategories on C such that C l = D and cocomplete functor operads on [ D , Set ], which restricts to the well-known [ ] equivalence between promonoidal structures on D op and closed monoidalstructures on [ D , Set ].We have recalled these facts in a very special case compared with the generalityat which this theory is developed in [ ]. In that work all structures are consideredas enriched over some nice symmetric monoidal closed base V , and moreover ratherthan D = C l as above, one has instead an identity on objects functor D → C l . The HE LIFTING THEOREM FOR MULTITENSORS 17 resulting combined setting is then what are called V -substitudes in [ ], and in the V = Set case the extra generality of the functor D → C l , corresponds at the level ofmultitensors, to the consideration of general closed multitensors on [ D , Set ] insteadof mere functor operads. In this section we shall recover standard convolution, forthe special case that we have described above, from the lifting theorem. Given a multicategory C we define the multi-tensor E on [ C , Set ] via the formula (cid:18) E ≤ i ≤ n X i (cid:19) ( C ) = a C ,...,C n C ( C , ..., C n ; C ) × Y ≤ i ≤ n X i ( C i ) using the unit and compositions for C in the evident way to give the unit u andsubstitution σ for E . When C has only one element, this is the multitensor on Set coming from the operad P described in [ ] and [ ], whose tensor product isgiven by the formula E ≤ i ≤ n X i = P n × X × ... × X n . An E -category with one object is exactly an algebra of the coloured operad P inthe usual sense. A general E -category amounts to a set X , sets X ( x , x )( C ) forall x , x ∈ X and C ∈ C , and functions(5) C ( C , ..., C n ; C ) × Y i X ( x i − , x i )( C i ) → X ( x , x n )( C )compatible in the evident way with the multicategory structure of C . On the otherhand an F -category amounts to a set X , sets X ( x , x )( C ) natural in C , andmaps as in (5) but which are natural in C , ..., C n , C , and compatible with C ’smulticategory structure. However this added naturality enjoyed by an F -categoryisn’t really an additional condition, because it follows from the compatibility withthe linear maps of C . Thus E and F -categories coincide, and one may easily extendthis to functors and so give E -Cat ∼ = F -Cat over G [ C , Set ].The unary part of E is given on objects by E ( X )( C ) = a D C l ( D, C ) × X ( D )which should be familiar – E is the monad on [ C , Set ] whose algebras are functors C l → Set , and may be recovered from left kan extension and restriction alongthe inclusion of objects C ֒ → C l . Thus the category of algebras of E may beidentified with the functor category [ C l , Set ]. Since the multitensor E is clearlycocomplete, it satisfies the hypotheses of theorem(7), and so one has a uniquefinitary distributive multitensor E ′ on [ C l , Set ] such that E -Cat ∼ = E ′ -Cat over G [ C , Set ]. By uniqueness we have Proposition 26. Let C be a multicategory, F be the standard convolution structureon [ C l , Set ] and E be the multitensor on [ C , Set ] defined above. Then one has anisomorphism F ∼ = E ′ of multitensors. In particular when C is a promonoidal category proposition(26) expresses clas-sical Day convolution, in the unenriched context, as a lift in the sense of theorem(7). Appendix A. Transfinite constructions in monad theoryA.1. Overview. Here we review some of the transfinite constructions in monadtheory that we will use in sections(2) and (3). An earlier reference for these mat-ters is [ ]. However due to the technical nature of this material, and our needfor its details when we come to making our constructions explicit, we feel that it isappropriate to give a rather thorough account of this background. A.2. Coequalisers in categories of algebras. Let T be a monad on acategory V that has filtered colimits and coequalisers and let( A, a ) ( B, b ) g / / f / / be morphisms in V T . We shall now construct morphisms v n : T Q n → Q n +1 q n : Q n → Q n +1 q T v $ $ JJJJJJJJ v $ $ JJJJJJJJJ T v $ $ JJJJJJJJ v $ $ JJJJJJJJJ T v $ $ JJJJJJJJ v $ $ JJJJJJJJJ Initial step . Define q < to be the identity, q to be the coequaliser of f and g , q < = q and v = q b . Note also that q = v η B . Inductive step . Assuming that v n , q n and q Let us write q m,n : Q m → Q n , q ′ m,n : T Q m → colim m If the sequence stabilises at n then it stabilises at any m ≥ n , andmoreover one has an isomorphism of sequences between the given sequence ( Q m , q m ) and the following one: Q ... Q n Q n ... q / / / / id / / id / / Proof. We show for m ≥ n that q m and q m +1 are isomorphisms, and pro-vide the component isomorphisms i m : Q m → Q n of the required isomorphism ofsequences, by transfinite induction on m . We define i m to be the identity when m ≤ n . In the initial step m = n , q m and q m +1 are isomorphisms by hypothesisand we define i n +1 = q n . In the inductive step when m ≥ n is a non-limit ordinal,we must show that q m +2 is an isomorphism and define i m +2 = q m +1 i m +1 . The keypoint is that(7) v m +1 µ = v m +1 T ( q − m +1 ) T ( v m +1 )because with this equation in hand one defines q ′ m +2 : Q m +3 → Q m +2 as the uniquemorphism satisfying q ′ m +2 v m +2 T ( q m +1 ) = v m +1 using the universal property of v m +2 , and then it is routine to verify that q ′ m +2 = q − m +2 . So for the inductive stepit remains to verify (7). But we have v m +1 µT ( q m ) = v m +1 T ( q m ) µ = v m +1 T ( v m ) = v m +1 T ( q − m +1 ) T ( q m +1 v m )= v m +1 T ( q − m +1 ) T ( v m +1 ) T ( q m )and so (7) follows since q m is an isomorphism. In the case where m is a limitordinal, we have stabilisation at m ′ established whenever n ≤ m ′ < m by theinduction hypothesis. Thus the colimit defining Q m is absolute (ie preserved by allfunctors) since its defining sequence from the position n onwards consists only ofisomorphisms. Thus q m and q m +1 are isomorphisms by lemma(27). By induction,the previously constructed i m ′ ’s provide a cocone on the defining diagram of Q m with vertex Q n , thus one induces the isomorphism i m compatible with the earlier i m ′ ’s and defines i m +1 = q m i m . (cid:3) Lemma 29. If the sequence stabilises at n then ( Q n , q − n v n ) is a T -algebra and q The unit law for ( Q n , q − n v n ) is immediate from the definition of q n and the associative law is the commutativity of the outside of the diagram on theleft T Q n T Q n Q n +1 Q n Q n +1 T Q n T Q n +1 Q n +2 T Q n +1 µ / / v n (cid:15) (cid:15) q − n (cid:15) (cid:15) / / q − n / / v n (cid:15) (cid:15) T q − n (cid:15) (cid:15) T v n T q n u u jjjjj v n +1 (cid:15) (cid:15) q − n +1 (cid:15) (cid:15) v n +1 , , YYYYY q − n +1 eeeee T B T Q n Q n +1 Q n B T q T q T Q m T Q nb ? ? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) q T b ggggggggg T q Theorem 30. Let V be a category with filtered colimits and coequalisers, T be amonad on V and ( A, a ) ( B, b ) g / / f / / be morphisms in V T . If T is λ -accessible for some regular cardinal λ , then q Take the smallest such ordinal n – it is necessarily a limit ordinal, and T and T by hypothesis preserve the defining colimit of Q n . Thus by lemmas(27)and (29) the result follows in this case, and in general by lemmas(28) and (29). (cid:3) Finally we mention the well-known special case when the above transfiniteconstruction is particularly simple, that will be worth remembering. Proposition 31. Let V be a category with filtered colimits and coequalisers, T bea monad on V and ( A, a ) ( B, b ) g / / f / / be morphisms in V T . If T and T preserve the coequaliser of f and g in V , thenthe sequence ( Q n , q n ) stabilises at . Denoting by w : T Q → Q the unique mapsuch that wT ( q ) = q b , q : ( B, b ) → ( Q , w ) is the coequaliser of f and g in V T . Proof. Refer to the diagram in V above that describes the first few steps ofthe construction of ( Q n , q n ). Since q and T q are epimorphisms, the T -algebraaxioms for ( Q , w ) follow from those for ( B, b ), and q is a T -algebra map bydefinition. Thus w is the coequaliser in V of µ Q and T w , and since T q is anepimorphism it is also the coequaliser of µ Q T ( q ) and T ( w ) T ( q ) = T v , but sois v , and so q is the canonical isomorphism between them. To see that q is alsoinvertible, apply the same argument with the composite q q in place of q . Theresult now follows by lemma(29). (cid:3) A.3. Monads induced by monad morphisms. Suppose that ( M, η M , µ M )and ( S, η S , µ S ) are monads on a category V , and φ : M → S is a morphism ofmonads. Then one has an obvious forgetful functor φ ∗ : V S → V M and one can askwhether φ ∗ has a left adjoint which, when it exists, we denote as φ ! . By the Dubucadjoint triangle theorem [ ], one may compute the value of φ ! at an M -algebra( X, x : M X → X ) as a reflexive coequaliser( SM X, µ SMX ) ( SX, µ SX ) φ ! ( X, x ) µ SX S ( φ X ) / / Sη MX o o Sx / / q ( X,x ) / / in V S , when this coequaliser exists. Thus by theorem(30) for the existence of φ ! itsuffices that V admit filtered colimits and coequalisers, and S be accessible. Withthe aid of section(A.2) we shall now give an explicit description of the composite U S φ ! under these hypotheses. To do so we shall construct morphisms v n,X,x : SQ n ( X, x ) → Q n +1 ( X, x ) q n,X,x : Q n ( X, x ) → Q n +1 ( X, x ) q Inductive step . Assuming that v n , q n and q Suppose that V admits filtered colimits and coequalisers, M and S are monads on V , φ : M → S is a morphism of monads, and ( X, x ) is an M -algebra.If moreover S is λ -accessible for some regular cardinal λ , then for any ordinal n such that | n | ≥ λ one may take φ ! ( X, x ) = ( Q n ( X, x ) , q − n v n ) q Suppose that under the hypotheses of corollary(32) that S and S preserve the coequaliser of µ SX S ( φ X ) and Sx in V . Then the sequence ( Q n , q n ) stabilises at , and writing w : SQ → Q for the unique map such that wS ( q ) = q µ SX , one may take φ ! ( X, x ) = ( Q ( X, x ) , w ) q : ( SX, µ X ) → ( Q ( X, x ) , w ) as an explicit definition of φ ! ( X, x ) and the associated coequalising map in V S . Remark 34. Here is a degenerate situation in which corollary(33) applies. Since U M φ ∗ = U S we have φ ! F M ∼ = F U , but another way to view this isomorphism asarising is to apply the corollary in the case where ( X, x ) is a free M -algebra, say( X, x ) = ( M Z, µ MZ ), for in this case one has the dotted arrows in SM Z SM Z SZ ( µ S M )( SφM ) / / Sµ M / / o o SMη M µ S S ( φ ) / / o o Sη M exhibiting µ SZ S ( φ Z ) as a split coequaliser, and thus absolute. Let us denote by ( T, η T , µ T ) the monad on V M induced by the adjunction φ ! ⊣ φ ∗ . We now give an explicit description of this monad. Let ( X, x ) be in V M , suppose S is λ -accessible and fix an ordinal n such that | n | ≥ λ . Then bycorollary(32) one may take T ( X, x ) = ( Q n ( X, x ) , a ( X, x ) φ Q n ( X,x ) ) a ( X, x ) = ( q − n ) Q n ( X,x ) ( v n ) Q n ( X,x ) as the definition of the endofunctor T . Note that ( Q n ( X, x ) , a ( X, x )) is just a morerefined notation for φ ! ( X, x ). Referring to the diagram M X M X XQ n SXSM X Mx / / µ MX / / x / / Sx / / µ SX S ( φ X ) / / q Since by definition µ T ( X,x ) underlies an S -algebra map, to finish the proof that ourdefinition really does describe the multiplication of T , it suffices by the universalproperty of η T to show that µ T ( X,x ) η TT ( X,x ) is the identity, and this is easily achievedusing the defining diagrams of µ T and η T together.Let us now describe η T and (especially) µ T in terms of the transfinite data thatgives Q n ( X, x ). To do so we shall for each ordinal m provide η ( m +1)( X,x ) : X → Q m +1 ( X, x ) µ ( m )( X,x ) : Q m ( Q n ( X, x ) , a ( X, x ) φ ) → Q n ( X, x )and µ ( m +1)( X,x ) in V such that µ ( m +1) v m = a ( X, x ) S ( µ ( m ) ), by transfinite induction on m . Initial step . Define µ (0)( X,x ) to be the identity, and η (1)( X,x ) and µ (1)( X,x ) as the uniquemorphisms such that η (1)( X,x ) x = ( q ) ( X,x ) φ X µ (1)( X,x ) ( q ) ( Q n ,aφ ) = a ( X, x )by the universal properties of x and q (as the evident coequalisers) respectively. Inductive step . Define η ( m +2) = q m +1 η ( m +1) and µ ( m +2) as the unique map HE LIFTING THEOREM FOR MULTITENSORS 25 satisfying µ ( m +2) v m +1 = a ( X, x ) S ( µ ( m +1) ) using the universal property of v m +1 asa coequaliser. Limit step . When m is a limit ordinal define η ( m )( X,x ) and µ ( m )( X,x ) as the mapsinduced by the η ( r ) and µ ( r ) for r < m and the universal property of Q m ( X, x ) asthe colimit of the sequence of the Q r for r < m . Then define η ( m +1) = q m η ( m ) and µ ( m +2) as the unique map satisfying µ ( m +2) v m +1 = a ( X, x ) S ( µ ( m +1) ) usingthe universal property of v m +1 as a coequaliser.The fact that the induction just given was obtained by unpacking the descriptionsof η T and µ T of the previous paragraph in terms of the transfinite construction ofthe endofunctor T (ie the Q m ( X, x )), is expressed by Corollary 35. Suppose that V admits filtered colimits and coequalisers, M and S are monads on V , φ : M → S is a morphism of monads, and ( X, x ) is an M -algebra.If moreover S is λ -accessible for some regular cardinal λ , then for any ordinal n such that | n | ≥ λ one may take T ( X, x ) = ( Q n ( X, x ) , a ( X, x ) φ Q n ( X,x ) ) η T ( X,x ) = η ( n )( X,x ) µ T ( X,x ) = µ ( n )( X,x ) as constructed above as an explicit description underlying endofunctor, unit andmultiplication of the monad generated by the adjunction φ ! ⊣ φ ∗ . and the simplification coming from proposition(31) gives Corollary 36. Under the hypotheses of corollary(35), if for ( X, x ) ∈ V M , S and S preserve the coequaliser of µ SX S ( φ X ) and Sx in V , then one may take T ( X, x ) = ( Q ( X, x ) , wφ ) η T ( X,x ) = η (1)( X,x ) µ T ( X,x ) = µ (1)( X,x ) with w as constructed in corollary(33). Acknowledgements We would like to acknowledge Clemens Berger, Richard Garner, Andr´e Joyal,Steve Lack, Joachim Kock, Jean-Louis Loday, Paul-Andr´e Melli`es, Ross Street andDima Tamarkin for interesting discussions on the substance of this paper. Wewould also like to acknowledge the Centre de Recerca Matem`atica in Barcelona forthe generous hospitality and stimulating environment provided during the thematicyear 2007-2008 on Homotopy Structures in Geometry and Algebra. The first authorwould like to acknowledge the financial support on different stages of this project ofthe Scott Russell Johnson Memorial Foundation, the Australian Research Council(grant No.DP0558372) and L’Universit´e Paris 13. The second author would liketo acknowledge the support of the ANR grant no. ANR-07-BLAN-0142. 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Available athttp://sites.google.com/site/markwebersmaths/, 2011. Department of Mathematics, Macquarie University E-mail address : [email protected] Departement des Mathematiques, Universit´e Paris 13 Villanteuse E-mail address : [email protected] Department of Mathematics, Macquarie University E-mail address ::