Iterated localization for operads and the n-categorical tensor product problem (cont.)
Michael Batanin – 2 December 1998
Let E be an analytic functor on the category of (n+1)-globular sets Gl_{n+1}. That is we have a cartesian natural transformation from E to the free (n+1)-category functor D. In particular this means that
E(X) ---> D(X) | | | | V V E(1) ---> D(1)= Tr_{n+1}
is a pullback. An (n+1)-operad is just a monoid in the monoidal (with respect to composition) category of analytic functors.
Definition. E is normalized if (E(1))_0 = 1.
This means that application of E to a globular set X does not change the objects of E. Let denote the monoidal category of normalized analytic functors by An^{norm}_{n+1}.
I now can introduce the notion of analytic functor of k-variables. This will be a functor
E_k:Gl_n x ... x Gl_n ---> Gl_n together with a cartesian natural transformation
E_k ---> D x ... x D.
Now consider the following category Seq(An_n). The objects are sequences {E_k, k>=0}, where E_k is an analytic functor of k-variables (for k=0 it is just an object of Gl_n). The morphisms are cartesian natural transformations.
This category has a monoidal structure
(E\otimes F)_m = \coprod_{k,j_1+...+j_k=m} E_k(F_{j_1},...,F_{j_k}).
This is an honest tensor product because analytic functors are additive on each variables (this is different from usual definition of analytic functors on Set).
Theorem. There is a strong monoidal equivalence of categories An^{norm}_{n+1} ~ Seq(An_n}.
To explain this theorem consider the following construction. Every n+1-globular set is just a graph in the category of n-globular set. Now let X_1,..., X_k be n-globular sets. One can consider the following graph G = G(X_1,...,X_k) of n-globular sets. Its set of objects is {0,...,n}. The only nontrivial globular sets of morphisms are G{j-1,j}= X_j. For j=0 we take G to have just 1 object 0 and empty set of morphisms.
Let E be an n+1 normalized analytic functor. Then for n-globular sets X_1,...,X_k put
E_k(X_1,...,X_k) = E(G(X_1,...,X_k))(0,k) The normalization condition is used here because E(G(X_1,...,X_k)) has {0,...,k} as the set of objects.
This generates a sequence of analytic functors. There is also an inverse functor and both these functors preserve tensor product.
Now we can give more general definition. Let C be a category and {E_k} be a sequence of functors
E_k:C x ... x C ---> C We call this sequence an operad on C (what is not very good name, I will be happy if somebody could suggest something else) if there are given a multiplication
E_k(E_j_1,...,E_j_k) ---> E_{j_1 + ... + j_k} and unit
Id ---> E_1
satisfying usual associativity and unitary conditions. (This is a strange mixture of operadic and monadic structure, a sort of "categorification" of operads).
Examples are monoid in Seq(An) and also any tensor product on C (put E_k(X_1,...,X_k)=X_1\otimes...\otimes X_k) and multiplication being asociativity isomorphism).
One can define "categories enriched over C with an operad" in the same way as one define categories enriched over monoidal categories. I denote this category as Cat(C,E).
Now for E\in An^{norm}_{n+1} we can define a corresponding sequence of analitic functors {E_k}. We also can normalize E_1. It means that I consider a suboperad of E_1 which consists of such operation e which have source and target in (E_1)_0 equal to 1 (which is a unit of a monoid (E_1)_0). Let this normalized operad be denoted by E^(norm).
Now for every k we have an action E_k(E^norm,...,E^norm)---> E_k(E_1,...,E_1) ---> E_k
This allows us to construct the following operad \bar{E} on the category of E^norm-algebras . \bar{E} is a coeqalizer in Alg-E^norm of the pair
E_k(E^norm(X_1),...,E^norm(X_k))===> E_k(X_1,...,X_k),
where the second morphism is induced by the structure E^norm-algebras morphisms.
Theorem: The category of E-algebras Alg-E is equivalent to Cat(Alg-E^norm,\bar{E}).
This shows that the theory of weak n-categories can be constructed in the usual iterated manner using this slightly more general notion of enrichment. How is this connected to the tensor product problem?
Let us call an operad E totally normalized if E_{Z^k U_r}=1 for all k and r. This guarantees that we can avoid the normalization in our localization process. In addition, \bar{E_1} is isomorphic to identity and E_0=1. The algebras of such an operad have strict units for all possible compositions in algebras.
Let us call a tensor product an (n+1)-operad E such that \bar{\mu}:\bar{E_k}(\bar{E_j_1},...,\bar{E_j_k} ---> ---> \bar{E_{j_1+...+j_k}} are isomorphisms. The morphisms of tensor products are the operadic morphisms.
So we have a subcategory Tens of Oper. My hypothesis is that the inclusion has a left adjoint. The only think that we need to show for this is that Tensor products are closed with respect to pullbacks. It would be sufficient for this to prove that the operation \bar{-} preserves some special pullbacks (it defenetely does not preserve all pullbacks). If it is true we can prove the following: Let Tot(Tens) be subcategory of Oper such that for every E from Tot(Tens) any of its iterated localizations (...((E_1)_1)_1...)_1 is a tensor product. Then the inclusion functor has a left adjoint L_{tot}
If we have this result then the problem of defining higher analogous of Gray-tensor product may be solved in a few steps. Consider a universal contractible totally normalized operad K (this may be constructed by the same process as I described in my paper). Then we have an operad L_{tot}(K) and this is a tensor product. Moreover, I think this operad is contractible and this is exactly what we need.
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