I am writing up material that has been taking shape over a number of years. The question is of the type "how much of this has been done before?" The shape of the math is that questions of coherence of categories with multiplication can be given a group theory flavor and so groups are injected in the middle of the discussion. I expect that little recognition will take place after the introduction of the groups, so my question focuses on what happens before the groups show up. I circulated this question a bit with the aid of a pdf file, but this list won't take attachments, so I will have to make do with some latex. The latex follows the signature. Any communication should be by direct email. I am far from a category theorist and do not follow this list. Thanks for any information, Matt Brin \documentclass[oneside]{amsart} \usepackage{amssymb} \usepackage[all]{xy} \DeclareMathAlphabet\EuScript{U}{eus}{m}{n} \newcommand{\scr}[1]{\EuScript{#1}} \begin{document} \newcommand{\ldoublet}{\xy(-4,-2); (0,2)**@{-}; (2,0)**@{-}; (-2,0); (0,-2)**@{-}; \endxy} \newcommand{\rdoublet}{\xy(4,-2); (0,2)**@{-}; (-2,0)**@{-}; (2,0); (0,-2)**@{-}; \endxy} If \(\scr{C}\) is a category with (functoral) mutliplication \(\otimes\), then inside the operad \(End_\scr{C}\) there is a suboperad \(\bigotimes\) derived from the multiplication \(\otimes\) and an obvious surjective map of operads \(h:\scr{T}\rightarrow \bigotimes\) whose domain is the operad of finite binary trees. This map will take, for example, the tree \(\rdoublet\) to the functor \begin{equation} \label {ExFunctA}(X,Y,Z)\mapsto X\otimes(Y\otimes Z)\end{equation} in \(End_{\scr{C}}\) and the tree \(\ldoublet\) to the functor \begin{equation} \label {ExFunctB} (X,Y,Z)\mapsto (X\otimes Y)\otimes Z \end{equation} in \(End_{\scr{C}}\). If there is a natural isomorphism \(\alpha\) given from the functor (\ref{ExFunctA}) to the functor (\ref{ExFunctB}) in \(End_{\scr{C}}\), then the isomorphisms generated in the usual way from (composites of expansions of instances of) \(\alpha\) and \(\alpha^{-1}\) and the identity isomorphisms on the functors in \(\bigotimes\) gives a category structure to \(\bigotimes\). There are now two category structures that we can put on the operad \(\scr{T}\) of finite binary trees. One is a ``pullback'' category structure that we get from the category structure on \(\bigotimes\) where we use \(h:\scr{T}\rightarrow \bigotimes\) to do the pullback. (Morphisms from \(T_1\) to \(T_2\) are just the morphisms from \(h(T_1)\) to \(h(T_2)\).) The other category structure on \(\scr{T}\) is the trivial structure in which every pair of trees with the same number of leaves gets a uniqe (iso)morphism between them in each direction. We let \(\bigotimes^h\) denote the pullback category and reuse the notation \(\scr{T}\) for the trivial category structure. There is a forgetful functor from \(\bigotimes^h\) to \(\scr{T}\) that is the identity on objects. The point of the coherence question is to ask whether this forgetful functor is an isomorphism. At this point we probably leave the realm that might seem familiar. However, I will press on in case the ``probably'' is wrong, and to tell what the point of all this is. Particularly pleasant properties of the operad \(\scr{T}\) allow one to compute two groups: one \(T(\bigotimes^h)\) from \(\bigotimes^h\) and another \(F\) from \(\scr{T}\). The second group is well known and is usually referred to as ``Thompson's group \(F\)'' so we have kept the letter \(F\) for it. There is a surjective homomorphism (call it a comparison homomorphism) \(\sigma\) from \(T(\bigotimes^h)\) to \(F\). The surjectivity is standard and the arguments are in MacLane paper noted below. Under the assumption that the multiplication \(\otimes\) has an identity (an object \(K\) in \(\scr{C}\) with a natural isomorphism \(\iota\) from the identity on \(\scr{C}\) to the functor \(X\mapsto X\otimes K\) with no further restrictions such as the satisfaction of a coherence property on the isomorphism \(\iota\)), then one proves easily that the associativity morphism \(\alpha\) makes the pentagonal diagrams commute if and only if the comparison homomorphism \(\sigma\) is an isomorphism. In fact, once a certain ``non-collapsing'' fact is proven from the existence of the identity \(K\), the rest is just a quote of definitions. Thus \(\scr{C}\) is a monoidal category if and only if the ``identity isomorphism'' \(\iota\) satisfies the usual coherence conditions on identities and the comparison homomorphism \(\sigma\) is an isomorphism. One can do exactly the same thing with symmetric, monoidal categories (in which case the comparison homomorphism is to a well known group known as Thompson's group \(V\)) and braided tensor categories (in which case the comparison homomorphism is to a group \(BV\) of mine that I call the braided version of \(V\)). In the case of symmetric, monoidal catetgories, the argument again boils down to a check of definitions once certain basic facts are established. In the case of braided tensor categories, there are real calculations that must be done since the definition of braided tensor categories reads very differently than it does for monoidal and symmetric, monoidal categories. This ends the summary. I can clarify my question a bit. I am familiar with the paper of MacLane in the Rice journal of 1963 on Natural associativity and commutativity. I am familiar with little else. This pretty much identifies the scope of my question. The language of operads does not appear in MacLane's paper and I am wondering how much of MacLane's results have been reworked to exploit operads and their structures. Referring to the summary above, I am curious about the structures that preceed the introduction of the group \(T(\bigotimes)\). \end{document} -- matt brin / math. dept / SUNY / Binghamton, NY 13902-6000 / (607)-777-2147 FAX: (607)-777-2450 EMAIL: matt@math.binghamton.edu WWW: http://math.binghamton.edu/matt