A question about literature on operads and coherence

[Matt Brin <matt@math.binghamton.edu>]

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

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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 \(T(\bigotimes)\).



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-- 
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