Subject: categories: no coherence assumptions

From: Noson Yanofsky <[email-masked]>
Date: Fri, 5 Feb 1999 14:03:37 -0500 (EST)

James Dolan wrote:
>what does "no coherence assumptions at all" mean in this context?
>does it mean that yanofsky is studying what i call "coarse
>n-categories", defined recursively as categories enriched over the
>cartesian closed category where the objects are the coarse
>[n-1]-categories and the morphisms are the enriched natural
>isomorphism classes of enriched functors?

I am not sure whether his "course 2-categories" is my associative 
categories. Here is the abstract of my paper available at

http://xxx.lanl.gov/abs/math.QA/9804106

Obstructions to Coherence:  Natural Noncoherent Associativity

Abstract
We study what happens when coherence fails. Categories
with a tensor product and a natural associativity isomorphism
that does not necessarily satisfy the pentagon coherence
requirements (called associative categories) are considered.
Categorical versions  of associahedra where naturality squares 
commute and pentagons do not are constructed (called Catalan 
groupoids, $\An$). These groupoids are used in the construction of the
free associative category. They are also used in the
construction of the theory of associative categories (given as a 2-sketch).
Generators and relations are given for the fundamental group, $\pi(\An)$, 
of the Catalan groupoids -- thought of as a simplicial complex. These 
groups are shown to be more than just free groups. Each associative 
category, $\bf B$, has related fundamental groups $\pi(\bf
B_n)$ and homomorphisms $\pi(P_n):\pi(\An) \longrightarrow \pi(\bf
B_n)$. If the images of the $\pi(P_n)$ are trivial, i.e. there is only
one associativity path between any two objects, then the category is
coherent. Otherwise the images of $\pi(P_n)$ are obstructions to
coherence. Some progress is made in classifying noncoherence of 
associative categories.

Prof. J. Lambek wrote:
>The tensor product automatically satisfies all functoriality, 
>associativity and coherence conditions, if it is introduced by a 
>universal property as by Bourbaki. 

Yes, but Bourbaki's universal properties required for the  isomorphism 
A(BC) ---> (AB)C is that a(bc) |---> (ab)c (see Algebre II.64). 
Mac Lane's classical counterexample a(bc)|--->(-1)(ab)c 
surely does not satisfy this coherence condition (page 163 of CWM). 
By the way, the fundamental group that corresponds to this
associative category is Z_2 since going around once does not give 
the identity but going around twice does.

John Baez wrote:
>Indeed, if you ask what are the "right" coherence laws, perhaps the 
>easiest answer is: the coherence laws automatically satisfied by universal 
>constructions! 

The whole point of my paper is that there are no "right" coherence laws.
Each structure has interesting theorems that can be proven about it.
Finding the "right" coherence axioms is like finding the "right" axioms to
study symmetry. Which one is "right": semi-groups, groups, Abelian groups
Lie groups etc.? If coherence theory is to be thought of as some type of
higher-dimensional algebra, then there are many structures that are of
interest.

Noson Yanofsky