Subject: categories: coherence

From: john baez <[email-masked]>
Date: Wed, 3 Feb 1999 18:11:46 -0800 (PST)

J. Lambek writes:

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

Vaughan Pratt writes:

>In view of this would it be fair to say that coherence is not a notion
>intrinsic to category theory, but rather arises from the traditional
>set theoretic presentation (or at least point of view) of category theory?

I think it's fair to say that operations automatically satisfy all
the right coherence laws if you define them using universal properties.
This is the idea behind Jim Dolan's and my definition of weak n-categories:
all the ways of composing cells are defined by means of universal properties,
so one doesn't need to explicitly list coherence laws - they're automatic.

Indeed, if you ask what are the "right" coherence laws, perhaps the 
easiest answer is: the coherence laws automatically satisfied by universal 
constructions!   (There are also some answers coming from homotopy theory
but probably deep down they are the same answer.)

We pound these points home with great rhetorical flourishes in the 
following paper:

Categorification, in Higher Category Theory, eds. Ezra Getzler and 
Mikhail Kapranov, American Mathematical Society, Providence, 1998, 
pp. 1-36.  

Also available electronically at http://math.ucr.edu/home/baez/cat.ps