Reading advice

Reading advice

[Prof. J. Lambek]

Concerning the question by Lindquist:

The tensor product automatically satisfies all functoriality, 
associativity and coherence conditions, if it is introduced by a 
universal property as by Bourbaki. This is shown for monoidal 
categories (bicategories with one object) e.g. in my paper 
``Multicategories revisited'', Contemporary Mathematics 92(1989). The 
same argument works for arbitrary bicategories provided, in defining 
a multicategory, one replaces the free monoid generated by a set by 
the free category generated by a graph.

Jim Lambek

Re: Reading advice

[Vaughan Pratt]

>From: "Prof. J. Lambek" <lambek@math.mcgill.ca>
>Subject: categories: Reading advice
>
>Concerning the question by Lindquist:
>
>The tensor product automatically satisfies all functoriality, 
>associativity and coherence conditions, if it is introduced by a 
>universal property as by Bourbaki.

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?

Much the same can surely be said of naturality, whose abstract essence
is that of 2-cells but which is standardly presented concretely, where
the interchange axiom becomes a not entirely trivial theorem.

Vaughan Pratt