Subject: categories: strict n-categories, equivalences

From: david carlton <[email-masked]>
Date: Thu, 5 Apr 2001 13:33:48 -0700 (PDT)

I'm trying to figure out the relation between the following notions of 
"equivalence":

We define a strict n-category to be a category enriched in strict
(n-1)-categories.  Then we can make a recursive definition of
"equivalence" as follows: we say that an arrow f: x -> y is an
equivalence if there exists an arrow g: y -> x such that there exist
arrows alpha: fg -> 1_x and beta: gf -> 1_y that are also
equivalences.

Also, we might want to say that a (strict) functor F: C -> D of strict
n-categories is an "equivalence" if:

1) For all objects d in D, there exists an object c of C and an
   equivalence F(c) -> d;
2) For all objects c, c' in C, the map Hom_C(c,c') -> Hom_D(F(c),F(c'))
   (which is a functor of (n-1)-categories) is an equivalence.

(To complete these definitions, I should also say that an arrow in a
1-category is an equivalence iff it's an isomorphism and that a
functor of 0-categories is an equivalence iff it's a bijection.)

Then, my main question is:

Q: Is it the case that an arrow f: x -> y is an equivalence if and only
   if, for all z, the Yoneda functor Hom(z,x) -> Hom(z,y) induced by f
   is an equivalence?

I think and hope that the answer is true, but I'm certainly not 100%
confident; it seems to me that its proof would involve a sufficiently
messy (and picky) simultaneous induction that I'd be quite happy if
somebody else had already worked this out.  If it's not true, are
there any other theorems along this line with improved definitions of
"equivalence" (either on the arrow side or the functor side)?

One warning: my terminology might be badly chosen, especially for
functors, because I don't think that it's the case that, if one
constructs the strict (n+1)-category of all n-categories, that the
functors that I'm calling equivalences are all equivilances as arrows
in that category.

In general, are there any good references for strict n-categories?
(E.g. one giving the details of the construction of nCat.)  I'm not as
conversant with the literature of category theory as I should be, and
I'd like to avoid reinventing the wheel more than necessary.

thanks,
david carlton
[email-masked]