Re: question

[Rory Lucyshyn-Wright <rorylw@mathstat.yorku.ca>]

Theorem 1 of IV.4 in Mac Lane's _Categories_for_the_Working_Mathematician_
shows in particular that for a functor S:A-->C, the following are
equivalent:
     (i) S is an equivalence of categories,
   (iii) S is full and faithful, and each object c \in C is isomorphic to
         Sa for some object a \in A.

The proof of the implication (iii)->(i) appears to depend, in general,
upon an analogue of the axiom of choice which is applied with respect to
functions between classes.

We say that S is essentially surjective on objects (e.s.o) if each object
c \in C is isomorphic to Sa for some object a \in A.

Thus, any full subcategory inclusion which is essentially surjective on
objects is an equivalence of categories, and, in particular, such an
inclusion is not only an equivalence but is also injective on objects.

If a full subcategory inclusion A --> C is e.s.o. and, moreover,
each isomorphism class of A is a singleton, then we say that A is a
skeleton of C.

Cheers,
Rory Lucyshyn-Wright

P.s. Perhaps someone on the list could provide a history of the different
formulations of equivalence of categories and the associated terminology,
with appropriate references.

On Sun, 20 Sep 2009, jim stasheff wrote:

> What do you call it when you have   one (small) category being a (full)
> subcategory of another , and every object in the big category is
> isomorphic to one in the small category ? This is the case for the
> category given by objects hom(S,A) ,and morphisms given by the equivalence
> relation hom(T,A) ,as a subcategory of stack(A) . Is there an
> equivalence of categories ?
>
> jim
>

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