Re: question

[Ross Street <street@ics.mq.edu.au>]

Jim:

I don't understand your context precisely. (I heard a talk the other  
day where the speaker used "category" to mean "A_{infinity}-category"  
without any explanation.) However I can tell a story which uses some  
of the words you have.

Without the axiom of choice (such as in a topos), there are two  
different conditions on a functor f : A --> X for it to be an  
"equivalence":

1) there is a functor g : X --> A such that f g and g f are isomorphic  
to identity functors; and,

2) f is full, faithful and essentially surjective on objects (this  
last means each object of X is isomorphic to a value of f).

Clearly 1) implies 2). The converse holds when epis split (Ax Choice)  
in the ambient world.

Stacks are designed not to see the difference between equivalences of  
types 1) and 2); that is, if you hom out of an equivalence of type 2)  
into a stack [for an appropriate topology], you get an equivalence of  
type 1).

See old papers of Paré, Bunge, Joyal, . . .

Ross

On 20/09/2009, at 11:21 PM, 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 ?


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