Re: question

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

I agree with John about the accepted meaning of "replete", but my
understanding of "skeletal" (supported by Mac Lane's "Categories
for the Working Mathematician") is that it means a category in which
every isomorphism class of objects has exactly one member. If I had
to find a word for a subcategory which meets every isomorphism
class of objects of the ambient category (but possibly in more than
one member) I'd call it "representative", or something like that.
A skeleton would then be a subcategory which is full, representative
and skeletal.

Regarding the question of whether such a subcategory is equivalent
to the ambient category, I recall that Peter Freyd once showed that
each of the following statements is equivalent to the axiom of choice:

(a) Every small category has a skeleton.

(b) A small category is equivalent to any of its skeletons.

(c) Any two skeletons of a given small category are isomorphic.

The first equivalence is trivial, but the other two require a bit of
ingenuity. I don't think he ever published this.

Peter Johnstone

On Tue, 22 Sep 2009, John Kennison wrote:

>
> My understanding of this ancient terminology ids that a replete subcategory=
> is one that is closed under the forming of isomorphic copies.A subcategory=
> which contains an isomorphic copy of every object in the containing catego=
> ry is called skeletal
> A subcategory ois both replete and skeletal if and only if it contains all =
> objects of the larger category.
>
> ---John
>
> On 9/22/09 3:04 AM, "Fred Linton" <flinton@wesleyan.edu> wrote:
>
> Jim Stasheff asked,
>
>> 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 ? ...
>
> One adjective that *had* been used for such a subcategory
> (whether small, or full, or not) was "replete". I'll defer
> to others on the question of whether that terminology is
> still in use today, or is ... um ... *deprecated* :-) .
>
> Cheers, -- Fred
>
>
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>


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