Yes, now that you mention it, I agree that this is how the word "skeletal" is used. John ________________________________________ From: Prof. Peter Johnstone [P.T.Johnstone@dpmms.cam.ac.uk] Sent: Wednesday, September 23, 2009 6:00 AM To: John Kennison Cc: Fred Linton; categories@mta.ca Subject: Re: categories: Re: question 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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