Following on from Fred Linton's comment, an easy example is groupoids. For a connected groupoid G, any vertex (object) group G(x)= G(x,x) is skeletal in G. See my book: www.bangor.ac.uk/r.brown/topgpds.html (since I do all the publicity, I have to take every opportunity....!) For many mathematicians, this meant that `groupoids reduced to groups'. But this reduction involves choices, and so cannot be made natural, which takes us back to the first paper on categories by E-M! Heller commented to me in the 1980s that on this reductionist basis, vector spaces reduce to a cardinality. But, as he said, the classification of vector spaces with n endomorphisms is interesting for n=1, hard for n=2, and unknown for n=3. I have not seen a classification of groupoids with one endomorphism! Ronnie ----- Original Message ----- From: "Galchin Vasili" <vigalchin@gmail.com> To: <categories@mta.ca> Sent: Thursday, June 29, 2006 6:29 AM Subject: categories: ramifications of Goldblatt's notion of a skeleton of a category
Hello,
Rob Goldblatt in section 9.2 of his book "Topoi: The Categorical Analysis of Logic" introduces the notion of a "skeleton of a category C" which he defines as a "full subcategory C-sub-zero of C that is skeletal, and such that each C-object is isomorphic to one and only one C-sub-zero object". This statement seems to imply that we can have an "operator":
skel: CAT -> CAT where CAT is the categories of (small) categories
such that
1) skel is idempotent on any member of C of CAT, i.e. ] skel (skel (C)) = skel (C)
2) skel (C) = a "maximal" skeleton of C.
I am struggling with
1) what "maximal" means in this case? E.g. is there some kind of order on all the skeletons of category C?
2) would the "operator" skel be a functor?
Kind regards, Bill Halchin
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