Vasili "Bill" [G|H]alchin, in
ramifications of Goldblatt's notion of a skeleton of a category
asks, in connection with
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"
, the following:
1) what "maximal" means in this case? ...some kind of order on all the skeletons of category C?
2) would the "operator" skel be a functor?
A category "is skeletal" if for each isomorphism A --> B the objects A and B are the same object (A = B). With this in mind, the phrase "and only one" in Golblatt's quoted definition is superfluous, being a consequence of (rather than a condition required for) the definition. Once one has a _choice_ of isomorphism from each object of C to the (unique) object of a skeleton of C that it's isomorphic to (but it may take the axiom of choice to be assured of such an iso), any two skeleta of C become isomorphic to each other. There's no inherent "order" among the various possible skeleta of C. Data making the inclusion into C of any full subcategory that is a skeleton of C an equivalence of categories IS precisely such a "choice of isomorphism from each object of C to the (unique) object of a skeleton of C that it's isomorphic to" mentioned above. There's little hope of skel becoming a functor without data of the sort just mentioned. Probably the strong temptation to "wish" that the full inclusion {SKEL} --> {CAT} (of the full subcategory of skeletal categories among all categories) were an equivalence of categories, or at least the inclusion of a full reflexive subcategory (with skel as inverse, or at least, reflection back down), is one to be steer clear of, if at all possible, in general. Cheers, -- Fred (Linton, and as of today, Emeritus from Wesleyan U. :-) ) [original post follows]
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