Fred E.J. Linton wrote:

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.

The situation is actually quite neat. Suppose that, for each category C in Cat, one chooses a skeleton C_0, and isomorphisms from each object to the choice of the skeletal object, as you say. Then, indeed, the operation Skel : Cat -> Cat which sends each category to its chosen skeleton can be made into a (necessarily weak, but thats the interesting part) 2-functor, and can can be extended to an equivalence of 2-categories. Its a special case of a more general notion : if you have a 2-category X, such that every object A in X has an associated object A' and an adjoint equivalence between A and A', then the operation T : X -> X, which (on objects) sends A to A', is in fact a weak 2-functor, and indeed a 2-equivalence. Its all rather nice if you draw it out in string diagrams. Regards, Bruce Bartlett 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