Subject: categories: Re: ramifications of Goldblatt's notion of a skeleton of a category

From: "Fred E.J. Linton" <[email-masked]>
Date: Sat, 01 Jul 2006 19:14:57 -0400

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
> 
> 
>