From: Bruce Bartlett <b.h.bartlett@sheffield.ac.uk>
To: categories@mta.ca
Subject: Re: ramifications of Goldblatt's notion of a skeleton of a category
Date: Tue, 04 Jul 2006 17:14:12 +0100 [thread overview]
Message-ID: <E1Fy5R9-0000u9-9Q@mailserv.mta.ca> (raw)
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
>
>
next reply other threads:[~2006-07-04 16:14 UTC|newest]
Thread overview: 4+ messages / expand[flat|nested] mbox.gz Atom feed top
2006-07-04 16:14 Bruce Bartlett [this message]
-- strict thread matches above, loose matches on Subject: below --
2006-07-03 11:07 Ronnie Brown
2006-07-01 23:14 Fred E.J. Linton
2006-06-29 5:29 Galchin Vasili
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