ramifications of Goldblatt's notion of a skeleton of a category

ramifications of Goldblatt's notion of a skeleton of a category

[Galchin Vasili]

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

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

[Bruce Bartlett]

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
>
>

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

[Ronnie Brown]

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
>
>
>
>
>
> --
> Internal Virus Database is out-of-date.
> Checked by AVG Free Edition.
> Version: 7.1.392 / Virus Database: 268.9.0/368 - Release Date: 16/06/2006
>

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

[Fred E.J. Linton]

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
> 
> 
>