question

question

[jim stasheff]

What do you call it when you have   one (small) category being a (full)
subcategory of another , and every object in the big category is
isomorphic to one in the small category ? This is the case for the
category given by objects hom(S,A) ,and morphisms given by the equivalence
relation hom(T,A) ,as a subcategory of stack(A) . Is there an
equivalence of categories ?

jim



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Re: question

[John Kennison]

Yes, now that you mention it, I agree that this is how the word "skeletal" is used.

John
________________________________________
From: Prof. Peter Johnstone [P.T.Johnstone@dpmms.cam.ac.uk]
Sent: Wednesday, September 23, 2009 6:00 AM
To: John Kennison
Cc: Fred Linton; categories@mta.ca
Subject: Re: categories: Re: question

I agree with John about the accepted meaning of "replete", but my
understanding of "skeletal" (supported by Mac Lane's "Categories
for the Working Mathematician") is that it means a category in which
every isomorphism class of objects has exactly one member. If I had
to find a word for a subcategory which meets every isomorphism
class of objects of the ambient category (but possibly in more than
one member) I'd call it "representative", or something like that.
A skeleton would then be a subcategory which is full, representative
and skeletal.

Regarding the question of whether such a subcategory is equivalent
to the ambient category, I recall that Peter Freyd once showed that
each of the following statements is equivalent to the axiom of choice:

(a) Every small category has a skeleton.

(b) A small category is equivalent to any of its skeletons.

(c) Any two skeletons of a given small category are isomorphic.

The first equivalence is trivial, but the other two require a bit of
ingenuity. I don't think he ever published this.

Peter Johnstone

On Tue, 22 Sep 2009, John Kennison wrote:

>
> My understanding of this ancient terminology ids that a replete subcategory=
> is one that is closed under the forming of isomorphic copies.A subcategory=
> which contains an isomorphic copy of every object in the containing catego=
> ry is called skeletal
> A subcategory ois both replete and skeletal if and only if it contains all =
> objects of the larger category.
>
> ---John
>
> On 9/22/09 3:04 AM, "Fred Linton" <flinton@wesleyan.edu> wrote:
>
> Jim Stasheff asked,
>
>> What do you call it when you have one (small) category being a (full)
>> subcategory of another, and every object in the big category is
>> isomorphic to one in the small category ? ...
>
> One adjective that *had* been used for such a subcategory
> (whether small, or full, or not) was "replete". I'll defer
> to others on the question of whether that terminology is
> still in use today, or is ... um ... *deprecated* :-) .
>
> Cheers, -- Fred
>

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Re: question

[Prof. Peter Johnstone]

I agree with John about the accepted meaning of "replete", but my
understanding of "skeletal" (supported by Mac Lane's "Categories
for the Working Mathematician") is that it means a category in which
every isomorphism class of objects has exactly one member. If I had
to find a word for a subcategory which meets every isomorphism
class of objects of the ambient category (but possibly in more than
one member) I'd call it "representative", or something like that.
A skeleton would then be a subcategory which is full, representative
and skeletal.

Regarding the question of whether such a subcategory is equivalent
to the ambient category, I recall that Peter Freyd once showed that
each of the following statements is equivalent to the axiom of choice:

(a) Every small category has a skeleton.

(b) A small category is equivalent to any of its skeletons.

(c) Any two skeletons of a given small category are isomorphic.

The first equivalence is trivial, but the other two require a bit of
ingenuity. I don't think he ever published this.

Peter Johnstone

On Tue, 22 Sep 2009, John Kennison wrote:

>
> My understanding of this ancient terminology ids that a replete subcategory=
> is one that is closed under the forming of isomorphic copies.A subcategory=
> which contains an isomorphic copy of every object in the containing catego=
> ry is called skeletal
> A subcategory ois both replete and skeletal if and only if it contains all =
> objects of the larger category.
>
> ---John
>
> On 9/22/09 3:04 AM, "Fred Linton" <flinton@wesleyan.edu> wrote:
>
> Jim Stasheff asked,
>
>> What do you call it when you have one (small) category being a (full)
>> subcategory of another, and every object in the big category is
>> isomorphic to one in the small category ? ...
>
> One adjective that *had* been used for such a subcategory
> (whether small, or full, or not) was "replete". I'll defer
> to others on the question of whether that terminology is
> still in use today, or is ... um ... *deprecated* :-) .
>
> Cheers, -- Fred
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: question

[John Kennison]

My understanding of this ancient terminology ids that a replete subcategory is one that is closed under the forming of isomorphic copies.A subcategory which contains an isomorphic copy of every object in the containing category is called skeletal
A subcategory ois both replete and skeletal if and only if it contains all objects of the larger category.

---John

On 9/22/09 3:04 AM, "Fred Linton" <flinton@wesleyan.edu> wrote:

Jim Stasheff asked,

> What do you call it when you have one (small) category being a (full)
> subcategory of another, and every object in the big category is
> isomorphic to one in the small category ? ...

One adjective that *had* been used for such a subcategory
(whether small, or full, or not) was "replete". I'll defer
to others on the question of whether that terminology is
still in use today, or is ... um ... *deprecated* :-) .

Cheers, -- Fred


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Re: question

[Robin Adams]

On Sunday 20 September 2009 14:21:13 jim stasheff wrote:
> What do you call it when you have   one (small) category being a (full)
> subcategory of another , and every object in the big category is
> isomorphic to one in the small category ? This is the case for the
> category given by objects hom(S,A) ,and morphisms given by the equivalence
> relation hom(T,A) ,as a subcategory of stack(A) .

In Adámek, Herrlich and Strecker's book "The Joy of Cats", the small category 
is said to be an "isomorphism-dense" subcategory of the big category.  I don't 
know how widespread this terminology is, though.

> Is there an equivalence of categories ?

Yes.  Whenever A is a full, isomorphism-dense subcategory of B, then the 
inclusion functor from A to B is an equivalence (Remark 4.10 in that book).

--
Robin Adams <robin@cs.rhul.ac.uk>
Royal Holloway, University of London


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Re: question

[Fred Linton]

Jim Stasheff asked,

> What do you call it when you have one (small) category being a (full)
> subcategory of another, and every object in the big category is
> isomorphic to one in the small category ? ...

One adjective that *had* been used for such a subcategory
(whether small, or full, or not) was "replete". I'll defer
to others on the question of whether that terminology is
still in use today, or is ... um ... *deprecated* :-) .

Cheers, -- Fred




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Re: question

[Ross Street]

Jim:

I don't understand your context precisely. (I heard a talk the other  
day where the speaker used "category" to mean "A_{infinity}-category"  
without any explanation.) However I can tell a story which uses some  
of the words you have.

Without the axiom of choice (such as in a topos), there are two  
different conditions on a functor f : A --> X for it to be an  
"equivalence":

1) there is a functor g : X --> A such that f g and g f are isomorphic  
to identity functors; and,

2) f is full, faithful and essentially surjective on objects (this  
last means each object of X is isomorphic to a value of f).

Clearly 1) implies 2). The converse holds when epis split (Ax Choice)  
in the ambient world.

Stacks are designed not to see the difference between equivalences of  
types 1) and 2); that is, if you hom out of an equivalence of type 2)  
into a stack [for an appropriate topology], you get an equivalence of  
type 1).

See old papers of Paré, Bunge, Joyal, . . .

Ross

On 20/09/2009, at 11:21 PM, jim stasheff wrote:

> What do you call it when you have   one (small) category being a  
> (full) subcategory of another , and every object in the big category  
> is isomorphic to one in the small category ? This is the case for  
> the category given by objects hom(S,A) ,and morphisms given by the  
> equivalence
> relation hom(T,A) ,as a subcategory of stack(A) . Is there an  
> equivalence of categories ?


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Re: question

[Rory Lucyshyn-Wright]

Theorem 1 of IV.4 in Mac Lane's _Categories_for_the_Working_Mathematician_
shows in particular that for a functor S:A-->C, the following are
equivalent:
     (i) S is an equivalence of categories,
   (iii) S is full and faithful, and each object c \in C is isomorphic to
         Sa for some object a \in A.

The proof of the implication (iii)->(i) appears to depend, in general,
upon an analogue of the axiom of choice which is applied with respect to
functions between classes.

We say that S is essentially surjective on objects (e.s.o) if each object
c \in C is isomorphic to Sa for some object a \in A.

Thus, any full subcategory inclusion which is essentially surjective on
objects is an equivalence of categories, and, in particular, such an
inclusion is not only an equivalence but is also injective on objects.

If a full subcategory inclusion A --> C is e.s.o. and, moreover,
each isomorphism class of A is a singleton, then we say that A is a
skeleton of C.

Cheers,
Rory Lucyshyn-Wright

P.s. Perhaps someone on the list could provide a history of the different
formulations of equivalence of categories and the associated terminology,
with appropriate references.

On Sun, 20 Sep 2009, jim stasheff wrote:

> What do you call it when you have   one (small) category being a (full)
> subcategory of another , and every object in the big category is
> isomorphic to one in the small category ? This is the case for the
> category given by objects hom(S,A) ,and morphisms given by the equivalence
> relation hom(T,A) ,as a subcategory of stack(A) . Is there an
> equivalence of categories ?
>
> jim
>

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RE: Question

[S.J.Vickers]

Dusko says:
> it's interesting that almost no one took michael healy's bait (attached).
> 

Here's a message I sent to Michael Healey privately:

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
There's a minefield of hotly held opinions behind this question, but let me
try to give a couple of pragmatic considerations.

1. What is the most appropriate characterization (for present purposes) of
collections? By elements or by functions? Set theory postulates that a
collection is entirely determined by its elements. If functions are better,
that's beginning to look more like a categorical approach.

Examples:
Topology - a space is not fully defined by saying what its points are. You
only capture continuity when you look at the maps between spaces.

Type theory - syntactic terms in general have free variables in them,
effectively parameters, and these really correspond to functions rather than
simple elements.

2. Do you want to consider a variety of logics? Set theory as such is
solidly classical. To relax that you need to consider different formal
systems, and then category theory usually provides tools for achieving
better presentation independence than more syntactic approaches.
<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<

In this hurried response to Michael's posting I was trying to get across
some sense of how category theory can help you see the wood for the trees -
the "better presentation independence" is what Dusko refers to as
"structuralist maths".

> for software engineers, foundations are the link with the meaning of their
> programs. having a slightly shorter history than math, they do not have
> languages as natural as arithmetic, or calculus, but have to chose between
> KIF and Ontolingua, and the various other versions of esperanto every day.
> categories dam the flood of structure in software engineering, just like
> they originally did in homology theory almost 60 years ago. some good math
> may come out of it if taken from a good side.

I agree.

I looked up keywords like SUO (Standard Upper Ontology) and KIF (Knowledge
Interchange Format) - they are about standard notations for expressing
information - and some key issues seemed to be rather basic things like the
meaning of first order logic.

I found this depressing. As we know, categorical logic has some very clear
accounts of this that have the great virtue of bringing out deep structure
over superficialities of the logical syntax. It also makes it easier to
question whether "self-evident" notation and axioms are actually meaningful
and valid.

But how can we bring these insights to the software engineering masses? It's
not enough to say "first learn about category theory and then look for
adjunctions, monads and <favourite keyword> everywhere". With this approach
it seems all too easy to prescribe people categorical logic as a cure for
their myopia, and then to find them trying to use it as reading glasses and
wondering why it doesn't work.

One of the things that makes Mac Lane's book such a masterpiece is the way
it uncovers category theory as something already understood rather than
presenting it as something new. The working mathematician is well familiar
with reasoning about adjunctions and monads in special cases, and it's
"just" a matter of uncovering the underlying pattern.

Speaking for myself, after all these years I still understand a lot of
category theory not in the abstract but through paradigms. Enriched
categories? They're just rings, really. Or, at least, ringoids. Presheaves?
They're just modules. Yoneda embedding? The free module on one generator is
just the ring itself.

The software engineer does not have the working mathematician's body of
knowledge. I think to give them category theory we first have to explain,
without mentioning categories, just why structure has to reside not inside
the objects but amongst the morphisms: to explain a collection by its
elements alone is not enough, even in a very basic logical framework.

Steve Vickers
Department of Pure Maths
Faculty of Maths and Computing
The Open University
-----------
Tel: 01908-653144
Fax: 01908-652140
Web: http://mcs.open.ac.uk/sjv22

Re: Question

[Michael J. Healy 425-865-3123]

I'd like to respond to Dusko's note with an open message to all; I'll try 
to be brief.  I'm grateful to Dusko for his posting, and also to the people 
who responded to me privately as I'd requested.  My only reason for requesting 
private replies was to avoid any intrusion people might feel if I started 
another thread such as the "battle" to which Dusko referred:
> 
> it's interesting that almost no one took michael healy's bait (attached).
> a couple of years ago, a similar question started a long battle between a
> group of categorists from this list, and a group of set theorists on
> another list. traces of that battle can still be found scattered on the
> web.
> 
I mean to send a compilation of the responses I received to all respondents 
because they've all expressed interest, either explicitly or implicitly by 
responding.  I've been short of time and haven't done this yet, partly 
because I also want to ask each individually if it's OK to use his/her 
name attached to the response (otherwise, said response will be included 
as "anonymous").  I will get to it within the next few weeks.  In the 
meantime, I've enjoyed this as a learning experience!

Thank you,
Mike

--

===========================================================================
                                         e	     
Michael J. Healy                          A
                                  FA ----------> GA
(425)865-3123                     |              |
FAX(425)865-2964                  |              |
                               Ff |              | Gf
c/o The Boeing Company            |              |   
PO Box 3707  MS 7L-66            \|/            \|/
Seattle, WA 98124-2207            '              '
USA                               FB ----------> GB
-or for priority mail-                   e             "I'm a natural man."
2760 160th Ave SE  MS 7L-66               B
Bellevue, WA 98008
USA

michael.j.healy@boeing.com          -or-            mjhealy@u.washington.edu

============================================================================

Re: Question

[Dusko Pavlovic]

it's interesting that almost no one took michael healy's bait (attached).
a couple of years ago, a similar question started a long battle between a
group of categorists from this list, and a group of set theorists on
another list. traces of that battle can still be found scattered on the
web.

i guess people got a bit tired. after all, for a working mathematician,
foundations are a bit like esperanto: ok, all math can be translated to
set theory, or to category theory, or to untyped lambda calculus --- so
what? do foundations help me calculate an integral? does anyone use von
neumann representation of numbers in arithmetic? no.

but guys, note that the question this time comes from cs.boeing.com! if
people at boeing think about categories, and sets, and foundations, then
this probably makes a difference for them, and helps them compute
something. might this not be worth your attention?

so let me say what i think. i think foundations mean different things for
different people.

for logicians, foundations are metamathematics: analyzing consistency,
independance etc of logical theories. this is what set theory was found to
be good for.

category theory, on the other hand, is not as handy for proving new
independence results, but it tells you to look for adjunctions everywhere,
or monads, or <the keyword from your last paper>. it is not
metamathematics, but perhaps *structuralist maths*: it displays abstract
structures... (i kno, this is getting to *just* the kind of philosophy you
were hoping to avoid by the synchronized silence. so let me make the
point.)

for software engineers, foundations are the link with the meaning of their
programs. having a slightly shorter history than math, they do not have
languages as natural as arithmetic, or calculus, but have to chose between
KIF and Ontolingua, and the various other versions of esperanto every day.
categories dam the flood of structure in software engineering, just like
they originally did in homology theory almost 60 years ago. some good math
may come out of it if taken from a good side.

-- dusko



"Michael J. Healy 425-865-3123" wrote:

> I'd like to ask category theorists how they would answer the attached message
> from a colleague here.  Both he and the person with whom he is corresponding
> are experts in the areas of knowledge representation within computer science
> (ontologies and the like).  I thought it best to hide their identities since
> I haven't asked permission to use them.  If you are interested, please respond
> to me privately if you would.
>
> Thank you,
> Mike Healy
> ------------------------------------------------------------------------------
>
> Message I received:----
>
> I would be delighted if there was no semantic conflict between
> category theory and set theory.  I kind of flagged this as a
> potential issue, but did not look  into it in detail, as it was
> not my main concern at the time.  However, I remain unconvinced.
> There has been some discussion of using set theory as the basis
> for a semantics for SUOKIF. If this is true, then I think it may
> be limiting to a CT based language. While it may be true that sets
> are common example of a catagory, my sense is that CT is much more
> powerful, and would be LIMITED if everything was forced into the
> single catagory of sets.
>
> Im a bit out of my element here, however, and need to defer to the
> formal expertise of others on this issue.
>
> Message to which the above was replying:---
>
> I agree that category theory is very powerful and could be
> an important basis for combining and sharing ontologies.
> But I disagree with the following point:
>
> >I think this idea has tremendous potential.  One problem is that the underlying
>
> >formal semantics of category theory is NOT set theory (which is what KIF uses),
>
> >furthermore, I think they may well be incompatible.
>
> First-order logic (including any and all notations for it,
> such as KIF, CGs, predicate calculus, existential graphs, etc.)
> is completely neutral with respect to set theory or category
> theory.  The version 3.0 of KIF did include a version of set
> theory, but that was removed in the KIF'99 version because it
> belongs to ontology rather than logic.
>
> And for that matter, there is no reason why you can't use both
> category theory and set theory together.  In fact, one of the
> most common examples of a category is the category of sets.
>
> Perhaps there may be incompatibilities between the methodology
> associated with Ontolingua and category-based techiques, but
> Ontolingua is not KIF.  Ontolingua simply uses KIF.
> --
>
> ===========================================================================
>                                          e
> Michael J. Healy                          A
>                                   FA ----------> GA
> (425)865-3123                     |              |
> FAX(425)865-2964                  |              |
>                                Ff |              | Gf
> c/o The Boeing Company            |              |
> PO Box 3707  MS 7L-66            \|/            \|/
> Seattle, WA 98124-2207            '              '
> USA                               FB ----------> GB
> -or for priority mail-                   e             "I'm a natural man."
> 2760 160th Ave SE  MS 7L-66               B
> Bellevue, WA 98008
> USA
>
> michael.j.healy@boeing.com          -or-            mjhealy@u.washington.edu
>
> ============================================================================

re: Question

[Joseph R. Kiniry]

Hello Michael,

I believe that category theory is an excellent foundation for ontology 
representation and manipulation -- I use it myself.

However, choosing this foundation comes at a price.  Unless significant 
work is done to hide this unfamiliar foundation, many users (of the theory, 
the system, the language, &c) will be biased against the work from 
minute-one.  This has as much to do with the unfamiliarity of CT as it does 
with certain unfortunate negative biases regularly expressed by many 
mathematicians and computer scientists - biases that, IMHO, are founded in 
ignorance and not reason.

Many computer scientists, mathematician, and users of knowledge 
representation systems are quite familiar (at least in use, but probably 
not in foundations or related complications) and comfortable with set 
theory.  This familiarity is an incentive rather than an obstacle to using 
related work.

Personally, I chose not to pursue a set theoretical foundation because of 
theoretical and representational complexity issues (e.g. witness the use of 
a set theoretical foundation for the Z specification language) as well as 
the unfortunate binding to a particular formalism that isn't necessarily 
congruent with others that I work in and apply my work to (e.g. type theory 
and programming languages).  To rephrase, I find using CT to be more clear 
and tractable than set theory and I feel that my work can, as a result, say 
and do more than it could if it had a set theory (plus some extra 
formalisms) bases.

Note that I also chose not to build my work (solely) on type theory and 
order sorted algebras for the same reason, though my work has elements of 
both of those fields as well.

The comments about Ontolingua and KIF are on-target in my experience.  I 
see no obstructions to the representation of CKML (the variant related to 
KIF and RDF that I happen to know well) with my work.

Finally, I should point out that I am but an infant in CT - I'm much more 
comfortable with TT, OSA, PL, and others.  I've only been learning and 
using CT for a few months and, while there have been some objection to my 
choice, I feel that a dissertation founded in these three major fields (CT, 
TT, and OSA) has significantly broader application and, implicitly, more to 
say about its author. <grin>

Best,
Joe Kiniry
-- 
Joseph R. Kiniry                   http://www.cs.caltech.edu/~kiniry/
California Institute of Technology       ID 78860581      ICQ 4344804


--On Tuesday, January 16, 2001 04:17:19 PM -0800 "Michael J. Healy 
425-865-3123" <mjhealy@redwood.rt.cs.boeing.com> wrote:

>
> I'd like to ask category theorists how they would answer the attached
> message  from a colleague here.  Both he and the person with whom he is
> corresponding  are experts in the areas of knowledge representation
> within computer science  (ontologies and the like).  I thought it best to
> hide their identities since  I haven't asked permission to use them.  If
> you are interested, please respond  to me privately if you would.
>
> Thank you,
> Mike Healy
> -------------------------------------------------------------------------
> -----
>
> Message I received:----
>
> I would be delighted if there was no semantic conflict between
> category theory and set theory.  I kind of flagged this as a
> potential issue, but did not look  into it in detail, as it was
> not my main concern at the time.  However, I remain unconvinced.
> There has been some discussion of using set theory as the basis
> for a semantics for SUOKIF. If this is true, then I think it may
> be limiting to a CT based language. While it may be true that sets
> are common example of a catagory, my sense is that CT is much more
> powerful, and would be LIMITED if everything was forced into the
> single catagory of sets.
>
> Im a bit out of my element here, however, and need to defer to the
> formal expertise of others on this issue.
>
>
> Message to which the above was replying:---
>
> I agree that category theory is very powerful and could be
> an important basis for combining and sharing ontologies.
> But I disagree with the following point:
>
>> I think this idea has tremendous potential.  One problem is that the
>> underlying
>
>> formal semantics of category theory is NOT set theory (which is what KIF
>> uses),
>
>> furthermore, I think they may well be incompatible.
>
> First-order logic (including any and all notations for it,
> such as KIF, CGs, predicate calculus, existential graphs, etc.)
> is completely neutral with respect to set theory or category
> theory.  The version 3.0 of KIF did include a version of set
> theory, but that was removed in the KIF'99 version because it
> belongs to ontology rather than logic.
>
> And for that matter, there is no reason why you can't use both
> category theory and set theory together.  In fact, one of the
> most common examples of a category is the category of sets.
>
> Perhaps there may be incompatibilities between the methodology
> associated with Ontolingua and category-based techiques, but
> Ontolingua is not KIF.  Ontolingua simply uses KIF.
> --
>
> Michael J. Healy

Question

[Michael J. Healy 425-865-3123]

I'd like to ask category theorists how they would answer the attached message 
from a colleague here.  Both he and the person with whom he is corresponding 
are experts in the areas of knowledge representation within computer science 
(ontologies and the like).  I thought it best to hide their identities since 
I haven't asked permission to use them.  If you are interested, please respond 
to me privately if you would.

Thank you,
Mike Healy
------------------------------------------------------------------------------

Message I received:----

I would be delighted if there was no semantic conflict between
category theory and set theory.  I kind of flagged this as a
potential issue, but did not look  into it in detail, as it was
not my main concern at the time.  However, I remain unconvinced.
There has been some discussion of using set theory as the basis
for a semantics for SUOKIF. If this is true, then I think it may
be limiting to a CT based language. While it may be true that sets
are common example of a catagory, my sense is that CT is much more
powerful, and would be LIMITED if everything was forced into the
single catagory of sets.

Im a bit out of my element here, however, and need to defer to the
formal expertise of others on this issue.


Message to which the above was replying:---

I agree that category theory is very powerful and could be
an important basis for combining and sharing ontologies.
But I disagree with the following point:

>I think this idea has tremendous potential.  One problem is that the underlying

>formal semantics of category theory is NOT set theory (which is what KIF uses),

>furthermore, I think they may well be incompatible.

First-order logic (including any and all notations for it,
such as KIF, CGs, predicate calculus, existential graphs, etc.)
is completely neutral with respect to set theory or category
theory.  The version 3.0 of KIF did include a version of set
theory, but that was removed in the KIF'99 version because it
belongs to ontology rather than logic.

And for that matter, there is no reason why you can't use both
category theory and set theory together.  In fact, one of the
most common examples of a category is the category of sets.

Perhaps there may be incompatibilities between the methodology
associated with Ontolingua and category-based techiques, but
Ontolingua is not KIF.  Ontolingua simply uses KIF.
--

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question

[adrian duma]

Dear Experts in Category Theory,

I am very interested in the following problem.
Let us consider the category Cat of small categories. I need for a "topology"
on Morph(Cat) (i.e., the functors between small categories), with the
following property:
For each u in Morph(Cat) and each D in Ob(Cat) there exist 
			v
		C------------>D
in Morph(Cat), a functor F:Cat----->Cat  with F(u) = v, and two "open
neighbourhoods" U and V of u and v, respectively, such that F (acting on
Morph(Cat)) is a "homeomorphism" between U and V.
I would be very grateful to you for any related comment.
With my best regards, I remain
Truly Yours
Adrian Duma.


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