Question

[mjhealy@redwood.rt.cs.boeing.com (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
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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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