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Re: category theory

From: Ioachim Drugus <sw@semanticsoft.net>
Date: Mon, 09 Jul 2007 15:19:57 -0700
Message-ID: <4692B48D.3020400@semanticsoft.net>
To: renato@ebi.ac.uk
CC: Ivan Herman <ivan@w3.org>, semantic-web@w3.org

As Ivan showed in concrete examples, OWL is at least same expressive as 
Category Theory meaning that the semantics associated with words in OWL 
vocabulary covers the expressiveness of *intra-category* terms, which 
describe  "local" properties like to be an injection (endomorphism),  
surjection (epimorphism), or bijection (isomorphism).

My view on how to use Category Theory in development of Semantic Web is 
based mostly on aspects *other* than expressiveness, which I will 
describe below. I believe, that Category Theory was not yet fully 
employed on Semantic Web, but it can contribute to development of its 
conceptuality and to overcoming certain difficulties which reasoners 
would encounter working with Full OWL.

1. As opposed to OWL, which  uses *intentional definitions* in 
semantics, the Category Theory uses *extensional definitions*. The local 
properties of "injection" were known since the beginnings of set theory. 
What category theory did was to express such "local" properties 
holistically - via the whole *context* of the universe of discource. I 
treat this reduction of local properties to the "whole" as category 
theory's distinctive feature among other theories (probably, Renato 
focus on category theory was due to the emphasis which he puts on 
*context* as I noticed in other threads).  Now, due to its intensional 
approach, OWL describes an open world model, which means that the 
"whole", described by a statement of category  theory, *changes*. 
Because the statements of category theory expressed in OWL would refer 
to the changing world, they might stop to be true on Semantic Web - this 
is one reason why they don't usually take statements of category theory 
into OWL. But when we assign in the *axioms* of DL certain 
"intentionality" to a term which refers to its behavior with respect to 
the whole universe of discourse (as when we state something in 
negative), then it is best to look for the formulations (definitions and 
statements) of category theory and write them in OWL. Therefore, I 
regard category theory as important for the development of semantics of 
Full OWL.

2. Category theory has *functors* which, simplistically said - map one 
category into another,  and it also has a very developed apparatus 
regarding correlations between functors. This could serve as a good 
conceptual basis for development of  *aggregation* and *integration* 
conceptuality and terms on these two activities.  I believe OWL already 
solved the task to provide the basic vocabulary for description of a 
domain, and I believe that in the versions to come it will add for us 
more words to talk about many domains and integration of knowledge. And 
then, the notion of *functor* and all associated with it (covariance, 
adjointness, duality, etc.)  might  get their counterparts in OWL 
semantics and some of them - in OWL syntax.

3. The two main philosophic disciplines, on which Semantic Web is based, 
are Ontology and Epistemology. But how come that something which just 
*exists* and is matter of Ontology, can convey *knowledge* which is 
matter of Epistemology!? I believe, it is Phenomenology, which serves as 
a link between the other two pillars of Semantic Web and it should serve 
as a third pillar. In another message (on clarification what is an 
"information resource") I exposed my view which I named MyPhenomenology, 
where I reduced cognition to *representation* and knowledge to 
*presentation* and stated that category theory is formalization of this 

I also have a 4th aspect of Category Theory strongly correlated with 
Semantic Web, but I would be ready to explose it later.

Main Architect

Renato golin wrote:
> Ivan Herman wrote:
>> Renato,
>> I think all the examples that you describe are covered by OWL. If you
>> use OWL for your ontology, you can specify that a predicate is
>> - transitive
>> - symmetric
>> - functional
>> - inverse of another predicate
>> - inverseFunctional
>> The terms injective and surjective are not used, these are equivalent (I
>> believe) to inverseFunctional and functonal, respectively (bijective is
>> simply a predicate that is both).
> Hi Ivan,
> I guess those cover my examples, will play with them for a while and 
> if I miss something will come back later... ;)
> thanks!
> --renato
Received on Monday, 9 July 2007 22:19:59 UTC

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