RE: rdf and category theory

Back a few years, emerging from the old IEEE Standard Upper Ontology group's work was Bob Kent's Information Flow Framework, an ontology framework (a meta-level framework) based on Barwise & Seligman's Information Flow Theory, itself an application of Category Theory. See, for example: http://arxiv.org/pdf/1109.0983v1.

Mainly folks have used Information Flow Theory or Goguen's notion of institutions as springboards from category theory to ontologies, especially for so-called "lattice of theories", ontology mapping, and semantic interoperability applications. Work includes Mossakowski's various papers: http://iws.cs.uni-magdeburg.de/~mossakow/.

For a short "position" paper, see:
Markus Kr¨otzsch, Pascal Hitzler, Marc Ehrig, York Sure. 2005. Category Theory in Ontology Research: Concrete Gain from an Abstract Approach. http://www.aifb.kit.edu/web/Techreport893.

For RDF and category theory, the only paper I know of addresses graph transformations for RDF:
Benjamin Braatz; Christoph Brandt. 2008. Graph Transformations for the Resource Description Framework. Proceedings of the Seventh International Workshop on Graph Transformation and Visual Modeling Techniques (GT-VMT 2008). http://journal.ub.tu-berlin.de/eceasst/article/view/158/142.

Admittedly most of the above are applications beyond logic itself and RDF, but might shed some light on how category theory is being used for ontologies.

Thanks,
Leo

From: henry.story@bblfish.net [mailto:henry.story@bblfish.net]
Sent: Wednesday, April 16, 2014 6:09 PM
To: Gregg Reynolds
Cc: Antoine Zimmermann; SW-forum Web; public-philoweb@w3.org
Subject: Re: rdf and category theory


On 11 Apr 2014, at 16:32, Gregg Reynolds <dev@mobileink.com<mailto:dev@mobileink.com>> wrote:


On Fri, Apr 11, 2014 at 8:30 AM, Antoine Zimmermann <antoine.zimmermann@emse.fr<mailto:antoine.zimmermann@emse.fr>> wrote:
There're a lot of resources available online and for free about category theory.

Some examples:
 - Jirí Adámek, Horst Herrlich, George E. Strecker. Abstract and Concrete Categories: The Joy of Cats (524 pages). http://katmat.math.uni-bremen.de/acc/acc.pdf
 - Maarten M. Fokkinga. A Gentle Introduction to Category Theory: the calculational approach. http://wwwhome.ewi.utwente.nl/~fokkinga/mmf92b.pdf (80 pages).
 - Jaap van Oosten. Basic Category Theory (88 pages). http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf

One of the best is Robert Goldblatt's Topoi : The Categorial Analysis of Logic <http://homepages.mcs.vuw.ac.nz/~rob/books.html> .  He pays special attention to linking CT concepts to both classic math and ordinary intuition.

I looked through Robert Goldblatt's Topoi quickly [1] and indeed it is the book that covers the subject probably most relevant to the semantic web community, since it aims to show how logic can be derived from Category Theory. In this area I found reading through the first part of Ralf Krömer's "Tool and Object: A History and Philosophy of Category Theory" to also be very interesting, as it gives an overview of the foundational debate in Mathematics  started by CT.

It's so odd that RDF is entirely about relations just as CT is ( except that RDF is one to many whereas CT arrows are functions). So I really look forward to understanding how these two domains fit together, and perhaps how they complement each other.


Henry

[1] Having read through half of  "Conceptual Mathematics" by Willima Lawvere and done most of the exercises there, I am starting to be able to read a lot of these books much more easily.




-Gregg

Social Web Architect
http://bblfish.net/