Where is the naming semantics in the Functorial view of RDF DBs?
Reading An Introduction to Description Logics I found the following passage on the semantics of individual names: This got me thinking about how this should be mapped to Functorial Semantics (such as Spivak's version mapping to Set or Patterson's Bicategories of Relations). This is how I noticed something that is not yet explained.
In the Functorial view we have a small category representing the Schema that is mapped by a Functor to an instance Category, either Set or Rel. The functor assigns to each object in the schema the set of instances. Morphism in the Schema are mapped to morphisms in the instance category.
But as we see in the Description Logic formalism (which I think is very close to that of RDF), we have sets of names for Relations, Concepts and Instances, which are then mapped by an interpretation function to a domain \Delta^I
of objects.
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So we could think of the objects in Rel as containing the names. The Functor is then thought of as the instance database. It's Grothendieck construction then gives us something very close to an RDF syntactic graph see answer to typed RDF. But then we'd need a functor from Rel or the Grothendieck construction to the object equivalent of
\Delta^I
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Alternatively we think of Set or Rel as sets of physical objects, but then we have to work out where the naming happens.
Also we need to work out how we name the Schame objects and relation. Furthermore in RDF URIs name both schema and instances, and what a URI names is discovered observationally (in the sense of coalgebraic observation).