What is the Semantics of Datasets? How can they modelled in CT?
The RDF 1.1 spec section on DataSets gives a simple structural definition of them
RDF datasets, defined in RDF11-CONCEPTS, package up zero or more named RDF graphs along with a single unnamed, default RDF graph. The graphs in a single dataset may share blank nodes. The association of graph name IRIs with graphs is used by SPARQL [SPARQL11-QUERY] to allow queries to be directed against particular graphs.
The following paragraphs feel a bit confused, which is not to say that they can not be given a good semantics, that respects the way they can be used.
The encoding of a Dataset in NQuads requires 4 elements, 1 more than the encoding of NTriples.
A Category, can be seen as consisting of 3 elements: an origin node, an arrow, and a target node.
If one names each arrow in a graph with an NTriple, then one can name the origin node with the subject of the NTriple, and the target with the object of the triple. It is thus easy to see that the structure of an RDF graph matches very well the structure of the Free Category that arises out of such a graph.
Formally we have a Graph where arrows are named by Triples
given by the two functions
source, target: Triples \to Nodes
When moving to DataSets this simple mapping no longer works. Something essentially different is being modeled. Indeed the passage on DataSets mentions modal logic
Other semantic extensions and entailment regimes MAY place further semantic conditions and restrictions on RDF datasets, just as with RDF graphs. One such extension, for example, could set up a modal-like interpretation structure so that entailment between datasets would require RDF graph entailments between the graphs with the same name (adding in empty graphs as required).
Datasets provide the tool for a notion of context, which is essential for use in modal logic (see literature). This question is therefore related but prior to How does one build a modal bicategory of relations?.
Category Theory allows one to give purely structural definitions of concepts, that can then be shown to have a number of semantics. See for example §6 of Pattersons Knowledge Representation in Bicategories of Relations where he shows how functorial semantics can allow one to give very different interpretations. The aim of an Cats view of RDF would be to allow one to show how for example Pat Hayes' 2007 Context Mereology is a coherent extension of the structure behind RDF 1.0 and of DataSets (see also his 2009 slides on blogic).