# What kind of Categorical object is an RDF Model?

RDF1.1 semantics defines an RDF Interpretation and mixes up the model and the mapping from the syntax to the model. I want to work out independently what its category of all models is. (In order to perhaps later be able to define an institution for it).

Luckily the model is extremely simple:

1. A non-empty set IR of resources, called the domain or universe of I.
2. A set IP, called the set of properties of I.
3. A mapping IEXT from IP into the powerset of $IR \times IR$ i.e. the set of sets of pairs $\langle x, y \rangle$ with x and y in IR .

It looks like one can map this structure nicely to a Quiver, by creating the set of arrows like this $AR =\{ \langle a,p,b\rangle \mid p \in IP \land \langle a,b \rangle \in IEXT(p) \}$. We can then define two functions $s,t : AR \to IR \cup IP$ as $s \langle a,p,b \rangle=a$ and ${t \langle a,p,b \rangle =b}$ which gives us the Quiver.

This is a subset of all Quivers I believe. Is it a special one? By mapping this to a Quiver this way I loose the type information, which does play a central role in RDF. So I was wondering if there is a structure that keeps that, so that I could know that all arrows that are part of the same $IEXT(p)$ powerset have something in common.

Do I need to add to the set of arrows some more arrows from an element of one of the relations selected by IEXT, ie to a subobject of $IR \times IR$. So we need to add $TP= \{ \langle rels,0,a \rangle, \langle rels,1,b \rangle \mid p \in IP \land IEXT(p) = rels \land \langle a,b \rangle \in rels \}$ ?

As this is used for logical modelling and we have powersets I was wondering if there is an obvious relation to topos theory perhaps...

It is often useful to convert set-valued mappings into bundles. You've nearly done this to $IEXT$.

I think the most natural repackaging of the data you list for a category theoretic point of view is that an "RDF-object" in a category with products consists of:

• An object $IR$
• An object $IP$
• A monomorphism $AR \hookrightarrow IR \times IP \times IR$

Then you have the fact that an RDF-object of Set is basically the same thing as what you define to be a model: the remaining part of the correspondence is that $IEXT$ is given by taking pullback squares: $$\require{AMScd} \begin{CD} IEXT(p) @>>> AR \\ @VVV @VVV \\1 @>p>> IP \end{CD}$$ where the arrow on the right is the suggested projection map, $1$ denotes a one-element set, and for $p \in IP$, I also write $p$ for the morphism $1 \to IP$ whose value is $p$.

The only essential difference between "RDF-object" and what you define to be a model is that you additionally require $IR \neq \varnothing$.

Incidentally, this abstract knowledge strongly suggests that you get a better behaved notion by dropping the $IR \neq \varnothing$ condition from the basic definitions — instead that condition should be viewed as an additional property that an RDF-object might have.

RDF-objects are models of a finite limit sketch. E.g. this means you can choose a category $\mathcal{T}$ and some additional data, and then an RDF-object in a category $\mathcal{C}$ is a functor $\mathcal{T} \to \mathcal{C}$ satisfying certain properties.

There is an implied notion of a morphism between two RDF-objects, and the category of RDF objects and said morphisms in Set is a particularly nice one. For example, it is locally finitely presentable.

From the point of view of categorical logic, this sketch describes the formal first-order theory described by

• There is a type $R$ whose members are called resources
• There is a type $P$ whose members are called properties
• There is a ternary relation $X$ whose arguments are of type $R$, $P$, $R$ respectively

There are no axioms for this theory; it's just the two types together with the ternary relation symbol.