There's a lot to unpack here and I don't know about Goguen's institutions. But perhaps I can give a partial answer to your question.

Let's start with "simple interpretations" of RDF, as defined by the spec, forgetting about richer languages like RDFS or OWL. If we ignore IRIs and literals, as in the first part of your question, then an RDF model is a simple object: an edge-labeled graph.

To view graphs as a category of models, we can use a variation of the graph indexing category, as discussed, for example, in David Spivak's book on category theory for the sciences. Define a category $\mathbf{RDF}_0$ with three objects $T$, $R$, and $P$, representing the triples, resources, and properties, and three morphisms $subj: T \to R$, $pred: T \to P$, $obj: T \to R$, representing the subject, predicate, and object of a triple. An RDF model is a functor $M: \mathbf{RDF}_0 \to \mathbf{Set}$. There is a functor category of all such models, whose morphisms are natural transformations. Under this definition, a morphism of RDF models is just a label-preserving graph homomorphism.

IRIs and literal can be incorporated within this general approach. Define a new category $\mathbf{RDF}$ consisting of the objects and morphisms of $\mathbf{RDF}_0$, plus two additional objects $IRI$ and $L$, representing the IRIs and literals, and two additional morphisms $irl: IRI \to R + P$ and $lit: L \to R$, representing interpretations of the IRLs and literals as resources and/or properties. Here the sum of objects ($+$) is a coproduct (disjoint union). We can handle the sum in multiple ways. We can say that $\mathbf{RDF}$ is the free cocartesian category generated by the above objects and morphisms. Or we can think of $\mathbf{RDF}$ as a sketch.

On this definition, an RDF model, as defined by the "simple interpretations" of the RDF 1.0 spec, is a (coproduct-preserving) functor $M: \mathbf{RDF} \to \mathbf{Set}$. As before, morphisms of RDF models are natural transformations between such functors.

That's all I have time to say right now. If you're interested, I can describe a different way of thinking about this problem that lends itself better to richer systems like RDFS or OWL.

There's a lot to unpack here and I don't know about Goguen's institutions. But perhaps I can give a partial answer to your question.

Let's start with "simple interpretations" of RDF, as defined by the spec, forgetting about richer languages like RDFS or OWL. If we ignore IRIs and literals, as in the first part of your question, then an RDF model is a simple object: an edge-labeled graph, whose vertices are resources, edges are triples, and edge labels are predicates.

To view RDF models as a category of models, we can use a variation of the graph indexing category, as discussed, for example, in David Spivak's book on category theory for the sciences. Define a category $\mathbf{RDF}_0$ with three objects $T$, $R$, and $P$, representing the triples, resources, and properties, and three morphisms $subj: T \to R$, $pred: T \to P$, $obj: T \to R$, representing the subject, predicate, and object of a triple. An RDF model is a functor $M: \mathbf{RDF}_0 \to \mathbf{Set}$. There is a functor category of all such models, whose morphisms are natural transformations. Under this definition, a morphism of RDF models is just a label-preserving graph homomorphism.

IRIs and literal can be incorporated within this general approach. Define a new category $\mathbf{RDF}$ consisting of the objects and morphisms of $\mathbf{RDF}_0$, plus two additional objects $IRI$ and $L$, representing the IRIs and literals, and two additional morphisms $iri: IRI \to R + P$ and $lit: L \to R$, representing interpretations of the IRIs and literals as resources and/or properties. Here the sum of objects ($+$) is a coproduct (disjoint union). We can handle the sum in multiple ways. We can say that $\mathbf{RDF}$ is the free cocartesian category generated by the above objects and morphisms. Or we can think of $\mathbf{RDF}$ as a sketch.

On this definition, an RDF model, as defined by the "simple interpretations" of the RDF 1.0 spec, is a (coproduct-preserving) functor $M: \mathbf{RDF} \to \mathbf{Set}$. As before, morphisms of RDF models are natural transformations between such functors.

That's all I have time to say right now. If you're interested, I can describe a different way of thinking about this problem that lends itself better to richer systems like RDFS or OWL.

There's a lot to unpack here and I don't know about Goguen's institutions. But perhaps I can give a partial answer to your question.

Let's start with "simple interpretations" of RDF, as defined by the spec, forgetting about richer languages like RDFS or OWL. If we ignore IRIs and literals, as in the first part of your question, then an RDF model is a very simple object: it's just a graph. The set $IR$ is the vertex set and $IP$ is the edge set.

To view graphs as a category of models, we can use the graph indexing category, as discussed, for example, in David Spivak's book on category theory for the sciences. In the present notation, we'll call this category $\mathbf{RDF}_0$. It has two objects $IR$ and $IP$, representing the resources and properties, and two morphisms $subj: IP \to IR$ and $obj: IP \to IR$, representing the subject and object of a triple. An RDF model (or graph) is a functor $M: \mathbf{RDF}_0 \to \mathbf{Set}$. There is a functor category of all such models, whose morphisms are natural transformations. Under this definition, a morphism of RDF models is just a graph homomorphism.

IRIs and literal can be incorporated within this general approach. Define a new category $\mathbf{RDF}$ consisting of the objects and morphisms of $\mathbf{RDF}_0$, plus two additional objects $IRI$ and $L$, representing the IRIs and literals, and two additional morphisms $irl: IRL \to IR + IP$ and $lit: L \to IR$, representing interpretations of the IRLs and literals as resources and/or properties. Here the sum of objects ($+$) is a coproduct (disjoint union). We can handle the sum in multiple ways. We can say that $\mathbf{RDF}$ is the free cocartesian category generated by the above objects and morphisms. Or we can think of $\mathbf{RDF}$ as a sketch.

On this definition, an RDF model, as defined by the "simple interpretations" of the RDF 1.0 spec, is a (coproduct-preserving) functor $M: \mathbf{RDF} \to \mathbf{Set}$. As before, morphisms of RDF models are natural transformations between such functors.

That's all I have time to say right now. If you're interested, I can describe a different way of thinking about this problem that lends itself better to richer systems like RDFS or OWL.

There's a lot to unpack here and I don't know about Goguen's institutions. But perhaps I can give a partial answer to your question.

Let's start with "simple interpretations" of RDF, as defined by the spec, forgetting about richer languages like RDFS or OWL. If we ignore IRIs and literals, as in the first part of your question, then an RDF model is a simple object: an edge-labeled graph.

To view graphs as a category of models, we can use a variation of the graph indexing category, as discussed, for example, in David Spivak's book on category theory for the sciences. Define a category $\mathbf{RDF}_0$ with three objects $T$, $R$, and $P$, representing the triples, resources, and properties, and three morphisms $subj: T \to R$, $pred: T \to P$, $obj: T \to R$, representing the subject, predicate, and object of a triple. An RDF model is a functor $M: \mathbf{RDF}_0 \to \mathbf{Set}$. There is a functor category of all such models, whose morphisms are natural transformations. Under this definition, a morphism of RDF models is just a label-preserving graph homomorphism.

IRIs and literal can be incorporated within this general approach. Define a new category $\mathbf{RDF}$ consisting of the objects and morphisms of $\mathbf{RDF}_0$, plus two additional objects $IRI$ and $L$, representing the IRIs and literals, and two additional morphisms $irl: IRI \to R + P$ and $lit: L \to R$, representing interpretations of the IRLs and literals as resources and/or properties. Here the sum of objects ($+$) is a coproduct (disjoint union). We can handle the sum in multiple ways. We can say that $\mathbf{RDF}$ is the free cocartesian category generated by the above objects and morphisms. Or we can think of $\mathbf{RDF}$ as a sketch.

On this definition, an RDF model, as defined by the "simple interpretations" of the RDF 1.0 spec, is a (coproduct-preserving) functor $M: \mathbf{RDF} \to \mathbf{Set}$. As before, morphisms of RDF models are natural transformations between such functors.

That's all I have time to say right now. If you're interested, I can describe a different way of thinking about this problem that lends itself better to richer systems like RDFS or OWL.