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1. a category of $Sig$ of Signatures $\Sigma$: For our example we could have a ground graph signature $\Sigma$ containing only a small subset of IRIs and some literals to correspond to the objects in our model. (No existential quantifiers (aka blank nodes) for example). IRIs are just the web equivalent of mathematical symbols, designed so that people can globally coin new symbols without fearing a name clash. They stand for International Resource Identifiers and are specified syntactically by RFC3987. IRI ⊇ URI ⊇ URL ∪ URN
2. a functor $Mod$ from $Sig^{op}$ to $Cat$ the category of categories selecting for each object in the category $\vert Sig \vert$ a category of models. So I guess that a signature with 4 IRIs and 2 literals can give us a signature to the category of Partially ordered models I drew above.
3. a functor $Sen$ from $Sig$ to sets of Sentences in $Set$ (here we need no ur-elements I think, since we can encode the Unicode alphabet with the empty set and sets thereof)
4. A relation $\models_\Sigma \space \subseteq \vert Mod(\Sigma) \vert \times Sen(\Sigma)$

1. a category of $Sig$ of Signatures $\Sigma$: For our example we could have a ground graph signature $\Sigma$ containing only a small subset of IRIs and some literals to correspond to the objects in our model. (No existential quantifiers (aka blank nodes) for example). IRIs are just the web equivalent of mathematical symbols, designed so that people can globally coin new symbols without fearing a name clash. They stand for International Resource Identifiers and are specified syntactically by RFC3987. IRI ⊇ URI ⊇ URL ∪ URN
2. a functor $Mod$ from $Sig^{op}$ to $Cat$ the category of categories selecting for each object in the category $\vert Sig \vert$ a category of models. So I guess that a signature with 4 IRIs and 2 literals can give us a signature to the category of Partially ordered models I drew above.
3. a functor $Sen$ from $Sig$ to sets of Sentences in $Set$ (here we need no ur-elements I think, since we can encode the Unicode alphabet with the empty set and sets thereof)
4. A relation $\models_\Sigma \space \subseteq \vert Mod(\Sigma) \vert \times Sen(\Sigma)$

1. Is such a Poset category $\Bbb{M}_{RP}$ what an RDF Institution would send a Signature $\Sigma$ to? Ie is this the right type $ Mod(\Sigma): \Bbb{M}_{RP}$ ?
2. Since there can be many such categories of different sizes depending on IP and IR (Appendix B: Finite Interpretations), it is likely that these categories of poset can form a category which we could call $\Bbb{M}$. (Somewhat like the category of boolean algebras). Is that the category that the functor $Sig$ has as range? Ie. $Mod(\Sigma): \Bbb{M}$
3. In the category $\Bbb{M}$ what is the smallest category (initial or final object?)
4. Any category $\Bbb{M}$ contains many models so for any such model some sentences will be satisfied others not. In modal logics those are thought of as possible worlds. But in David Lewis' sense those are complete objects. It looks like possible worlds must be fixed points - or final objects - in a category of models of ever increasing size. Is there a way to capture these purely category theoretically? (Also can one call those mini models possible worlds?)
5. Is there a way to state how the open world assumption works in this context?

1. Is such a Poset category $\Bbb{M}_{RP}$ what an RDF Institution would send a Signature $\Sigma$ to? Ie is this the right type $ Mod(\Sigma): \Bbb{M}_{RP}$ ?
2. Since there can be many such categories of different sizes depending on IP and IR (Appendix B: Finite Interpretations), it is likely that these categories of poset can form a category which we could call $\Bbb{M}$. (Somewhat like the category of boolean algebras). Is that the category that the functor $Mod$ has as range? Ie. $Mod: Sig^{op} \to \Bbb{M}$
3. In the category $\Bbb{M}$ what is the smallest category (initial or final object?)
4. Any category $\Bbb{M}$ contains many models so for any such model some sentences will be satisfied others not. In modal logics those are thought of as possible worlds. But in David Lewis' sense those are complete objects. It looks like possible worlds must be fixed points - or final objects - in a category of models of ever increasing size. Is there a way to capture these purely category theoretically? (Also can one call those mini models possible worlds?)
5. Is there a way to state how the open world assumption works in this context?

In yellow above are all the models that fit the actual world. In light blue we have different possible worlds/models (I may be able to have different shades for different incompatible possible worlds there). The biggest graph before Top J7 models worlds where Tim and Vint have the same age, worlds which presumably would require quite a lot of change to the actual one - some complex scenario involving Vint being frozen for over a decade or so before being born... In pink we have models that we don't really want to allow at all, such as model X1 which states that Tim is older than the literal "Vint Cerf", or X2 where Tim is related by the name relation to Vint (but Vint Cerf is a person not a name). X3 is the union of that nonsense (are these what one would term impossible worlds?) RDFS and Description Logic Based OWL vocabularies allow one to exclude those as models, but I think the basic minimal RDF framework does not.

In yellow above are all the models that fit the actual world. The most comprehensive such model is M7. In light blue we have different possible worlds/models (I may be able to have different shades for different incompatible possible worlds there). The biggest graph before Top N8 models worlds where Tim and Vint have the same age, worlds which presumably would require quite a lot of change to the actual one - some complex scenario involving Vint being frozen for over a decade or so before being born... In pink we have models that we don't really want to allow at all, such as model X1 which states that Tim is older than the literal "Vint Cerf", or X2 where Tim is related by the name relation to Vint (but Vint Cerf is a person not a name). X3 is the union of that nonsense (are these what one would term impossible worlds?) RDFS and Description Logic Based OWL vocabularies allow one to exclude those as models, but I think the basic minimal RDF framework does not.

We can see that moving up any arrows into a model (eg M7) is the union of the models into it, which form a logical conjunction or a categorical coproduct. Moving in the opposite direction, say looking at the opposite category we have arrows out of a model such as M7 form its logical disjunctions, or categorical products.