Model -> Model family

[olynch]

I propose that we use the term "model system" for what we previously called a "model" of a GAT. Instances of a model system struct are individual models. Some model systems consist of just a single model (i.e. zero field structs), but in general a model system might encompass many values of the struct, and thus many "models".

[epatters]

I agree that it is good to distinguish a particular model of the GAT from a struct that specifies many different models at once. I'm not sure about the term "model system" though. Maybe "model family," being short for "a family of models"?

[olynch]

I like model family; family is more specific to the usage of "system" that I had in mind (i.e., analogous to "solar system").

[jpfairbanks]

So is model family taking the place of the logician's Model. And model is now referring to the entities in the model family?

[KevinDCarlson]

I feel like that's not what Owen meant but it's also not clear to me. Is it instead that a model family is going to be a struct such that any instance of the struct is an actual model of the theory, e.g. struct FinSet int n end is a model family for the GAT of sets?

[olynch]

No, the logician's model corresponds to specific instances of the struct that corresponds to the model family. For instance, we might have a model family

struct SliceCat{Ob, Hom, C<:Model{Category, Tuple{Ob, Hom}} <: Model{Category, Tuple{Tuple{Ob, Hom}, Hom}}
  cat::C
  over::Ob
end

Each value of SliceCat corresponds to a specific model. This also says that an object of the slice category consists of an object and morphism in the base category, and a morphism in the slice category is just a morphism in the base category.

[jpfairbanks]

Ok so SliceCat is a model family. And then what that type signature says is that each instance of SliceCat is a model of the theory of Categories.

What do we call the structs that will store the objects and morphisms in a particular model (instance of SliceCat)?

[olynch]

Ok so SliceCat is a model family. And then what that type signature says is that each instance of SliceCat is a model of the theory of Categories.

Yes, exactly.

What do we call the structs that will store the objects and morphisms in a particular model (instance of SliceCat)?

Good question, I don't yet have a name for that. Any suggestions?