I’m working on this idea of considering theory signatures as contexts of some dependent type theory and theory presentations as a category fibred over the category of contexts. A model of a theory presentation corresponds to a point of said object, that is a morphism from the terminal theory presentation (i.e. the empty context with no axioms) to said object. The fibration projects models down their underlying structure, which is a point in the corresponding theory signature, that is a morphisms from the terminal theory signature (i.e. the empty context) to said theory signature, which in turn corresponds to a term fulfilling the theory signature.

Given that two structures for signature S are arrows M N : 1 → S, it seems natural to represent homomorphisms from M to N as a 2-cell F : M ⇒ N.

If our category of theory signatures is going to actually be a 2-category then we need to consider 2-fibrations, which seem to have been worked out in Claudio Hermida’s paper entitled "Some Properties of Fib as a fibred 2-category".

More importantly I need to work out what a 2-cell in the category of contexts would be exactly. Some things are pretty clear; for the context containing x:Type and when the corresponding components of the structures, M and N, are ⊢ s:Type and ⊢ t:Type, we want the corresponding component F : M ⇒ N to be a term ⊢ F_{x} : s → t. If the context contains an operation f : x → x, and the corresponding components of the structures are ⊢ g : s → s and ⊢ h : t → t, then we want the corresponding component of F : M ⇒ N to be the constraint ⊢ F_{f} : F_{x}∘g =_{s → t} h∘F_{x}. But to work it out for the general case of dependent type boggles my mind.

There is an interesting downward transformation going on here where types in the signature turn are realized by (function) terms in the 2-cells, and terms in the signatures are realized by (equality) propositions in the 2-cells, and propositions (in theory presentations) will come for free in the 2-cells (is this always true?). Perhaps kinds will realized by types?

This all vaguely resembles Reynold's abstraction theorem.

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