Is there a notion in the literature of a theory which is locally consistent, in some sense wherein choosing a "vantage" yields a theory which is consistent, but the entire theory may not be consistent? I don't really know how to formalize this question, but my intuition is that vantages are like objects in a category, and looking at the theory from a vantage amounts to something like passing to the slice category. Maybe one way to formalize this is in terms of locally cartesian closed $n$-categories?
I imagine it would be interesting to have a system which coheres globally, but has "holes" in it. One can still perform deductions from individual vantages, but to compare local theorems one must choose a consistent "path" between vantages that facilitates the transport of theorems. Clearly there is a strong geometric analogy here.
I apologize if this is not appropriate for MathOverflow, but I wanted to pick the brains of researchers in logic and model theory to get as enlightening answers as possible. Thank you.
