# Functorial comparison of Different Models of Set Theory

I'm very much a novice to Model Theory/Categorical Model Theory (though I very much would like to learn). I apologize if my question is improperly stated.

Fix a small (or locally small if desired) category $$\mathcal{C}$$.

From what I've understood of Model Theory, is one chiefly looks at the individual characteristics of each model of a theory.

Different models of Set Theory (say we denote them as $$S_1, S_2,\dots$$ generate different categories of Sets (which I'll denote $$\textbf{Set}_{S_1},\textbf{Set}_{S_2}, \dots)$$. What is known about the presheaf categories $$[\mathcal{C}^{op},\textbf{Set}_{S_i}]$$? Or the functor categories $$[\textbf{Set}_{S_i}, \textbf{Set}_{S_j}]$$ for the different models $$S_i, S_j$$?

My motivation is that it seems that results in Model Theory are not derived from comparing the theories (functorically or otherwise), and I'd like to see various literature on the subject. I'm primarily concerned about Category Theoretic results, but I'm interested in general.

• "My motivation is that it seems that results in Model Theory are not derived from comparing the theories (functorically or otherwise)" Can you clarify what you mean by that? Theories of what? – Noah Schweber Mar 26 at 1:02
• You might be interested in this question: math.stackexchange.com/questions/1000097/… – Eric Wofsey Mar 26 at 3:37