Leírás
Title: Model theory in the category of sheaves
Abstract:
We can consider models of theories in categories other than the category of sets. For example, if E is a category with finite products, then we can talk about group objects in E. Alternatively, if T is a coherent theory (every axiom is an implication between two positive existential formulas; at the price of extending the signature, any first-order theory is of this form) and E is a coherent category (a category with finite limits, pullback-stable image factorizations, and pullback-stable finite unions), then we can talk about T-models in E. In particular, if E=Sh(X) is the category of sheaves over a topological space X (or more generally, if E is a Grothendieck topos) then we can talk about T-models in E.
Models in Sh(X) admit an explicit description: a model is a sheaf of L-structures such that over each point the stalk is a model. Informally, a model in Sh(X) is a "continuous family of models parametrized by the space X". For example, if T is the theory of local rings, then T-models in Sh(X) are the locally ringed spaces with underlying space X (something important in geometry). Another example: if B is a complete Boolean algebra, then models in the category of sheaves over B are the same as B-valued models (something that appears in forcing).
These examples motivate the question: how much of model theory can be lifted to the context of models internal to a Grothendieck topos? The main tool for answering that question is to translate model theory into the language of category theory. This is an alternative to algebraic logic: instead of using cylindric algebras, we identify coherent theories with small coherent categories, models with Set-valued coherent functors, and homomorphisms with natural transformations between these functors. So now we can try the following; pick a theorem of model theory, express it as a claim about Set-valued coherent functors, prove that claim in the language of category theory, then try to make the proof work for topos-valued coherent functors (instead of Set-valued ones). I will show some examples where this works, and some examples where models internal to a topos behave differently from models internal to Set.