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first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.

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Categorical semantics of universe levels in dependent type theory

I know that locally closed cartesian categories provide categorical semantics for type theories with dependent products. What kind of categories model type theories with infinite universe hierarchies …
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