Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic.
It is also notable for its connections to theoretical computer science.
[1] In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor.
The categorical framework provides a rich conceptual background for logical and type-theoretic constructions.
The categorical semantics of a logic consists in describing a category of structured categories that is related to the category of theories in that logic by an adjunction, where the two functors in the adjunction give the internal language of a structured category on the one hand, and the term model of a theory on the other.