A biequivalence of path categories and axiomatic Martin-L\"of type theories

The semantics of extensional type theory has an elegant categorical description: models of Sigma-types and extensional Id-types are biequivalent to finitely complete categories. We establish a similar result for intensional type theories: weak models of Sigma-types and axiomatic Id-types are biequivalent to path categories. These axiomatic Id-types only satisfy beta-reduction rule as a propositional equality, and appear in cubical type theory, as well as axiomatic type theory (type theory without definitional equality). Path categories take inspiration from homotopy theory and simplify the structure of type theory using a primitive notion of equivalence. Our biequivalence allows us to turn path categories into actual models of type theory: we turn them into weak models, where substitution is only specified up to isomorphism, which we can in turn strictify using the left adjoint splitting. In addition, we introduce a more fine-grained notion: that of a display path category, and prove a similar biequivalence. These display path categories still model axiomatic Id-types, but not the intensional notion of Sigma-types. We show how they can be extended with axiomatic notions of Sigma-types and Pi-types, allowing us to model axiomatic type theory.