Modeling Martin-Loef type theory in categories

摘要

We present a model of Martin-Loef type theory that includes both dependent products and the identity type. It is based on the category of small categories, with cloven Grothendieck bifibrations used to model dependent types. The identity type is modeled by a path functor that seems to have independent interest from the point of view of homotopy theory. We briefly describe this model's strengths and limitations.