Section: New Results

Martin-Löf Identity Type in the Category of Small Categories

Participant : François Lamarche.

For the last five or six years there has been a surge of interest in finding models for the identity type in Martin-Löf type theory, and it has been clear for some time that there was a tight connection with path objects in abstract homotopy theory. A lot of proposals have been made, but there are very few semantics that fit the necessary requirements of having dependent products and also an identity type which is fully stable under substitution. The most famous model of the sort is the one proposed by Voevodsky, in his Univalent Foundations project, which uses for base category the category of simplicial sets and models dependent types by the means of Kan Fibrations. In [13] François Lamarche proposes another such model, where the base category is the category of small categories, and dependent types are modelled with Grothendieck bifibrations (maps between categories that are Grothendieck fibrations as well as their duals between the opposite categories). The full requirements of modelling Martin-Löf type theory are met. Calculations show that the model shows some amount of degeneracy “in dimensions above 2” for the associativity of equality (which should not be strict in any dimension), which is a great improvement over the models on strict groupoids and strict $\omega$-groupoids. The construction that models the identity type is a concrete path functor for categories. It is showing itself to be very useful in homotopy theory.