nLab contextual fibrancy

Idea

In a categorical model of dependent type theory, a type $T$ in context $\Gamma$ is called fibrant if the projection morphism $p : \Gamma.T \to \Gamma$ from the extended context, is a fibration.

When we want to study or use fibrancy within type theory, we may be interested in the following properties, which do not necessarily hold:

1. The Π-type is fibrant if its codomain is,

2. The fibrant replacement monad commutes with substitution,

3. A type in context $\Gamma$ is fibrant if and only if its [pullback]] (substitution) along any morphism/cell $\gamma : y W \to \Gamma$ from a representable context $y W$, is fibrant.

The first property is just practical. The second property allows the fibrant replacement to be internalized as a monad on types, so that fibrancy can be defined internally as being an algebra for that monad. The third property allows a direct construction of a Hofmann-Streicher universe? (Hofmann and Streicher) of fibrant types. It should be straightforward to prove that this is equivalent with the second property, and with the property that the fibrant replacement can be taken cellwise.

One way to guarantee that properties 1-2 hold, is by defining fibrations as the right class of an algebraic weak factorization system generated by a class of generating left maps whose pullback is again a left map (Nuyts 2020). (So a fibration is any morphism that right-lifts any generating left map, and a left map is any morphism that left-lifts any fibration.)

Important notions of fibrancy violate the properties above:

• Kan fibrancy as used in models of HoTT. The problem is that a filling of a horn or open box takes place above a cell that is not part of that horn or open box.
• Segal fibrancy as one would like to use in models of directed type theory, for the same reason.

For this reason, Boulier and Tabareau introduced contextual Kan fibrancy. A type $T$ in a context $\Gamma.\Theta$ is contextually Kan fibrant in context $\Gamma$ if it has fillers for open boxes whose “open” dimension is $\Gamma$-homogeneous, i.e. if it lifts diagrams of the following shape:

This is an instance of a damped algebraic weak factorization system (Nuyts 2020). We can also build a damped AWFS for the Segal condition, where at most one of the morphisms-to-be-composed can be $\Gamma$-heterogeneous: Both damped AWFSs have the property that they are generated by a class of generating left “damped arrows” (which are just sequences of two composable arrows) whose pullbacks are again left damped arrows, and the result is that we get a form of properties 1-2 (Nuyts 2018, 2020) for these notions of contextual fibrancy.

References

• Andreas Nuyts, Contributions to Multimode and Presheaf Type Theory, chapter 8: Fibrancy, PhD thesis, KU Leuven, Belgium, 2020

• Andreas Nuyts, Robust Notions of Contextual Fibrancy, HoTT/UF, 2018

• Martin Hofmann, Thomas Streicher, Lifting Grothendieck Universes , ms. University of Darmstadt (unpublished). (pdf)

• Simon Boulier?, Nicolas Tabareau, Model structure on the universe of all types in interval type theory, MSCS, vol. 31, 2021 (doi:10.1017/S0960129520000213)