This is an archived version pinned as of the submission of my master's thesis. An up-to-date version may be found online.

Flat dependent function types

{-# OPTIONS --cohesion --flat-split #-}

module modal-type-theory.flat-dependent-function-types where
Imports 
open import foundation.universe-levels

open import modal-type-theory.flat-modality

Idea

We study interactions between the flat modality and dependent function types.

Properties

Flat distributes over dependent function types

module _
  {@♭ l1 l2 : Level} {@♭ A : UU l1} {@♭ B : A → UU l2}
  where

  map-crisp-distributive-flat-Π : ♭ ((x : A) → B x) → ((@♭ x : A) → ♭ (B x))
  map-crisp-distributive-flat-Π (cons-flat f) x = cons-flat (f x)

module _
  {@♭ l1 l2 : Level} {@♭ A : UU l1} {@♭ B : UU l2}
  where

  map-crisp-distributive-flat-function-types : ♭ (A → B) → (@♭ A → ♭ B)
  map-crisp-distributive-flat-function-types = map-crisp-distributive-flat-Π

  map-distributive-flat-function-types : ♭ (A → B) → (♭ A → ♭ B)
  map-distributive-flat-function-types f (cons-flat x) =
    map-crisp-distributive-flat-function-types f x

See also

References

[Lic]
Dan Licata. Dlicata335/cohesion-agda. GitHub repository. URL: https://github.com/dlicata335/cohesion-agda.
[Shu18]
Michael Shulman. Brouwer's fixed-point theorem in real-cohesive homotopy type theory. Mathematical Structures in Computer Science, 28(6):856–941, 06 2018. arXiv:1509.07584, doi:10.1017/S0960129517000147.