open import Cat.Displayed.Cartesian
open import Cat.Displayed.Fibre
open import Cat.Displayed.Total
open import Cat.Displayed.Base
open import Cat.Prelude

import Cat.Functor.Reasoning
import Cat.Reasoning

module Cat.Displayed.Cartesian.Street where

open Cartesian-fibration
open Cartesian-lift
open is-cartesian
open Displayed

Street fibrations🔗

In classical category theory, a fibration is defined to be a certain functor $P : \mathcal{E} \to \mathcal{B}$, the idea being that $\mathcal{E}$ is really the total space of a certain displayed category, and $P$ is really the first projection functor $\pi^f$, which sends each displayed object to the object it is displayed over. But can we go the other way? If we have a functor $P : \mathcal{E} \to \mathcal{B}$, can we create a category displayed $\mathcal{E}'$ over $\mathcal{B}$, such that $\int \mathcal{E}' \cong \mathcal{E}$?

module _ {o ℓ o′ ℓ′} {E : Precategory o ℓ} {B : Precategory o′ ℓ′} (P : Functor E B) where
  private
    module E = Cat.Reasoning E
    module B = Cat.Reasoning B
    module P = Cat.Functor.Reasoning P
  open B.HLevel-instance
  open E.HLevel-instance

  functor→displayed : Displayed B (o ⊔ ℓ′) (ℓ ⊔ ℓ′)
  functor→displayed .Ob[_] x = Σ[ u ∈ E.Ob ] (P.₀ u B.≅ x)

Following (Sterling and Angiuli 2022), we define such a category by defining the space of objects over $x$ to be “left corners”. Strictly speaking, a left corner is given by an object $u$ together with an isomorphism $\phi : P(u) \cong x$. But visually, we depict them as

whence the name “left corner”. The maps lying over $f$ consist of those maps $u \to_f v$ which commute with the mediating isomorphisms:

  functor→displayed .Hom[_] f (u , φ) (v , ψ) =
    Σ[ h ∈ E.Hom u v ] (B.to ψ B.∘ P.₁ h ≡ f B.∘ B.to φ)

This fits in a diagram like the one below. Note that the commutativity condition is for the lower shape, which is a distorted square.

The axioms for a displayed category are evident: all that matters are the maps in the total category $\mathcal{E}$, since the rest of the data is property (rather than data).

  functor→displayed .Hom[_]-set f a b = hlevel 
  functor→displayed .id′ = E.id , B.elimr P.F-id ∙ B.introl refl
  functor→displayed ._∘′_ (f , φ) (g , ψ) = f E.∘ g ,
    ap₂ B._∘_ refl (P.F-∘ f g) ∙ B.pulll φ ∙ B.pullr ψ ∙ B.assoc _ _ _
  functor→displayed .idr′ f′ = Σ-pathp-dep (E.idr _) $
    is-set→squarep (λ _ _ → hlevel ) _ _ _ _
  functor→displayed .idl′ f′ = Σ-pathp-dep (E.idl _) $
    is-set→squarep (λ _ _ → hlevel ) _ _ _ _
  functor→displayed .assoc′ f′ g′ h′ = Σ-pathp-dep (E.assoc _ _ _) $
    is-set→squarep (λ _ _ → hlevel ) _ _ _ _

We call a functor that gives rise to a Cartesian fibration through this process a Street fibration. It is routine to verify that our notion of Street fibration corresponds to.. well, Street’s.