open import 1Lab.HLevel
open import 1Lab.Path
open import 1Lab.Type hiding (id ; _∘_)

open import Cat.Base

module Cat.Displayed.Base where

Displayed CategoriesπŸ”—

The core idea behind displayed categories is that we want to capture the idea of being able to place extra structure over some sort of β€œbase” category. For instance, we can think of categories of algebraic objects (monoids, groups, rings, etc) as being extra structure placed atop the objects of Set, and extra conditions placed atop the morphisms of Set.

We start by defining a displayed category over some sort of β€œbase,” which will act as the category we add the extra structure to.

record Displayed {o β„“} (B : Precategory o β„“)
                 (oβ€² β„“β€² : Level) : Type (o βŠ” β„“ βŠ” lsuc oβ€² βŠ” lsuc β„“β€²) where

  open Precategory B

For each object of the base category, we associate a type of objects. Going back to our original example of algebraic structures + Set, this would be something like Monoid-on : Set β†’ Type. This highlights an important point for intuition: we should think of the objects of the displayed category as structures over the objects of the base.

    Ob[_] : Ob β†’ Type oβ€²

We do a similar thing for morphisms: For each morphism f : Hom x y in the base category, there is a set of morphisms between objects in the displayed category. Keeping with our running example, given a function f : X β†’ Y and monoid structures M : Monoid-on X, N : Monoid-on Y, then Hom[ f ] M N is the proposition that β€œf is a monoid homomorphism.” Again, we should best think of these as structures over morphisms.

    Hom[_] : βˆ€ {x y} β†’ Hom x y β†’ Ob[ x ] β†’ Ob[ y ] β†’ Type β„“β€²
    Hom[_]-set : βˆ€ {a b} (f : Hom a b)
                 β†’ (x : Ob[ a ]) β†’ (y : Ob[ b ]) β†’ is-set (Hom[ f ] x y)

We also have identity and composition of displayed morphisms, but this is best thought of as witnessing that the identity morphism in the base has some structure, and that composition preserves that structure. For monoids, this would be a proof that the identity function is a monoid homomorphism, and that the composition of homomorphisms is indeed a homomorphism.

    idβ€² : βˆ€ {a} {x : Ob[ a ]} β†’ Hom[ id ] x x
    _βˆ˜β€²_ : βˆ€ {a b c x y z} {f : Hom b c} {g : Hom a b}
           β†’ Hom[ f ] y z β†’ Hom[ g ] x y β†’ Hom[ f ∘ g ] x z

Now, for the difficult part of displayed category theory: equalities. If we were to naively try to write out the right-identity law, we would immediately run into trouble. The problem is that fβ€² βˆ˜β€² idβ€² : Hom[ f ∘ id ] x y, but fβ€² : Hom [ f ] x y! IE: the laws only hold relative to equalities in the base category. Therefore, instead of using _≑_, we must use PathP. Let’s provide some helpful notation for doing so.

  infixr 40 _βˆ˜β€²_

  _≑[_]_ : βˆ€ {a b x y} {f g : Hom a b} β†’ Hom[ f ] x y β†’ f ≑ g β†’ Hom[ g ] x y β†’ Type β„“β€²
  _≑[_]_ {a} {b} {x} {y} fβ€² p gβ€² = PathP (Ξ» i β†’ Hom[ p i ] x y) fβ€² gβ€²

  infix 30 _≑[_]_

Finally, the laws. These are mostly what one would expect, just done over the equalities in the base.

    idrβ€² : βˆ€ {a b x y} {f : Hom a b} β†’ (fβ€² : Hom[ f ] x y) β†’ (fβ€² βˆ˜β€² idβ€²) ≑[ idr f ] fβ€²
    idlβ€² : βˆ€ {a b x y} {f : Hom a b} β†’ (fβ€² : Hom[ f ] x y) β†’ (idβ€² βˆ˜β€² fβ€²) ≑[ idl f ] fβ€²
    assocβ€² : βˆ€ {a b c d w x y z} {f : Hom c d} {g : Hom b c} {h : Hom a b}
             β†’ (fβ€² : Hom[ f ] y z) β†’ (gβ€² : Hom[ g ] x y) β†’ (hβ€² : Hom[ h ] w x)
             β†’ fβ€² βˆ˜β€² (gβ€² βˆ˜β€² hβ€²) ≑[ assoc f g h ] ((fβ€² βˆ˜β€² gβ€²) βˆ˜β€² hβ€²)