open import Cat.Instances.Comma
open import Cat.Functor.Base
open import Cat.Univalent
open import Cat.Prelude

import Cat.Functor.Reasoning as Func
import Cat.Reasoning

  {xo xβ„“ yo yβ„“ zo zβ„“}
  {X : Precategory xo xβ„“} {Y : Precategory yo yβ„“} {Z : Precategory zo zβ„“}
  (F : Functor Y X) (G : Functor Z X)
  (xuniv : is-category X)
  (yuniv : is-category Y)
  (zuniv : is-category Z)

Comma categories preserve univalenceπŸ”—

Theorem. Let Yβ†’FX←GZ\ca{Y} \xto{F} \ca{X} \xot{G} \ca{Z} be a cospan of functors between three univalent categories. Then the comma category F↓GF \downarrow G is also univalent.

It suffices to establish that, given an isomorphism f:oβ‰…oβ€²f : o \cong o' in F↓GF \downarrow G, one gets an identification fΛ†:o≑oβ€²\^f : o \equiv o', over which ff is the identity map. Since Y\ca{Y} and Z\ca{Z} are both univalent categories, we get (from the components fΞ±f_\alpha, fΞ²f_\beta of ff) identifications fΛ†Ξ±:ox≑oxβ€²\^f_\alpha : o_x \equiv o'_x and fΛ†Ξ²:oy≑oyβ€²\^f_\beta : o_y \equiv o'_y.

Comma-is-category : is-category (F ↓ G)
Comma-is-category ob .centre = ob , F↓
Comma-is-category ob .paths (obβ€² , isom) = Ξ£-pathp objs maps where
  module isom = F↓G._β‰…_ isom

  x-is-x : ob .x Y.β‰… obβ€² .x
  y-is-y : ob .y Z.β‰… obβ€² .y

  x-is-x = Y.make-iso ( .Ξ±) (isom.from .Ξ±) (ap Ξ± isom.invl) (ap Ξ± isom.invr)
  y-is-y = Z.make-iso ( .Ξ²) (isom.from .Ξ²) (ap Ξ² isom.invl) (ap Ξ² isom.invr)

Observe that, over fΛ†Ξ±\^f_\alpha and fΛ†Ξ²\^f_\beta, the map components omo_m and omβ€²o'_m are equal (we say β€œequal” rather than β€œidentified” because Hom-sets are sets). This follows by standard abstract nonsense: in particular, functors between univalent categories respect isomorphisms in a strong sense (F-map-path).

This lets us reduce statements about FF and GG’s object-part action on paths to statements about their morphism parts F1F_1, G1G_1 on the components of the isomorphisms fΞ±f_\alpha and fΞ²f_\beta. But then we have

G1(fΞ²)omF1(fΞ±βˆ’1)=omβ€²F1(fΞ±)F1(fΞ±βˆ’1)=omβ€²F1(fΞ±fΞ±βˆ’1)=omβ€², \begin{split} G_1(f_\beta) o_m F_1(f_\alpha^{-1}) &= o'_m F_1(f_\alpha) F_1(f_\alpha^{-1}) \\ &= o'_m F_1(f_\alpha f_\alpha^{-1}) \\ &= o'_m\text{,} \end{split}

so over these isomorphisms the map parts become equal, thus establishing an identification o≑oβ€²o \equiv o'.

  objs : ob ≑ obβ€²
  objs i .x = iso→path Y yuniv x-is-x i
  objs i .y = iso→path Z zuniv y-is-y i
  objs i .map = lemmaβ€² i where
    lemmaβ€² : PathP (Ξ» i β†’ X.Hom (F.β‚€ (objs i .x)) (G.β‚€ (objs i .y)))
              (ob .map) (obβ€² .map)
    lemmaβ€² = transport
      (Ξ» i β†’ PathP (Ξ» j β†’ X.Hom (F-map-path F x-is-x yuniv xuniv (~ i) j)
                                (F-map-path G y-is-y zuniv xuniv (~ i) j))
                   (ob .map) (obβ€² .map)) $
      Hom-pathp-iso X xuniv $
        X.pulll   (sym ( .sq)) βˆ™
        X.cancelr (F.annihilate (ap Ξ± isom.invl))

It still remains to show that, over this identification, the isomorphism ff is equal to the identity function. But this is simply a matter of pushing the identifications down to reach the β€œleaf” morphisms.

  maps : PathP (Ξ» i β†’ ob F↓G.β‰… objs i) _ isom
  maps = F↓G.β‰…-pathp _ _
    (↓Hom-pathp _ _ (Hom-pathp-reflr-iso Y yuniv (Y.idr _))
                    (Hom-pathp-reflr-iso Z zuniv (Z.idr _)))