Isomorphisms in noncoherent wild higher precategories

Content created by Fredrik Bakke and Vojtěch Štěpančík.

Created on 2024-06-16.
Last modified on 2024-07-10.

{-# OPTIONS --guardedness #-}

module wild-category-theory.isomorphisms-in-noncoherent-wild-higher-precategories where

Imports

open import foundation.dependent-pair-types
open import foundation.universe-levels

open import wild-category-theory.noncoherent-wild-higher-precategories

Idea

Consider a noncoherent wild higher precategory 𝒞. An isomorphism^¶ in 𝒞 is a morphism f : x → y in 𝒞 equipped with

a morphism s : y → x
a $2$ -morphism is-split-epi : f ∘ s → id, where ∘ and id denote composition of morphisms and the identity morphism given by the transitive and reflexive structure on the underlying globular type, respectively
a proof is-iso-is-split-epi : is-iso is-split-epi, which shows that the above $2$ -morphism is itself an isomorphism
a morphism r : y → x
a $2$ -morphism is-split-mono : r ∘ f → id
a proof is-iso-is-split-mono : is-iso is-split-mono.

This definition of an isomorphism mirrors the definition of biinvertible maps between types.

It would be in the spirit of the library to first define what split epimorphisms and split monomorphisms are, and then define isomorphisms as those morphisms which are both. When attempting that definition, one runs into the problem that the $2$ -morphisms in the definitions should still be isomorphisms. Alternatively, one could require that the morphisms compose to the identity morphism up to identification of morphisms, instead of up to a higher isomorphism.

Note that a noncoherent wild higher precategory is the most general setting that allows us to define isomorphisms in wild categories, but because of the missing coherences, we cannot show any of the expected properties. For example we cannot show that all identities are isomorphisms, or that isomorphisms compose.

Definitions

The predicate on a morphism of being an isomorphism

record
  is-iso-Noncoherent-Wild-Higher-Precategory
  {l1 l2 : Level} (𝒞 : Noncoherent-Wild-Higher-Precategory l1 l2)
  {x y : obj-Noncoherent-Wild-Higher-Precategory 𝒞}
  (f : hom-Noncoherent-Wild-Higher-Precategory 𝒞 x y) : UU l2
  where
  coinductive
  field
    hom-section-is-iso-Noncoherent-Wild-Higher-Precategory :
      hom-Noncoherent-Wild-Higher-Precategory 𝒞 y x
    is-split-epi-is-iso-Noncoherent-Wild-Higher-Precategory :
      2-hom-Noncoherent-Wild-Higher-Precategory 𝒞
        ( comp-hom-Noncoherent-Wild-Higher-Precategory 𝒞
          ( f)
          ( hom-section-is-iso-Noncoherent-Wild-Higher-Precategory))
        ( id-hom-Noncoherent-Wild-Higher-Precategory 𝒞)
    is-iso-is-split-epi-is-iso-Noncoherent-Wild-Higher-Precategory :
      is-iso-Noncoherent-Wild-Higher-Precategory
        ( hom-noncoherent-wild-higher-precategory-Noncoherent-Wild-Higher-Precategory
          ( 𝒞)
          ( y)
          ( y))
        ( is-split-epi-is-iso-Noncoherent-Wild-Higher-Precategory)

    hom-retraction-is-iso-Noncoherent-Wild-Higher-Precategory :
      hom-Noncoherent-Wild-Higher-Precategory 𝒞 y x
    is-split-mono-is-iso-Noncoherent-Wild-Higher-Precategory :
      2-hom-Noncoherent-Wild-Higher-Precategory 𝒞
        ( comp-hom-Noncoherent-Wild-Higher-Precategory 𝒞
          ( hom-retraction-is-iso-Noncoherent-Wild-Higher-Precategory)
          ( f))
        ( id-hom-Noncoherent-Wild-Higher-Precategory 𝒞)
    is-iso-is-split-mono-is-iso-Noncoherent-Wild-Higher-Precategory :
      is-iso-Noncoherent-Wild-Higher-Precategory
        ( hom-noncoherent-wild-higher-precategory-Noncoherent-Wild-Higher-Precategory
          ( 𝒞)
          ( x)
          ( x))
        ( is-split-mono-is-iso-Noncoherent-Wild-Higher-Precategory)

open is-iso-Noncoherent-Wild-Higher-Precategory public

Isomorphisms in a noncoherent wild higher precategory

iso-Noncoherent-Wild-Higher-Precategory :
  {l1 l2 : Level} (𝒞 : Noncoherent-Wild-Higher-Precategory l1 l2)
  (x y : obj-Noncoherent-Wild-Higher-Precategory 𝒞) →
  UU l2
iso-Noncoherent-Wild-Higher-Precategory 𝒞 x y =
  Σ ( hom-Noncoherent-Wild-Higher-Precategory 𝒞 x y)
    ( is-iso-Noncoherent-Wild-Higher-Precategory 𝒞)

Components of an isomorphism in a noncoherent wild higher precategory

module _
  {l1 l2 : Level} {𝒞 : Noncoherent-Wild-Higher-Precategory l1 l2}
  {x y : obj-Noncoherent-Wild-Higher-Precategory 𝒞}
  (f : iso-Noncoherent-Wild-Higher-Precategory 𝒞 x y)
  where

  hom-iso-Noncoherent-Wild-Higher-Precategory :
    hom-Noncoherent-Wild-Higher-Precategory 𝒞 x y
  hom-iso-Noncoherent-Wild-Higher-Precategory = pr1 f

  is-iso-hom-iso-Noncoherent-Wild-Higher-Precategory :
    is-iso-Noncoherent-Wild-Higher-Precategory 𝒞
      ( hom-iso-Noncoherent-Wild-Higher-Precategory)
  is-iso-hom-iso-Noncoherent-Wild-Higher-Precategory = pr2 f

  hom-section-iso-Noncoherent-Wild-Higher-Precategory :
    hom-Noncoherent-Wild-Higher-Precategory 𝒞 y x
  hom-section-iso-Noncoherent-Wild-Higher-Precategory =
    hom-section-is-iso-Noncoherent-Wild-Higher-Precategory
      ( is-iso-hom-iso-Noncoherent-Wild-Higher-Precategory)

  is-split-epi-iso-Noncoherent-Wild-Higher-Precategory :
    2-hom-Noncoherent-Wild-Higher-Precategory 𝒞
      ( comp-hom-Noncoherent-Wild-Higher-Precategory 𝒞
        ( hom-iso-Noncoherent-Wild-Higher-Precategory)
        ( hom-section-iso-Noncoherent-Wild-Higher-Precategory))
      ( id-hom-Noncoherent-Wild-Higher-Precategory 𝒞)
  is-split-epi-iso-Noncoherent-Wild-Higher-Precategory =
    is-split-epi-is-iso-Noncoherent-Wild-Higher-Precategory
      ( is-iso-hom-iso-Noncoherent-Wild-Higher-Precategory)

  is-iso-is-split-epi-iso-Noncoherent-Wild-Higher-Precategory :
    is-iso-Noncoherent-Wild-Higher-Precategory
      ( hom-noncoherent-wild-higher-precategory-Noncoherent-Wild-Higher-Precategory
        ( 𝒞)
        ( y)
        ( y))
      ( is-split-epi-iso-Noncoherent-Wild-Higher-Precategory)
  is-iso-is-split-epi-iso-Noncoherent-Wild-Higher-Precategory =
    is-iso-is-split-epi-is-iso-Noncoherent-Wild-Higher-Precategory
      ( is-iso-hom-iso-Noncoherent-Wild-Higher-Precategory)

  iso-is-split-epi-iso-Noncoherent-Wild-Higher-Precategory :
    iso-Noncoherent-Wild-Higher-Precategory
      ( hom-noncoherent-wild-higher-precategory-Noncoherent-Wild-Higher-Precategory
        ( 𝒞)
        ( y)
        ( y))
      ( comp-hom-Noncoherent-Wild-Higher-Precategory 𝒞
        ( hom-iso-Noncoherent-Wild-Higher-Precategory)
        ( hom-section-iso-Noncoherent-Wild-Higher-Precategory))
      ( id-hom-Noncoherent-Wild-Higher-Precategory 𝒞)
  pr1 iso-is-split-epi-iso-Noncoherent-Wild-Higher-Precategory =
    is-split-epi-iso-Noncoherent-Wild-Higher-Precategory
  pr2 iso-is-split-epi-iso-Noncoherent-Wild-Higher-Precategory =
    is-iso-is-split-epi-iso-Noncoherent-Wild-Higher-Precategory

  hom-retraction-iso-Noncoherent-Wild-Higher-Precategory :
    hom-Noncoherent-Wild-Higher-Precategory 𝒞 y x
  hom-retraction-iso-Noncoherent-Wild-Higher-Precategory =
    hom-retraction-is-iso-Noncoherent-Wild-Higher-Precategory
      ( is-iso-hom-iso-Noncoherent-Wild-Higher-Precategory)

  is-split-mono-iso-Noncoherent-Wild-Higher-Precategory :
    2-hom-Noncoherent-Wild-Higher-Precategory 𝒞
      ( comp-hom-Noncoherent-Wild-Higher-Precategory 𝒞
        ( hom-retraction-iso-Noncoherent-Wild-Higher-Precategory)
        ( hom-iso-Noncoherent-Wild-Higher-Precategory))
      ( id-hom-Noncoherent-Wild-Higher-Precategory 𝒞)
  is-split-mono-iso-Noncoherent-Wild-Higher-Precategory =
    is-split-mono-is-iso-Noncoherent-Wild-Higher-Precategory
      ( is-iso-hom-iso-Noncoherent-Wild-Higher-Precategory)

  is-iso-is-split-mono-iso-Noncoherent-Wild-Higher-Precategory :
    is-iso-Noncoherent-Wild-Higher-Precategory
      ( hom-noncoherent-wild-higher-precategory-Noncoherent-Wild-Higher-Precategory
        ( 𝒞)
        ( x)
        ( x))
      ( is-split-mono-iso-Noncoherent-Wild-Higher-Precategory)
  is-iso-is-split-mono-iso-Noncoherent-Wild-Higher-Precategory =
    is-iso-is-split-mono-is-iso-Noncoherent-Wild-Higher-Precategory
      ( is-iso-hom-iso-Noncoherent-Wild-Higher-Precategory)

  iso-is-split-mono-iso-Noncoherent-Wild-Higher-Precategory :
    iso-Noncoherent-Wild-Higher-Precategory
      ( hom-noncoherent-wild-higher-precategory-Noncoherent-Wild-Higher-Precategory
        ( 𝒞)
        ( x)
        ( x))
      ( comp-hom-Noncoherent-Wild-Higher-Precategory 𝒞
        ( hom-retraction-iso-Noncoherent-Wild-Higher-Precategory)
        ( hom-iso-Noncoherent-Wild-Higher-Precategory))
      ( id-hom-Noncoherent-Wild-Higher-Precategory 𝒞)
  pr1 iso-is-split-mono-iso-Noncoherent-Wild-Higher-Precategory =
    is-split-mono-iso-Noncoherent-Wild-Higher-Precategory
  pr2 iso-is-split-mono-iso-Noncoherent-Wild-Higher-Precategory =
    is-iso-is-split-mono-iso-Noncoherent-Wild-Higher-Precategory

Recent changes

2024-07-10. Fredrik Bakke. chore: Fix some typos in the wild-category-theory module (#1158).
2024-06-16. Fredrik Bakke and Vojtěch Štěpančík. Noncoherent wild higher precategories (#1099).

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