# Isomorphisms in noncoherent large wild higher precategories

Content created by Fredrik Bakke and Vojtěch Štěpančík.

Created on 2024-06-16.

{-# OPTIONS --guardedness #-}

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

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

open import wild-category-theory.isomorphisms-in-noncoherent-wild-higher-precategories
open import wild-category-theory.noncoherent-large-wild-higher-precategories


## Idea

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

• a morphism s : y → x
• a -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 -morphism is itself an isomorphism
• a morphism r : y → x
• a -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 -morphisms in the definitions should still be isomorphisms.

Note that a noncoherent large wild higher precategory is the most general setting that allows us to define isomorphisms in large 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 morphisms of being an isomorphism

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

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

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


### Isomorphisms in a noncoherent large wild higher precategory

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


### Components of an isomorphism in a noncoherent large wild higher precategory

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

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

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

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

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

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

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

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

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

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

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