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.noncoherent-wild-higher-precategories where
Imports
open import category-theory.precategories open import foundation.action-on-identifications-binary-functions open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.sets open import foundation.strictly-involutive-identity-types open import foundation.universe-levels open import structured-types.globular-types open import structured-types.reflexive-globular-types open import structured-types.transitive-globular-types
Idea
It is an important open problem known as the coherence problem to define a fully coherent notion of -category in univalent type theory. The subject of wild category theory attempts to recover some of the benefits of -category theory without tackling this problem. We introduce, as one of our basic building blocks in this subject, the notion of a noncoherent wild higher precategory.
A noncoherent wild higher precategory 𝒞
is a structure that attempts at
capturing the structure of a higher precategory to the 'th order. It consists
of in some sense all of the operations and none of the coherence of a higher
precategory. Thus, it is defined as a
globular type with families of
-morphisms labeled as "identities"
id-hom : (x : 𝑛-Cell 𝒞) → (𝑛+1)-Cell 𝒞 x x
and a composition operation at every dimension
comp-hom :
{x y z : 𝑛-Cell 𝒞} → (𝑛+1)-Cell 𝒞 y z → (𝑛+1)-Cell 𝒞 x y → (𝑛+1)-Cell 𝒞 x z.
Entirely concretely, we define a noncoherent wild higher precategory¶ to be a reflexive and transitive globular type. We call the 0-cells the objects, the 1-cells the morphisms and the higher cells the -morphisms. The reflexivities are called the identity morphisms, and the transitivity operations are branded as composition of morphisms.
Definitions
Noncoherent wild higher precategories
Noncoherent-Wild-Higher-Precategory : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) Noncoherent-Wild-Higher-Precategory l1 l2 = Σ ( UU l1) ( λ obj-Noncoherent-Wild-Higher-Precategory → Σ ( globular-structure l2 obj-Noncoherent-Wild-Higher-Precategory) ( λ hom-globular-structure-Noncoherent-Wild-Higher-Precategory → ( is-reflexive-globular-structure ( hom-globular-structure-Noncoherent-Wild-Higher-Precategory)) × ( is-transitive-globular-structure ( hom-globular-structure-Noncoherent-Wild-Higher-Precategory)))) make-Noncoherent-Wild-Higher-Precategory : {l1 l2 : Level} → (obj-Noncoherent-Wild-Higher-Precategory : UU l1) (hom-globular-structure-Noncoherent-Wild-Higher-Precategory : globular-structure l2 obj-Noncoherent-Wild-Higher-Precategory) → ( is-reflexive-globular-structure hom-globular-structure-Noncoherent-Wild-Higher-Precategory) → ( is-transitive-globular-structure hom-globular-structure-Noncoherent-Wild-Higher-Precategory) → Noncoherent-Wild-Higher-Precategory l1 l2 make-Noncoherent-Wild-Higher-Precategory obj hom id comp = ( obj , hom , id , comp) {-# INLINE make-Noncoherent-Wild-Higher-Precategory #-} module _ {l1 l2 : Level} (𝒞 : Noncoherent-Wild-Higher-Precategory l1 l2) where obj-Noncoherent-Wild-Higher-Precategory : UU l1 obj-Noncoherent-Wild-Higher-Precategory = pr1 𝒞 hom-globular-structure-Noncoherent-Wild-Higher-Precategory : globular-structure l2 obj-Noncoherent-Wild-Higher-Precategory hom-globular-structure-Noncoherent-Wild-Higher-Precategory = pr1 (pr2 𝒞) id-hom-globular-structure-Noncoherent-Wild-Higher-Precategory : is-reflexive-globular-structure ( hom-globular-structure-Noncoherent-Wild-Higher-Precategory) id-hom-globular-structure-Noncoherent-Wild-Higher-Precategory = pr1 (pr2 (pr2 𝒞)) comp-hom-globular-structure-Noncoherent-Wild-Higher-Precategory : is-transitive-globular-structure ( hom-globular-structure-Noncoherent-Wild-Higher-Precategory) comp-hom-globular-structure-Noncoherent-Wild-Higher-Precategory = pr2 (pr2 (pr2 𝒞)) globular-type-Noncoherent-Wild-Higher-Precategory : Globular-Type l1 l2 pr1 globular-type-Noncoherent-Wild-Higher-Precategory = obj-Noncoherent-Wild-Higher-Precategory pr2 globular-type-Noncoherent-Wild-Higher-Precategory = hom-globular-structure-Noncoherent-Wild-Higher-Precategory
We record some common projections for noncoherent wild higher precategories.
hom-Noncoherent-Wild-Higher-Precategory : obj-Noncoherent-Wild-Higher-Precategory → obj-Noncoherent-Wild-Higher-Precategory → UU l2 hom-Noncoherent-Wild-Higher-Precategory = 1-cell-globular-structure ( hom-globular-structure-Noncoherent-Wild-Higher-Precategory) id-hom-Noncoherent-Wild-Higher-Precategory : {x : obj-Noncoherent-Wild-Higher-Precategory} → hom-Noncoherent-Wild-Higher-Precategory x x id-hom-Noncoherent-Wild-Higher-Precategory = refl-1-cell-is-reflexive-globular-structure ( id-hom-globular-structure-Noncoherent-Wild-Higher-Precategory) comp-hom-Noncoherent-Wild-Higher-Precategory : {x y z : obj-Noncoherent-Wild-Higher-Precategory} → hom-Noncoherent-Wild-Higher-Precategory y z → hom-Noncoherent-Wild-Higher-Precategory x y → hom-Noncoherent-Wild-Higher-Precategory x z comp-hom-Noncoherent-Wild-Higher-Precategory = comp-1-cell-is-transitive-globular-structure ( comp-hom-globular-structure-Noncoherent-Wild-Higher-Precategory) hom-globular-type-Noncoherent-Wild-Higher-Precategory : (x y : obj-Noncoherent-Wild-Higher-Precategory) → Globular-Type l2 l2 pr1 (hom-globular-type-Noncoherent-Wild-Higher-Precategory x y) = hom-Noncoherent-Wild-Higher-Precategory x y pr2 (hom-globular-type-Noncoherent-Wild-Higher-Precategory x y) = globular-structure-1-cell-globular-structure ( hom-globular-structure-Noncoherent-Wild-Higher-Precategory) ( x) ( y) hom-noncoherent-wild-higher-precategory-Noncoherent-Wild-Higher-Precategory : (x y : obj-Noncoherent-Wild-Higher-Precategory) → Noncoherent-Wild-Higher-Precategory l2 l2 hom-noncoherent-wild-higher-precategory-Noncoherent-Wild-Higher-Precategory x y = make-Noncoherent-Wild-Higher-Precategory ( hom-Noncoherent-Wild-Higher-Precategory x y) ( globular-structure-1-cell-globular-structure ( hom-globular-structure-Noncoherent-Wild-Higher-Precategory) ( x) ( y)) ( is-reflexive-globular-structure-1-cell-is-reflexive-globular-structure ( id-hom-globular-structure-Noncoherent-Wild-Higher-Precategory) ( x) ( y)) ( is-transitive-globular-structure-1-cell-is-transitive-globular-structure ( comp-hom-globular-structure-Noncoherent-Wild-Higher-Precategory) ( x) ( y))
2-hom-Noncoherent-Wild-Higher-Precategory : {x y : obj-Noncoherent-Wild-Higher-Precategory} → hom-Noncoherent-Wild-Higher-Precategory x y → hom-Noncoherent-Wild-Higher-Precategory x y → UU l2 2-hom-Noncoherent-Wild-Higher-Precategory = 2-cell-globular-structure ( hom-globular-structure-Noncoherent-Wild-Higher-Precategory) id-2-hom-Noncoherent-Wild-Higher-Precategory : {x y : obj-Noncoherent-Wild-Higher-Precategory} {f : hom-Noncoherent-Wild-Higher-Precategory x y} → 2-hom-Noncoherent-Wild-Higher-Precategory f f id-2-hom-Noncoherent-Wild-Higher-Precategory = refl-2-cell-is-reflexive-globular-structure ( id-hom-globular-structure-Noncoherent-Wild-Higher-Precategory) comp-2-hom-Noncoherent-Wild-Higher-Precategory : {x y : obj-Noncoherent-Wild-Higher-Precategory} {f g h : hom-Noncoherent-Wild-Higher-Precategory x y} → 2-hom-Noncoherent-Wild-Higher-Precategory g h → 2-hom-Noncoherent-Wild-Higher-Precategory f g → 2-hom-Noncoherent-Wild-Higher-Precategory f h comp-2-hom-Noncoherent-Wild-Higher-Precategory = comp-2-cell-is-transitive-globular-structure ( comp-hom-globular-structure-Noncoherent-Wild-Higher-Precategory)
3-hom-Noncoherent-Wild-Higher-Precategory : {x y : obj-Noncoherent-Wild-Higher-Precategory} {f g : hom-Noncoherent-Wild-Higher-Precategory x y} → 2-hom-Noncoherent-Wild-Higher-Precategory f g → 2-hom-Noncoherent-Wild-Higher-Precategory f g → UU l2 3-hom-Noncoherent-Wild-Higher-Precategory = 3-cell-globular-structure ( hom-globular-structure-Noncoherent-Wild-Higher-Precategory) id-3-hom-Noncoherent-Wild-Higher-Precategory : {x y : obj-Noncoherent-Wild-Higher-Precategory} {f g : hom-Noncoherent-Wild-Higher-Precategory x y} {H : 2-hom-Noncoherent-Wild-Higher-Precategory f g} → 3-hom-Noncoherent-Wild-Higher-Precategory H H id-3-hom-Noncoherent-Wild-Higher-Precategory = refl-3-cell-is-reflexive-globular-structure ( id-hom-globular-structure-Noncoherent-Wild-Higher-Precategory) comp-3-hom-Noncoherent-Wild-Higher-Precategory : {x y : obj-Noncoherent-Wild-Higher-Precategory} {f g : hom-Noncoherent-Wild-Higher-Precategory x y} {H K L : 2-hom-Noncoherent-Wild-Higher-Precategory f g} → 3-hom-Noncoherent-Wild-Higher-Precategory K L → 3-hom-Noncoherent-Wild-Higher-Precategory H K → 3-hom-Noncoherent-Wild-Higher-Precategory H L comp-3-hom-Noncoherent-Wild-Higher-Precategory = comp-3-cell-is-transitive-globular-structure ( comp-hom-globular-structure-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).