Synthetic categories
Content created by Egbert Rijke and Ivan Kobe.
Created on 2024-09-01.
Last modified on 2024-09-25.
{-# OPTIONS --guardedness #-} module synthetic-category-theory.synthetic-categories where
Imports
open import foundation.dependent-pair-types open import foundation.universe-levels open import structured-types.globular-types
Idea
Synthetic categories¶ are defined by establishing the rules on the type of all synthetic categories. In particular, synthetic categories are not defined to be types of objects equipped with hom-sets and so on, but they are simply elements of the type of synthetic categories, which is given sufficient structure so that we can work with its elements as if they are categories.
The primitive notions of synthetic categories are:
- Synthetic categories
- Functors between them,
- Natural isomorphisms between them,
- Natural isomorphisms between those,
- and so on.
The type of synthetic categories is furthermore postulated to have the following structure:
- A terminal category
- An initial category
- Cartesian product categories
- Coproduct categories
- Pullbacks of categories
- Functor categories
- A representing arrow
- A representing commuting triangle
Furthermore, coproducts are assumed to be universal, there is a Segal axiom and a Rezk axiom, and some that we haven’t listed here.
The theory of synthetic categories is not intended to be infinitely coherent. Similar to wild category theory, some higher coherences, such as the Mac Lane pentagon and higher associahedra, are missing. Nevertheless, the theory is strong enough to embody a large amount of higher category theory.
Definitions
The type of synthetic categories
The language of synthetic categories
In synthetic category theory we may speak of categories, functors, isomorphisms between them, isomorphisms between those, and so forth. The sorts in the language of synthetic category theory are therefore organized in a globular type.
module _ (l : Level) where language-Synthetic-Category-Theory : UU (lsuc l) language-Synthetic-Category-Theory = Globular-Type l l
The sort of categories in the language of synthetic category theory
The sort of categories in the language of synthetic category theory is the type
of 0-cells in the globular type of sorts of the language of synthetic category
theory.
module _ {l : Level} where category-Synthetic-Category-Theory : language-Synthetic-Category-Theory l → UU l category-Synthetic-Category-Theory = 0-cell-Globular-Type
The sort of functors in the language of synthetic category theory
The sort of functors from C to D in the language of synthetic category
theory is the type of 1-cells between C and D in the globular type of
sorts of the language of synthetic category theory.
module _ {l : Level} where functor-Synthetic-Category-Theory : (κ : language-Synthetic-Category-Theory l) → (C D : category-Synthetic-Category-Theory κ) → UU l functor-Synthetic-Category-Theory = 1-cell-Globular-Type
The source and target of a functor
module _ {l : Level} where source-functor-Synthetic-Category-Theory : (κ : language-Synthetic-Category-Theory l) {C D : category-Synthetic-Category-Theory κ} → functor-Synthetic-Category-Theory κ C D → category-Synthetic-Category-Theory κ source-functor-Synthetic-Category-Theory κ {C = C} f = C target-functor-Synthetic-Category-Theory : (κ : language-Synthetic-Category-Theory l) {C D : category-Synthetic-Category-Theory κ} → functor-Synthetic-Category-Theory κ C D → category-Synthetic-Category-Theory κ target-functor-Synthetic-Category-Theory κ {D = D} f = D
The globular type of functors between categories
The globular type of functors from C to D in the language of synthetic
category theory is the globular type of 1-cells between C and D in the
globular type of sorts of the language of synthetic category theory.
module _ {l : Level} where functor-globular-type-Synthetic-Category-Theory : (κ : language-Synthetic-Category-Theory l) → (C D : category-Synthetic-Category-Theory κ) → Globular-Type l l functor-globular-type-Synthetic-Category-Theory = 1-cell-globular-type-Globular-Type
The sort of isomorphisms between functors in the language ofsynthetic category theory
The sort of isomorphisms between functors F and G in the language of
synthetic category theory is the type of 2-cells between F and G in the
globular type of sorts of the language of synthetic category theory.
module _ {l : Level} where isomorphism-Synthetic-Category-Theory : (κ : language-Synthetic-Category-Theory l) {C D : category-Synthetic-Category-Theory κ} (F G : functor-Synthetic-Category-Theory κ C D) → UU l isomorphism-Synthetic-Category-Theory = 2-cell-Globular-Type
Isomorphism between isomorphisms between functors in the language of synthetic category theory
module _ {l : Level} where 3-isomorphism-Synthetic-Category-Theory : (κ : language-Synthetic-Category-Theory l) {C D : category-Synthetic-Category-Theory κ} {F G : functor-Synthetic-Category-Theory κ C D} (α β : isomorphism-Synthetic-Category-Theory κ F G) → UU l 3-isomorphism-Synthetic-Category-Theory = 3-cell-Globular-Type
The structure of identity morphisms in the language of synthetic category theory
In the language of synthetic category theory we may speak of the identity functor between categories, the identity isomorphism between functors, and so on. The structure of identity morphisms is therefore a coinductive record, where the base type is the type of identity functors between synthetic categories, and coinductively the structure of identity morphisms in the globular type of functors between any two synthetic categories.
module _ {l : Level} where record identity-Synthetic-Category-Theory (κ : language-Synthetic-Category-Theory l) : UU l where coinductive field id-functor-Synthetic-Category-Theory : (C : category-Synthetic-Category-Theory κ) → functor-Synthetic-Category-Theory κ C C identity-isomorphism-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} → identity-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C D) open identity-Synthetic-Category-Theory public id-iso-Synthetic-Category-Theory : {κ : language-Synthetic-Category-Theory l} (ι : identity-Synthetic-Category-Theory κ) {C D : category-Synthetic-Category-Theory κ} (F : functor-Synthetic-Category-Theory κ C D) → isomorphism-Synthetic-Category-Theory κ F F id-iso-Synthetic-Category-Theory ι = id-functor-Synthetic-Category-Theory ( identity-isomorphism-Synthetic-Category-Theory ι)
Composition operators in the language of synthetic category theory
The language of synthetic category theory is equipped with composition operators at each level.
module _ {l : Level} where record composition-Synthetic-Category-Theory (κ : language-Synthetic-Category-Theory l) : UU l where coinductive field comp-functor-Synthetic-Category-Theory : {C D E : category-Synthetic-Category-Theory κ} → functor-Synthetic-Category-Theory κ D E → functor-Synthetic-Category-Theory κ C D → functor-Synthetic-Category-Theory κ C E composition-isomorphism-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} → composition-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C D) open composition-Synthetic-Category-Theory public comp-iso-Synthetic-Category-Theory : {κ : language-Synthetic-Category-Theory l} (μ : composition-Synthetic-Category-Theory κ) → {C D : category-Synthetic-Category-Theory κ} {F G H : functor-Synthetic-Category-Theory κ C D} → isomorphism-Synthetic-Category-Theory κ G H → isomorphism-Synthetic-Category-Theory κ F G → isomorphism-Synthetic-Category-Theory κ F H comp-iso-Synthetic-Category-Theory μ = comp-functor-Synthetic-Category-Theory ( composition-isomorphism-Synthetic-Category-Theory μ)
Inverses of isomorphisms
Isomorphisms between functors, as well as higher isomorphisms, are invertible.
module _ {l : Level} where record inverse-Synthetic-Category-Theory (κ : language-Synthetic-Category-Theory l) (μ : composition-Synthetic-Category-Theory κ) (ι : identity-Synthetic-Category-Theory κ) : UU l where coinductive field inv-iso-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} → {F G : functor-Synthetic-Category-Theory κ C D} → (isomorphism-Synthetic-Category-Theory κ F G) → (isomorphism-Synthetic-Category-Theory κ G F) inv-iso-left-inv-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} → {F G : functor-Synthetic-Category-Theory κ C D} → (α : isomorphism-Synthetic-Category-Theory κ F G) → isomorphism-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C D) ( comp-iso-Synthetic-Category-Theory μ ( inv-iso-Synthetic-Category-Theory α) α) ( id-iso-Synthetic-Category-Theory ι F) inv-iso-right-inv-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} → {F G : functor-Synthetic-Category-Theory κ C D} → (α : isomorphism-Synthetic-Category-Theory κ F G) → isomorphism-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C D) ( comp-iso-Synthetic-Category-Theory μ α ( inv-iso-Synthetic-Category-Theory α)) ( id-iso-Synthetic-Category-Theory ι G) inverse-isomorphism-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} → inverse-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C D) ( composition-isomorphism-Synthetic-Category-Theory μ) ( identity-isomorphism-Synthetic-Category-Theory ι) open inverse-Synthetic-Category-Theory public
Left unit law operators in the language of synthetic category theory
module _ {l : Level} where record left-unit-law-composition-Synthetic-Category-Theory (κ : language-Synthetic-Category-Theory l) (ι : identity-Synthetic-Category-Theory κ) (μ : composition-Synthetic-Category-Theory κ) : UU l where coinductive field left-unit-law-comp-functor-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} (F : functor-Synthetic-Category-Theory κ C D) → isomorphism-Synthetic-Category-Theory κ ( comp-functor-Synthetic-Category-Theory μ ( id-functor-Synthetic-Category-Theory ι D) ( F)) ( F) left-unit-law-composition-isomorphism-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} → left-unit-law-composition-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C D) ( identity-isomorphism-Synthetic-Category-Theory ι) ( composition-isomorphism-Synthetic-Category-Theory μ) open left-unit-law-composition-Synthetic-Category-Theory public
Right unit law operators in the language of synthetic category theory
record right-unit-law-composition-Synthetic-Category-Theory (κ : language-Synthetic-Category-Theory l) (ι : identity-Synthetic-Category-Theory κ) (μ : composition-Synthetic-Category-Theory κ) : UU l where coinductive field right-unit-law-comp-functor-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} (F : functor-Synthetic-Category-Theory κ C D) → isomorphism-Synthetic-Category-Theory κ ( comp-functor-Synthetic-Category-Theory μ ( F) ( id-functor-Synthetic-Category-Theory ι C)) ( F) right-unit-law-composition-isomorphism-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} → right-unit-law-composition-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C D) ( identity-isomorphism-Synthetic-Category-Theory ι) ( composition-isomorphism-Synthetic-Category-Theory μ) open right-unit-law-composition-Synthetic-Category-Theory public
Associators in the language of synthetic category theory
module _ {l : Level} where record associative-composition-Synthetic-Category-Theory (κ : language-Synthetic-Category-Theory l) (μ : composition-Synthetic-Category-Theory κ) : UU l where coinductive field associative-comp-functor-Synthetic-Category-Theory : {C1 C2 C3 C4 : category-Synthetic-Category-Theory κ} (H : functor-Synthetic-Category-Theory κ C3 C4) (G : functor-Synthetic-Category-Theory κ C2 C3) (F : functor-Synthetic-Category-Theory κ C1 C2) → isomorphism-Synthetic-Category-Theory κ ( comp-functor-Synthetic-Category-Theory μ ( comp-functor-Synthetic-Category-Theory μ H G) ( F)) ( comp-functor-Synthetic-Category-Theory μ ( H) ( comp-functor-Synthetic-Category-Theory μ G F)) associative-comp-isomorphism-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} → associative-composition-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C D) ( composition-isomorphism-Synthetic-Category-Theory μ) open associative-composition-Synthetic-Category-Theory public
Horizontal composition operators in the language of synthetic category theory
module _ {l : Level} where record horizontal-composition-Synthetic-Category-Theory (κ : language-Synthetic-Category-Theory l) (μ : composition-Synthetic-Category-Theory κ) : UU l where coinductive field horizontal-comp-iso-Synthetic-Category-Theory : {C D E : category-Synthetic-Category-Theory κ} {H I : functor-Synthetic-Category-Theory κ D E} {F G : functor-Synthetic-Category-Theory κ C D} → isomorphism-Synthetic-Category-Theory κ H I → isomorphism-Synthetic-Category-Theory κ F G → isomorphism-Synthetic-Category-Theory κ ( comp-functor-Synthetic-Category-Theory μ H F) ( comp-functor-Synthetic-Category-Theory μ I G) horizontal-composition-isomorphism-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} → horizontal-composition-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C D) ( composition-isomorphism-Synthetic-Category-Theory μ) open horizontal-composition-Synthetic-Category-Theory public
We can prove that taking an inverse is an idempotent operation
module _ {l : Level} where inverse-idempotent-Synthetic-Category-Theory : (κ : language-Synthetic-Category-Theory l) (μ : composition-Synthetic-Category-Theory κ) (ι : identity-Synthetic-Category-Theory κ) (ν : inverse-Synthetic-Category-Theory κ μ ι) (Α : associative-composition-Synthetic-Category-Theory κ μ) (Χ : horizontal-composition-Synthetic-Category-Theory κ μ) (Λ : left-unit-law-composition-Synthetic-Category-Theory κ ι μ) (Ρ : right-unit-law-composition-Synthetic-Category-Theory κ ι μ) {C D : category-Synthetic-Category-Theory κ} {F G : functor-Synthetic-Category-Theory κ C D} (α : isomorphism-Synthetic-Category-Theory κ F G) → isomorphism-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C D) ( inv-iso-Synthetic-Category-Theory ν ( inv-iso-Synthetic-Category-Theory ν α)) ( α) inverse-idempotent-Synthetic-Category-Theory κ μ ι ν Α Χ Λ Ρ α = comp-iso-Synthetic-Category-Theory ( composition-isomorphism-Synthetic-Category-Theory μ) ( left-unit-law-comp-functor-Synthetic-Category-Theory ( left-unit-law-composition-isomorphism-Synthetic-Category-Theory Λ) ( α)) ( comp-iso-Synthetic-Category-Theory ( composition-isomorphism-Synthetic-Category-Theory μ) ( horizontal-comp-iso-Synthetic-Category-Theory ( horizontal-composition-isomorphism-Synthetic-Category-Theory Χ) ( inv-iso-left-inv-Synthetic-Category-Theory ν ( inv-iso-Synthetic-Category-Theory ν α)) ( id-iso-Synthetic-Category-Theory ( identity-isomorphism-Synthetic-Category-Theory ι) ( α))) ( comp-iso-Synthetic-Category-Theory ( composition-isomorphism-Synthetic-Category-Theory μ) ( inv-iso-Synthetic-Category-Theory ( inverse-isomorphism-Synthetic-Category-Theory ν) ( associative-comp-functor-Synthetic-Category-Theory ( associative-comp-isomorphism-Synthetic-Category-Theory Α) ( inv-iso-Synthetic-Category-Theory ν ( inv-iso-Synthetic-Category-Theory ν α)) ( inv-iso-Synthetic-Category-Theory ν α) ( α))) ( comp-iso-Synthetic-Category-Theory ( composition-isomorphism-Synthetic-Category-Theory μ) ( horizontal-comp-iso-Synthetic-Category-Theory ( horizontal-composition-isomorphism-Synthetic-Category-Theory Χ) ( id-iso-Synthetic-Category-Theory ( identity-isomorphism-Synthetic-Category-Theory ι) ( inv-iso-Synthetic-Category-Theory ν ( inv-iso-Synthetic-Category-Theory ν α))) ( inv-iso-Synthetic-Category-Theory ( inverse-isomorphism-Synthetic-Category-Theory ν) ( inv-iso-left-inv-Synthetic-Category-Theory ν α))) ( inv-iso-Synthetic-Category-Theory ( inverse-isomorphism-Synthetic-Category-Theory ν) ( right-unit-law-comp-functor-Synthetic-Category-Theory ( right-unit-law-composition-isomorphism-Synthetic-Category-Theory Ρ) ( inv-iso-Synthetic-Category-Theory ν ( inv-iso-Synthetic-Category-Theory ν α)))))))
Identity preservation operators for horizontal composition in the language of synthetic category theory
module _ {l : Level} where record preserves-identity-horizontal-composition-Synthetic-Category-Theory (κ : language-Synthetic-Category-Theory l) (ι : identity-Synthetic-Category-Theory κ) (μ : composition-Synthetic-Category-Theory κ) (ν : horizontal-composition-Synthetic-Category-Theory κ μ) : UU l where coinductive field preserves-identity-horizontal-comp-iso-Synthetic-Category-Theory : {C D E : category-Synthetic-Category-Theory κ} {G : functor-Synthetic-Category-Theory κ D E} {F : functor-Synthetic-Category-Theory κ C D} → isomorphism-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C E) ( horizontal-comp-iso-Synthetic-Category-Theory ν ( id-iso-Synthetic-Category-Theory ι G) ( id-iso-Synthetic-Category-Theory ι F)) ( id-iso-Synthetic-Category-Theory ι ( comp-functor-Synthetic-Category-Theory μ G F)) preserves-identity-horizontal-composition-isomorphism-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} → preserves-identity-horizontal-composition-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C D) ( identity-isomorphism-Synthetic-Category-Theory ι) ( composition-isomorphism-Synthetic-Category-Theory μ) ( horizontal-composition-isomorphism-Synthetic-Category-Theory ν) open preserves-identity-horizontal-composition-Synthetic-Category-Theory public
Interchange operators for composition and horizontal composition in the language of synthetic category theory
module _ {l : Level} where record interchange-composition-Synthetic-Category-Theory (κ : language-Synthetic-Category-Theory l) (μ : composition-Synthetic-Category-Theory κ) (ν : horizontal-composition-Synthetic-Category-Theory κ μ) : UU l where coinductive field interchange-comp-functor-Synthetic-Category-Theory : {C D E : category-Synthetic-Category-Theory κ} {G1 G2 G3 : functor-Synthetic-Category-Theory κ D E} {F1 F2 F3 : functor-Synthetic-Category-Theory κ C D} (δ : isomorphism-Synthetic-Category-Theory κ G2 G3) (γ : isomorphism-Synthetic-Category-Theory κ G1 G2) (β : isomorphism-Synthetic-Category-Theory κ F2 F3) (α : isomorphism-Synthetic-Category-Theory κ F1 F2) → isomorphism-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C E) ( horizontal-comp-iso-Synthetic-Category-Theory ν ( comp-iso-Synthetic-Category-Theory μ δ γ) ( comp-iso-Synthetic-Category-Theory μ β α)) ( comp-iso-Synthetic-Category-Theory μ ( horizontal-comp-iso-Synthetic-Category-Theory ν δ β) ( horizontal-comp-iso-Synthetic-Category-Theory ν γ α)) interchange-composition-isomorphism-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} → interchange-composition-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C D) ( composition-isomorphism-Synthetic-Category-Theory μ) ( horizontal-composition-isomorphism-Synthetic-Category-Theory ν) open interchange-composition-Synthetic-Category-Theory public
Commuting triangles of functors in the language of synthetic category theory
module _ {l : Level} where commuting-triangle-functors-Synthetic-Category-Theory : (κ : language-Synthetic-Category-Theory l) (μ : composition-Synthetic-Category-Theory κ) {A B X : category-Synthetic-Category-Theory κ} (f : functor-Synthetic-Category-Theory κ A X) (g : functor-Synthetic-Category-Theory κ B X) (h : functor-Synthetic-Category-Theory κ A B) → UU l commuting-triangle-functors-Synthetic-Category-Theory κ μ f g h = isomorphism-Synthetic-Category-Theory κ ( f) ( comp-functor-Synthetic-Category-Theory μ g h)
Commuting squares of functors in the language of synthetic category theory
module _ {l : Level} where commuting-square-functors-Synthetic-Category-Theory : (κ : language-Synthetic-Category-Theory l) (μ : composition-Synthetic-Category-Theory κ) {A B X Y : category-Synthetic-Category-Theory κ} (f : functor-Synthetic-Category-Theory κ A B) (j : functor-Synthetic-Category-Theory κ B Y) (i : functor-Synthetic-Category-Theory κ A X) (g : functor-Synthetic-Category-Theory κ X Y) → UU l commuting-square-functors-Synthetic-Category-Theory κ μ f j i g = isomorphism-Synthetic-Category-Theory κ ( comp-functor-Synthetic-Category-Theory μ j f) ( comp-functor-Synthetic-Category-Theory μ g i)
Pasting of commutative squares of functors
module _ {l : Level} where pasting-commuting-squares-functors-Synthetic-Category-Theory : (κ : language-Synthetic-Category-Theory l) (μ : composition-Synthetic-Category-Theory κ) (ι : identity-Synthetic-Category-Theory κ) (ν : inverse-Synthetic-Category-Theory κ μ ι) (α : associative-composition-Synthetic-Category-Theory κ μ) (Χ : horizontal-composition-Synthetic-Category-Theory κ μ) {A B C X Y Z : category-Synthetic-Category-Theory κ} (f : functor-Synthetic-Category-Theory κ A B) (g : functor-Synthetic-Category-Theory κ B C) (h : functor-Synthetic-Category-Theory κ C Z) (r : functor-Synthetic-Category-Theory κ A X) (s : functor-Synthetic-Category-Theory κ X Y) (t : functor-Synthetic-Category-Theory κ Y Z) (u : functor-Synthetic-Category-Theory κ B Y) → (commuting-square-functors-Synthetic-Category-Theory κ μ f u r s) → (commuting-square-functors-Synthetic-Category-Theory κ μ g h u t) → commuting-square-functors-Synthetic-Category-Theory κ μ ( comp-functor-Synthetic-Category-Theory μ g f) ( h) ( r) ( comp-functor-Synthetic-Category-Theory μ t s) pasting-commuting-squares-functors-Synthetic-Category-Theory κ μ ι ν α Χ f g h r s t u τ σ = comp-iso-Synthetic-Category-Theory μ ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory α t s r)) ( comp-iso-Synthetic-Category-Theory μ ( horizontal-comp-iso-Synthetic-Category-Theory Χ (id-iso-Synthetic-Category-Theory ι t) τ) ( comp-iso-Synthetic-Category-Theory μ ( associative-comp-functor-Synthetic-Category-Theory α t u f) ( comp-iso-Synthetic-Category-Theory μ ( horizontal-comp-iso-Synthetic-Category-Theory Χ ( σ) ( id-iso-Synthetic-Category-Theory ι f)) ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory α h g f)))))
Commutative squares of isomorphisms in the language of synthetic category theory
module _ {l : Level} where commuting-square-isomorphisms-Synthetic-Category-Theory : (κ : language-Synthetic-Category-Theory l) (μ : composition-Synthetic-Category-Theory κ) {C D : category-Synthetic-Category-Theory κ} {k l f g : functor-Synthetic-Category-Theory κ C D} (p : isomorphism-Synthetic-Category-Theory κ k l) (q : isomorphism-Synthetic-Category-Theory κ l g) (r : isomorphism-Synthetic-Category-Theory κ k f) (s : isomorphism-Synthetic-Category-Theory κ f g) → UU _ commuting-square-isomorphisms-Synthetic-Category-Theory κ μ = commuting-square-functors-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ _ _) ( composition-isomorphism-Synthetic-Category-Theory μ)
Pasting commutative squares of isomorphisms
module _ {l : Level} where pasting-commuting-squares-isomorphisms-Synthetic-Category-Theory : (κ : language-Synthetic-Category-Theory l) (μ : composition-Synthetic-Category-Theory κ) (ι : identity-Synthetic-Category-Theory κ) (ν : inverse-Synthetic-Category-Theory κ μ ι) (Α : associative-composition-Synthetic-Category-Theory κ μ) (Χ : horizontal-composition-Synthetic-Category-Theory κ μ) {C D : category-Synthetic-Category-Theory κ} {f g h r s t : functor-Synthetic-Category-Theory κ C D} (α : isomorphism-Synthetic-Category-Theory κ f g) (β : isomorphism-Synthetic-Category-Theory κ g h) (γ : isomorphism-Synthetic-Category-Theory κ h t) (δ : isomorphism-Synthetic-Category-Theory κ f r) (ε : isomorphism-Synthetic-Category-Theory κ r s) (φ : isomorphism-Synthetic-Category-Theory κ s t) (ξ : isomorphism-Synthetic-Category-Theory κ g s) → (commuting-square-isomorphisms-Synthetic-Category-Theory κ μ α ξ δ ε) → (commuting-square-isomorphisms-Synthetic-Category-Theory κ μ β γ ξ φ) → commuting-square-isomorphisms-Synthetic-Category-Theory κ μ ( comp-iso-Synthetic-Category-Theory μ β α) ( γ) ( δ) ( comp-iso-Synthetic-Category-Theory μ φ ε) pasting-commuting-squares-isomorphisms-Synthetic-Category-Theory κ μ ι ν Α Χ = pasting-commuting-squares-functors-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ _ _) ( composition-isomorphism-Synthetic-Category-Theory μ) ( identity-isomorphism-Synthetic-Category-Theory ι) ( inverse-isomorphism-Synthetic-Category-Theory ν) ( associative-comp-isomorphism-Synthetic-Category-Theory Α) ( horizontal-composition-isomorphism-Synthetic-Category-Theory Χ)
Left unit law preservation operators for horizontal composition
module _ {l : Level} where record preserves-left-unit-law-composition-horizontal-composition-Synthetic-Category-Theory (κ : language-Synthetic-Category-Theory l) (ι : identity-Synthetic-Category-Theory κ) (μ : composition-Synthetic-Category-Theory κ) (η : left-unit-law-composition-Synthetic-Category-Theory κ ι μ) (ν : horizontal-composition-Synthetic-Category-Theory κ μ) : UU l where coinductive field preserves-left-unit-law-comp-functor-horizontal-comp-iso-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} {F G : functor-Synthetic-Category-Theory κ C D} (τ : isomorphism-Synthetic-Category-Theory κ F G) → commuting-square-isomorphisms-Synthetic-Category-Theory κ μ ( left-unit-law-comp-functor-Synthetic-Category-Theory η F) ( τ) ( horizontal-comp-iso-Synthetic-Category-Theory ν ( id-iso-Synthetic-Category-Theory ι ( id-functor-Synthetic-Category-Theory ι D)) ( τ)) ( left-unit-law-comp-functor-Synthetic-Category-Theory η G) preserves-left-unit-law-composition-horizontal-composition-isomorphism-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} → preserves-left-unit-law-composition-horizontal-composition-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C D) ( identity-isomorphism-Synthetic-Category-Theory ι) ( composition-isomorphism-Synthetic-Category-Theory μ) ( left-unit-law-composition-isomorphism-Synthetic-Category-Theory η) ( horizontal-composition-isomorphism-Synthetic-Category-Theory ν) open preserves-left-unit-law-composition-horizontal-composition-Synthetic-Category-Theory public
Right unit law preservation operators for horizontal composition
module _ {l : Level} where record preserves-right-unit-law-composition-horizontal-composition-Synthetic-Category-Theory (κ : language-Synthetic-Category-Theory l) (ι : identity-Synthetic-Category-Theory κ) (μ : composition-Synthetic-Category-Theory κ) (η : right-unit-law-composition-Synthetic-Category-Theory κ ι μ) (ν : horizontal-composition-Synthetic-Category-Theory κ μ) : UU l where coinductive field preserves-right-unit-law-comp-functor-horizontal-comp-iso-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} {F G : functor-Synthetic-Category-Theory κ C D} (τ : isomorphism-Synthetic-Category-Theory κ F G) → commuting-square-isomorphisms-Synthetic-Category-Theory κ μ ( right-unit-law-comp-functor-Synthetic-Category-Theory η F) ( τ) ( horizontal-comp-iso-Synthetic-Category-Theory ν ( τ) ( id-iso-Synthetic-Category-Theory ι ( id-functor-Synthetic-Category-Theory ι C))) ( right-unit-law-comp-functor-Synthetic-Category-Theory η G) preserves-right-unit-law-composition-horizontal-composition-isomorphism-Synthetic-Category-Theory : {C D : category-Synthetic-Category-Theory κ} → preserves-right-unit-law-composition-horizontal-composition-Synthetic-Category-Theory ( functor-globular-type-Synthetic-Category-Theory κ C D) ( identity-isomorphism-Synthetic-Category-Theory ι) ( composition-isomorphism-Synthetic-Category-Theory μ) ( right-unit-law-composition-isomorphism-Synthetic-Category-Theory η) ( horizontal-composition-isomorphism-Synthetic-Category-Theory ν) open preserves-right-unit-law-composition-horizontal-composition-Synthetic-Category-Theory public
Associator preservation operators for horizontal composition
module _ {l : Level} where record preserves-associativity-composition-horizontal-composition-Synthetic-Category-Theory (κ : language-Synthetic-Category-Theory l) (μ : composition-Synthetic-Category-Theory κ) (η : associative-composition-Synthetic-Category-Theory κ μ) (ν : horizontal-composition-Synthetic-Category-Theory κ μ) : UU l where coinductive field preserves-associativity-comp-functor-horizontal-comp-iso-Synthetic-Category-Theory : {C1 C2 C3 C4 : category-Synthetic-Category-Theory κ} {H H' : functor-Synthetic-Category-Theory κ C3 C4} {G G' : functor-Synthetic-Category-Theory κ C2 C3} {F F' : functor-Synthetic-Category-Theory κ C1 C2} (γ : isomorphism-Synthetic-Category-Theory κ H H') (β : isomorphism-Synthetic-Category-Theory κ G G') (α : isomorphism-Synthetic-Category-Theory κ F F') → commuting-square-isomorphisms-Synthetic-Category-Theory κ μ ( associative-comp-functor-Synthetic-Category-Theory η H G F) ( horizontal-comp-iso-Synthetic-Category-Theory ν ( γ) ( horizontal-comp-iso-Synthetic-Category-Theory ν β α)) ( horizontal-comp-iso-Synthetic-Category-Theory ν ( horizontal-comp-iso-Synthetic-Category-Theory ν γ β) ( α)) ( associative-comp-functor-Synthetic-Category-Theory η H' G' F') open preserves-associativity-composition-horizontal-composition-Synthetic-Category-Theory public
We can prove that the inverse of the associator preservers horizontal composition
module _ {l : Level} where preserves-associativity-comp-functor-horizontal-comp-iso-inv-Synthetic-Category-Theory : (κ : language-Synthetic-Category-Theory l) (μ : composition-Synthetic-Category-Theory κ) (ι : identity-Synthetic-Category-Theory κ) (ν : inverse-Synthetic-Category-Theory κ μ ι) (Α : associative-composition-Synthetic-Category-Theory κ μ) (Χ : horizontal-composition-Synthetic-Category-Theory κ μ) (Λ : left-unit-law-composition-Synthetic-Category-Theory κ ι μ) (Ρ : right-unit-law-composition-Synthetic-Category-Theory κ ι μ) (Ξ : preserves-associativity-composition-horizontal-composition-Synthetic-Category-Theory κ μ Α Χ) {C1 C2 C3 C4 : category-Synthetic-Category-Theory κ} (H H' : functor-Synthetic-Category-Theory κ C3 C4) (G G' : functor-Synthetic-Category-Theory κ C2 C3) (F F' : functor-Synthetic-Category-Theory κ C1 C2) (γ : isomorphism-Synthetic-Category-Theory κ H H') (β : isomorphism-Synthetic-Category-Theory κ G G') (α : isomorphism-Synthetic-Category-Theory κ F F') → commuting-square-isomorphisms-Synthetic-Category-Theory κ μ ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H G F)) ( horizontal-comp-iso-Synthetic-Category-Theory Χ ( horizontal-comp-iso-Synthetic-Category-Theory Χ γ β) ( α)) ( horizontal-comp-iso-Synthetic-Category-Theory Χ γ ( horizontal-comp-iso-Synthetic-Category-Theory Χ β α)) ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H' G' F')) preserves-associativity-comp-functor-horizontal-comp-iso-inv-Synthetic-Category-Theory κ μ ι ν Α Χ Λ Ρ Ξ H H' G G' F F' γ β α = comp-iso-Synthetic-Category-Theory ( composition-isomorphism-Synthetic-Category-Theory μ) ( right-unit-law-comp-functor-Synthetic-Category-Theory ( right-unit-law-composition-isomorphism-Synthetic-Category-Theory Ρ) ( comp-iso-Synthetic-Category-Theory μ ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H' G' F')) ( horizontal-comp-iso-Synthetic-Category-Theory Χ ( γ) ( horizontal-comp-iso-Synthetic-Category-Theory Χ β α)))) ( comp-iso-Synthetic-Category-Theory ( composition-isomorphism-Synthetic-Category-Theory μ) ( horizontal-comp-iso-Synthetic-Category-Theory ( horizontal-composition-isomorphism-Synthetic-Category-Theory Χ) ( id-iso-Synthetic-Category-Theory ( identity-isomorphism-Synthetic-Category-Theory ι) ( comp-iso-Synthetic-Category-Theory μ ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H' G' F')) ( horizontal-comp-iso-Synthetic-Category-Theory Χ γ ( horizontal-comp-iso-Synthetic-Category-Theory Χ β α)))) ( inv-iso-right-inv-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H G F))) ( comp-iso-Synthetic-Category-Theory ( composition-isomorphism-Synthetic-Category-Theory μ) ( associative-comp-functor-Synthetic-Category-Theory ( associative-comp-isomorphism-Synthetic-Category-Theory Α) ( comp-iso-Synthetic-Category-Theory μ ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H' G' F')) ( horizontal-comp-iso-Synthetic-Category-Theory Χ ( γ) ( horizontal-comp-iso-Synthetic-Category-Theory Χ β α))) ( associative-comp-functor-Synthetic-Category-Theory Α H G F) ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H G F))) ( comp-iso-Synthetic-Category-Theory ( composition-isomorphism-Synthetic-Category-Theory μ) ( horizontal-comp-iso-Synthetic-Category-Theory ( horizontal-composition-isomorphism-Synthetic-Category-Theory Χ) ( inv-iso-Synthetic-Category-Theory ( inverse-isomorphism-Synthetic-Category-Theory ν) ( associative-comp-functor-Synthetic-Category-Theory ( associative-comp-isomorphism-Synthetic-Category-Theory Α) ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H' G' F')) ( horizontal-comp-iso-Synthetic-Category-Theory Χ ( γ) ( horizontal-comp-iso-Synthetic-Category-Theory Χ β α)) ( associative-comp-functor-Synthetic-Category-Theory Α H G F))) ( id-iso-Synthetic-Category-Theory ( identity-isomorphism-Synthetic-Category-Theory ι) ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H G F)))) ( comp-iso-Synthetic-Category-Theory ( composition-isomorphism-Synthetic-Category-Theory μ) ( horizontal-comp-iso-Synthetic-Category-Theory ( horizontal-composition-isomorphism-Synthetic-Category-Theory Χ) ( horizontal-comp-iso-Synthetic-Category-Theory ( horizontal-composition-isomorphism-Synthetic-Category-Theory Χ) ( id-iso-Synthetic-Category-Theory ( identity-isomorphism-Synthetic-Category-Theory ι) ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H' G' F'))) ( inv-iso-Synthetic-Category-Theory ( inverse-isomorphism-Synthetic-Category-Theory ν) ( preserves-associativity-comp-functor-horizontal-comp-iso-Synthetic-Category-Theory Ξ γ β α))) ( id-iso-Synthetic-Category-Theory ( identity-isomorphism-Synthetic-Category-Theory ι) ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H G F)))) ( comp-iso-Synthetic-Category-Theory ( composition-isomorphism-Synthetic-Category-Theory μ) ( horizontal-comp-iso-Synthetic-Category-Theory ( horizontal-composition-isomorphism-Synthetic-Category-Theory Χ) ( associative-comp-functor-Synthetic-Category-Theory ( associative-comp-isomorphism-Synthetic-Category-Theory Α) ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H' G' F')) ( associative-comp-functor-Synthetic-Category-Theory Α H' G' F') ( horizontal-comp-iso-Synthetic-Category-Theory Χ ( horizontal-comp-iso-Synthetic-Category-Theory Χ γ β) ( α))) ( id-iso-Synthetic-Category-Theory ( identity-isomorphism-Synthetic-Category-Theory ι) ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H G F)))) ( comp-iso-Synthetic-Category-Theory ( composition-isomorphism-Synthetic-Category-Theory μ) ( horizontal-comp-iso-Synthetic-Category-Theory ( horizontal-composition-isomorphism-Synthetic-Category-Theory Χ) ( horizontal-comp-iso-Synthetic-Category-Theory ( horizontal-composition-isomorphism-Synthetic-Category-Theory Χ) ( inv-iso-Synthetic-Category-Theory ( inverse-isomorphism-Synthetic-Category-Theory ν) ( inv-iso-left-inv-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H' G' F'))) ( id-iso-Synthetic-Category-Theory ( identity-isomorphism-Synthetic-Category-Theory ι) ( horizontal-comp-iso-Synthetic-Category-Theory Χ ( horizontal-comp-iso-Synthetic-Category-Theory Χ γ β) (α)))) ( id-iso-Synthetic-Category-Theory ( identity-isomorphism-Synthetic-Category-Theory ι) ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H G F)))) ( horizontal-comp-iso-Synthetic-Category-Theory ( horizontal-composition-isomorphism-Synthetic-Category-Theory Χ) ( inv-iso-Synthetic-Category-Theory ( inverse-isomorphism-Synthetic-Category-Theory ν) ( left-unit-law-comp-functor-Synthetic-Category-Theory ( left-unit-law-composition-isomorphism-Synthetic-Category-Theory Λ) ( horizontal-comp-iso-Synthetic-Category-Theory Χ ( horizontal-comp-iso-Synthetic-Category-Theory Χ γ β) ( α)))) ( id-iso-Synthetic-Category-Theory ( identity-isomorphism-Synthetic-Category-Theory ι) ( inv-iso-Synthetic-Category-Theory ν ( associative-comp-functor-Synthetic-Category-Theory Α H G F))))))))))
Preservation of isomorphisms of natural isomorphisms by horizontal composition
We have to assume as an additional axiom that given natural isomorphism α, α’, β, β’ with appropriate domains and codomains, together with isomorphisms α ≅ α’ and β ≅ β’, there exists an isomorphisms between the horizontal composites β _ α ≅ β’ _ α’.
module _ {l : Level} where record preserves-isomorphism-horizontal-composition-Synthetic-Category-Theory (κ : language-Synthetic-Category-Theory l) (ι : identity-Synthetic-Category-Theory κ) (μ : composition-Synthetic-Category-Theory κ) (Χ : horizontal-composition-Synthetic-Category-Theory κ μ) : UU l where coinductive field preserves-isomorphism-horizontal-comp-iso-Synthetic-Category-Theory : {C D E : category-Synthetic-Category-Theory κ} {F F' : functor-Synthetic-Category-Theory κ C D} {G G' : functor-Synthetic-Category-Theory κ D E} {α β : isomorphism-Synthetic-Category-Theory κ F F'} {γ δ : isomorphism-Synthetic-Category-Theory κ G G'} → 3-isomorphism-Synthetic-Category-Theory κ α β → 3-isomorphism-Synthetic-Category-Theory κ γ δ → 3-isomorphism-Synthetic-Category-Theory κ ( horizontal-comp-iso-Synthetic-Category-Theory Χ γ α) ( horizontal-comp-iso-Synthetic-Category-Theory Χ δ β) open preserves-isomorphism-horizontal-composition-Synthetic-Category-Theory public
Recent changes
- 2024-09-25. Ivan Kobe. Pullbacks of synthetic categories (#1183).
- 2024-09-09. Ivan Kobe. defined sections, retractions and equivalences and proved lemma 1.1.6. (#1182).
- 2024-09-03. Egbert Rijke. Some infrastructure for dependent globular types (#1176).
- 2024-09-01. Egbert Rijke. Cisinski’s Formalization of Higher Categories (#1171).