module HoTT where

The HoTT Book?🔗

While the 1Lab has not been consciously designed as a project to formalise the HoTT book, in the course of our explorations into formalised univalent mathematics, we have formalised a considerable subset of the first part, and most of chapter 9. The vast majority of the 1Lab is material that was not covered in the HoTT book.

Part 1: Foundations🔗

Chapter 2: Homotopy type theory🔗

2.1: Types are higher groupoids🔗

Lemma 2.1.1: sym
Lemma 2.1.2: _∙_
Lemma 2.1.4:
1. ∙-idl, ∙-idr
2. ∙-invl, ∙-invr
3. Definitional in cubical type theory
4. ∙-assoc
Theorem 2.1.6: Ωⁿ⁺²-is-abelian-group
Definition 2.1.7: Type∙
Definition 2.1.8: Ωⁿ

2.2: Functions are functors🔗

Lemma 2.2.1: ap
Lemma 2.2.2:
1. ap-∙
2. Definitional in cubical type theory
3. Definitional in cubical type theory
4. Definitional in cubical type theory

2.3: Type families are fibrations🔗

Lemma 2.3.1: subst
Lemma 2.3.2: Σ-pathp
Lemma 2.3.4: Our ap is dependent by nature.
Lemma 2.3.5: transport-refl
Lemma 2.3.9: subst-∙
Lemma 2.3.10: Definitional in cubical type theory

2.4: Homotopies and equivalences🔗

Lemma 2.4.3: homotopy-natural
Lemma 2.4.4: homotopy-invert
Lemma 2.4.6: is-iso
Example 2.4.9: transport⁻transport
Lemma 2.4.12: id-equiv, Equiv.inverse, _∙e_

2.7: Cartesian product types🔗

Theorem 2.7.2: Σ-pathp-iso
Theorem 2.7.3: Agda has definitional η equality for records.

2.9: Π-types and function extensionality🔗

Theorem 2.9.3: funext (no longer an axiom)
Lemma 2.9.6: funext-dep (no longer an axiom)

2.10: Universes and univalence🔗

Lemma 2.10.1: path→equiv
Theorem 2.10.3: univalence
- ua, uaβ
- ua-id-equiv, ua∙, sym-ua

2.11: Identity type🔗

Theorem 2.11.1: is-equiv→is-embedding
Lemma 2.11.2: subst-path-left, subst-path-right, transport-path
Theorem 2.11.5: commutes→square

2.12: Coproducts🔗

Theorem 2.12.5: ⊎Path.Code≃Path

Exercises🔗

Exercise 2.10: Σ-assoc

Chapter 3: Sets and Logic🔗

3.1: Sets and n-types🔗

Definition 3.1.1: is-set
- Example 3.1.2: Definitional in cubical type theory.
- Example 3.1.4: Nat-is-set
- Example 3.1.5: ×-is-hlevel (special case)
- Example 3.1.6: Π-is-hlevel (special case)
Definition 3.1.7: is-groupoid
Lemma 3.1.8: is-hlevel-suc (special case)

3.2: Propositions as types?🔗

Theorem 3.2.2: ¬DNE∞
Corollary 3.2.7: ¬LEM∞

3.3: Mere propositions🔗

Definition 3.3.1: is-prop
Lemma 3.3.3: prop-ext
Lemma 3.3.4: is-prop→is-set
Lemma 3.3.5: is-prop-is-prop, is-hlevel-is-prop

3.4: Classical vs. intuitionistic logic🔗

Definition 3.4.1: LEM
Definition 3.4.2: DNE
Definition 3.4.3:
- 1. Dec
- 1. Discrete

3.5: Subsets and propositional resizing🔗

Lemma 3.5.1: Σ-prop-path
Axiom 3.5.5: Ω, □.

3.7: Propositional truncation🔗

The type itself is defined as a higher-inductive type ∥_∥. We also define ∃ as a shorthand for the truncation of Σ.

3.8: The axiom of choice🔗

Definition 3.8.3: Axiom-of-choice

3.9: The principle of unique choice🔗

Lemma 3.9.1: is-prop→equiv∥-∥
Corollary 3.9.2: Implicit in e.g. ∥-∥-univ, ∥-∥-out

3.11: Contractibility🔗

Definition 3.11.1: is-contr
Definition 3.11.4: is-contr-is-prop
Definition 3.11.7: retract→is-contr
Definition 3.11.8: Singleton-is-contr
Lemma 3.11.9: Σ-contract, Σ-contr-eqv

Exercises🔗

Exercise 3.1: equiv→is-hlevel
Exercise 3.2: ⊎-is-hlevel
Exercise 3.3: Σ-is-hlevel
Exercise 3.5: is-contr-if-inhabited→is-prop, is-prop∙→is-contr
Exercise 3.6: H-Level-Dec
Exercise 3.7: disjoint-⊎-is-prop
Exercise 3.11: ¬global-choice
Exercise 3.17: ∥-∥-elim
Exercise 3.18: LEM≃DNE
Exercise 3.19: ℕ-well-ordered
Exercise 3.20: Σ-contr-eqv
Exercise 3.21: is-prop≃equiv∥-∥
Exercise 3.22: Finite-choice

Chapter 4: Equivalences🔗

4.2: Half adjoint equivalences🔗

Definition 4.2.1: is-half-adjoint-equiv
Definition 4.2.3: is-iso→is-half-adjoint-equiv
Definition 4.2.4: fibre
Lemma 4.2.5: fibre-paths
Theorem 4.2.6: is-half-adjoint-equiv→is-equiv
Definition 4.2.7: linv, rinv
Lemma 4.2.8: is-equiv→pre-is-equiv, is-equiv→post-is-equiv
Lemma 4.2.9: is-iso→is-contr-linv, is-iso→is-contr-rinv

4.3: Bi-invertible maps🔗

Definition 4.3.1: is-biinv
Theorem 4.3.2: is-biinv-is-prop

4.4: Contractible fibres🔗

Note

This is our “default” definition of equivalence, but we generally use it through the interface of half-adjoint equivalences.

Definition 4.4.1: is-equiv
Theorem 4.4.3: is-equiv→is-half-adjoint-equiv
Lemma 4.4.4: is-equiv-is-prop

4.6: Surjections and embeddings🔗

Definition 4.6.1:
1. is-surjective
2. is-embedding, embedding→cancellable
Definition 4.6.2: injective
Theorem 4.6.3: is-equiv→is-surjective, is-equiv→is-embedding, embedding-surjective→is-equiv

4.8: The object classifier🔗

Lemma 4.8.1: Fibre-equiv
Lemma 4.8.2: Total-equiv
Theorem 4.8.3: Map-classifier

Chapter 5: Induction🔗

5.3: W-types🔗

W-types: W

5.4: Inductive types are initial algebras🔗

Theorem 5.4.7: W-initial

Chapter 6: Higher inductive types🔗

6.2: Induction principles and dependent paths🔗

Remark 6.2.3: to-pathp, from-pathp
Induction principle for $S^{1}$ : by pattern matching.
Lemma 6.2.5: S¹-rec
Lemma 6.2.9: Ωⁿ≃Sⁿ-map for n = 1

6.3: The interval🔗

This is the higher inductive type [0,1], not the interval type I.

Lemma 6.3.1: interval-contractible

6.4: Circles and spheres🔗

Lemma 6.4.1: refl≠loop
Lemma 6.4.2: always-loop
Lemma 6.4.4: ap-square

6.5: Suspensions🔗

The suspension: Susp
Lemma 6.5.1: SuspS⁰≡S¹
Definition 6.5.2: Sⁿ⁻¹
Lemma 6.5.4: Σ-map∙≃map∙-Ω

6.6: Cell complexes🔗

The torus: T².

6.9: Truncations🔗

Lemma 6.9.1: ∥-∥₀-elim

6.10: Quotients🔗

We define the quotient _/_ in terms of coequalisers Coeq.

Lemma 6.10.3: Coeq-univ.

6.11: Algebra🔗

Definition 6.11.1: Monoid-on, Group-on
Definition 6.11.4: πₙ₊₁
Lemma 6.11.5: Monoid.Free-monoid⊣Forget
Lemma 6.11.6: Group.make-free-group

Exercises🔗

Exercise 6.3: T²≃S¹×S¹

Chapter 7: Homotopy n-types🔗

7.1: Definition of n-types🔗

Definition 7.1.1: is-hlevel
Theorem 7.1.4: retract→is-hlevel
Corollary 7.1.5: equiv→is-hlevel
Theorem 7.1.7: is-hlevel-suc
Theorem 7.1.8: Σ-is-hlevel
Theorem 7.1.9: Π-is-hlevel
Theorem 7.1.10: is-hlevel-is-prop
Theorem 7.1.11: n-Type-is-hlevel

7.2: Uniqueness of identity proofs and Hedberg’s theorem🔗

Theorem 7.2.2: set-identity-system, identity-system→hlevel
Theorem 7.2.5: Discrete→is-set
Theorem 7.2.6: Discrete-Nat
Theorem 7.2.7: hubs-and-spokes→hlevel, hlevel→hubs-and-spokes

7.3: Truncations🔗

Lemma 7.3.1: n-Tr-is-hlevel
Lemma 7.3.2: n-Tr-elim
Theorem 7.3.12: n-Tr-path-equiv

7.5: Connectedness🔗

Definition 7.5.1: is-n-connected, is-n-connected-Tr
Lemma 7.5.7: relative-n-type-const
Corollary 7.5.9: is-n-connected→n-type-const, n-type-const→is-n-connected
Lemma 7.5.11: is-n-connected-point, point-is-n-connected

Exercises🔗

Exercise 7.6: is-n-connected≃∥-∥

Part 2: Mathematics🔗

Chapter 8: Homotopy theory🔗

The only non-trivial result worth mentioning from Chapter 8 is the fundamental group of the circle.

8.1: π₁(S¹)🔗

Definition 8.1.1: S¹Path.Cover
Definition 8.1.5: S¹Path.encode
Definition 8.1.6: S¹Path.decode
Lemma 8.1.7: S¹Path.encode-decode
Lemma 8.1.8: S¹Path.encode-loopⁿ
Corollary 8.1.10: ΩS¹≃integers
Corollary 8.1.11: π₁S¹≡ℤ, πₙ₊₂S¹≡0

8.2: Connectedness of suspensions🔗

Theorem 8.2.1: Susp-is-connected
Corollary 8.2.2: Sⁿ⁻¹-is-connected

8.6: The Freudenthal suspension theorem🔗

Lemma 8.6.1: relative-n-type-const-plus
Lemma 8.6.2: Wedge.elim

Chapter 9: Category theory🔗

Since a vast majority of the 1Lab’s mathematics consists of pure category theory, or mathematics done with a very categorical inclination, this is our most complete chapter.

9.1: Categories and Precategories🔗

Cat.Morphism, Cat.Univalent.

Definition 9.1.1: Precategory
Definition 9.1.2: is-invertible, _≅_
Lemma 9.1.3: is-invertible-is-prop
Lemma 9.1.4: path→iso
Example 9.1.5: Sets
Definition 9.1.6¹: is-category
Example 9.1.7: Sets-is-category
Lemma 9.1.8: Univalent.Ob-is-groupoid
Lemma 9.1.9: Hom-transport (9.1.10), path→to-sym (9.1.11), path→to-∙ (9.1.12/9.1.13)
Example 9.1.14: Poset
Example 9.1.15: is-gaunt
Example 9.1.16: Disc
Example 9.1.19: Rel

9.2: Functors and Transformations🔗

Definition 9.2.1: Functor
Definition 9.2.2: _=>_
- The paragraph immediately after 9.2.2 is Nat-pathp and Nat-is-set
- The one after that is Functor-path.
Definition 9.2.3: Cat[_,_]
Lemma 9.2.4: If: invertible→invertibleⁿ; Only if: isoⁿ→iso
Theorem 9.2.5: Functor-is-category
Theorem 9.2.6: _F∘_
Definition 9.2.7: _◂_, _▸_
Lemma 9.2.9: F∘-assoc
Lemma 9.2.10: See the definition of Prebicategory.pentagon for Cat.
Lemma 9.2.11: See the definition of Prebicategory.triangle for Cat.

9.3: Adjunctions🔗

Lemma 9.3.1: _⊣_
Lemma 9.3.2: is-left-adjoint-is-prop

9.4: Equivalences🔗

Definition 9.4.1: is-equivalence
Definition 9.4.3: is-faithful, is-full, is-fully-faithful
Definition 9.4.4: is-split-eso
Lemma 9.4.5: ff+split-eso→is-equivalence
Definition 9.4.6: is-eso
Lemma 9.4.7: Essential-fibre-between-cats-is-prop, ff+eso→is-equivalence
Definition 9.4.8: is-precat-iso
Lemma 9.4.14: is-equivalence→is-precat-iso, is-precat-iso→is-equivalence
Lemma 9.4.15: Precategory-identity-system
Theorem 9.4.16: Category-identity-system

9.5: The Yoneda lemma🔗

Definition 9.5.1: _^op
Definition 9.5.2: _×ᶜ_
Lemma 9.5.3: Curry, Uncurry
- The $Hom -functor:$ Hom[-,-]
- The Yoneda embedding: よ
Corollary 9.5.6: よ-is-fully-faithful
Corollary 9.5.7: As a corollary of Representation-is-prop
Definition 9.5.8: Representation
Theorem 9.5.9: Representation-is-prop
Lemma 9.5.10: Adjoints in terms of representability

9.6: Strict categories🔗

This chapter is mostly text.

Definition 9.6.1: Strict precategories

9.7: Dagger categories🔗

We do not have a type of dagger-categories, but note that we do have the closely-related notion of allegory.

9.8: The structure identity principle🔗

Definition 9.8.1: Thin-structure-over, generalised into Displayed
Theorem 9.8.2: Structured-objects-is-category, generalised into is-category-total

9.9: The Rezk completion🔗

Lemma 9.9.1: eso→pre-faithful
Lemma 9.9.2: eso-full→pre-ff
Lemma 9.9.4: weak-equiv→pre-equiv, weak-equiv→pre-iso
Theorem 9.9.5: Rezk-completion, Rezk-completion-is-category, complete, complete-is-ff, complete-is-eso.

Exercises🔗

Exercise 9.1: Slice, slice-is-category
Exercise 9.2: Total-space, Total-space-is-ff, Total-space-is-eso
Exercise 9.3: is-equivalence.F⁻¹⊣F
Exercise 9.4: Prebicategory

Chapter 10: Set theory🔗

10.1: The category of sets🔗

10.1.1 Limits and colimits: Sets-is-complete, Sets-is-cocomplete
Theorem 10.1.5: Sets-regular, surjective→regular-epi, epi→surjective
10.1.3 Quotients: Sets-effective-congruences
Lemma 10.1.8: Congruence.is-effective
Lemma 10.1.13: Susp-prop-is-set, Susp-prop-path
Theorem 10.1.14: AC→LEM

10.5: The cumulative hierarchy🔗

Definition 10.5.1: V
Lemma 10.5.6: presentation
Definition 10.5.7: Presentation.members
Theorem 10.5.8:
- 1. extensionality
- 1. empty-set
- 1. pairing
- 1. zero∈ℕ, suc∈ℕ
- 1. union
- 1. ∈-induction
- 1. replacement
- 1. separation

We use a slightly different definition of univalence for categories. It is shown equivalent to the usual formulation by equiv-path→identity-system↩︎