* A major difference is that in this library we heavily use pathovers [D. Licata, G. Brunerie. A Cubical Approach to Synthetic Homotopy Theory]. This means that we need less theorems about transports, but instead corresponding theorems about pathovers. These are in [init.pathover](init/pathover.hlean). For higher paths there are [squares](cubical/square.hlean), [squareovers](cubical/squareover.hlean), and the rudiments of [cubes](cubical/cube.hlean) and [cubeovers](cubical/cubeover.hlean).
* The category theory library is more extensive than what is presented in the book. For example, we have [limits](algebra/category/limits/limits.md).
- 1.1 (Type theory versus set theory): no formalizable content.
- 1.2 (Function types): no formalizable content. Related: [init.function](init/function.hlean)
- 1.3 (Universes and families): no formalizable content (Lean also has a hierarchy of universes `Type.{i} : Type.{i + 1}`, but they are *not* cumulative).
- 1.4 (Dependent function types (Π-types)): no formalizable content. Related: [init.function](init/function.hlean)
- 1.5 (Product types): declaration in [init.datatypes](init/datatypes.hlean), notation in [init.types](init/types.hlean)
- 1.6 (Dependent pair types (Σ-types)): declaration in [init.datatypes](init/datatypes.hlean), notation in [init.types](init/types.hlean)
- 1.7 (Coproduct types): declaration in [init.datatypes](init/datatypes.hlean), notation in [init.types](init/types.hlean)
- 1.8 (The type of booleans): declaration in [init.datatypes](init/datatypes.hlean), notation in [init.bool](init/bool.hlean)
- 1.9 (The natural numbers): [init.nat](init/nat.hlean) (declaration in [init.datatypes](init/datatypes.hlean))
- 1.10 (Pattern matching and recursion): no formalizable content (we can use the "pattern matching" notation using the function definition package, which are reduced to applying recursors).
- 1.11 (Propositions as types): some logic is in [init.logic](init/logic.hlean) and [init.types](init/types.hlean).
- 1.12 (Identity types): declaration in [init.datatypes](init/datatypes.hlean), more in [init.logic](init/logic.hlean)
Chapter 2: Homotopy type theory
---------
- 2.1 (Types are higher groupoids): [init.path](init/path.hlean) (pointed types and loop spaces in [types.pointed](types/pointed.hlean))
- 2.2 (Functions are functors): [init.path](init/path.hlean)
- 2.3 (Type families are fibrations): [init.path](init/path.hlean)
- 2.4 (Homotopies and equivalences): homotopies in [init.path](init/path.hlean) and equivalences in [init.equiv](init/equiv.hlean)
- 2.5 (The higher groupoid structure of type formers): no formalizable content
- 2.9 (Π-types and the function extensionality axiom): [init.funext](init/funext.hlean), [types.pi](types/pi.hlean) and [types.arrow](types/arrow.hlean)
- 2.11 (Identity type): [init.equiv](init/equiv.hlean) (ap is an equivalence), [types.eq](types/eq.hlean) and [cubical.square](cubical/square.hlean) (characterization of pathovers in equality types)
- 3.5 (Subsets and propositional resizing): Lemma 3.5.1 is subtype_eq in [types.sigma](types/sigma.hlean). We have not declared propositional resizing as an axiom.
- 3.6 (The logic of mere propositions): in the corresponding file in the [types](types/types.md) folder. (is_trunc_prod is defined in [types.sigma](types/sigma.hlean))
- 3.7 (Propositional truncation): [init.hit](init/hit.hlean) and [hit.trunc](hit/trunc.hlean)
- 4.7 (Closure properties of equivalences): 4.7.1 in [init.equiv](init/equiv.hlean); 4.7.2 in [function](function.hlean); 4.7.3 and 4.7.4 in [types.arrow_2](types/arrow_2.hlean) 4.7.5 in [types.sigma](types/sigma.hlean) (sigma_functor is a generalization of total(f)); 4.7.6 in [types.fiber](types/fiber.hlean) and 4.7.7 in [types.equiv](types/equiv.hlean).
- 5.8 (Identity types and identity systems): 5.8.1-5.8.4 not formalized, 5.8.5 in [init.ua](init/ua.hlean) and 5.8.6 in [init.funext](init/funext.hlean)
We have two primitive HITs in Lean, the computation rules are manually added to the Lean-HoTT kernel. The primitive HITs are the n-truncation and the quotient (not to be confused with the set-quotient). See [init.hit](init/hit.hlean).
- 6.2 (Induction principles and dependent paths): dependent paths in [init.pathover](init/pathover.hlean), circle in [homotopy.circle](homotopy/circle.hlean)
- 6.3 (The interval): [homotopy.interval](homotopy/interval.hlean)
- 6.4 (Circles and spheres): [homotopy.sphere](homotopy/sphere.hlean) and [homotopy.circle](homotopy/circle.hlean)
- 6.5 (Suspensions): [homotopy.suspension](homotopy/susp.hlean) (we define the circle to be the suspension of bool, but Lemma 6.5.1 is similar to proving the ordinary induction principle for the circle in [homotopy.circle](homotopy/circle.hlean)) and a bit in [homotopy.sphere](homotopy/sphere.hlean) and [types.pointed](types/pointed.hlean)
- 6.6 (Cell complexes): We define the torus using a two quotient, which in turn is defined in terms of the quotient, see [homotopy.torus](homotopy/torus.hlean).
- 6.7 (Hubs and spokes): We define the two quotient using only the quotient in [hit.two_quotient](hit/two_quotient.hlean). This is slightly different than what is done in section 6.7, because the HIT in section 6.7 is not a quotient.
- 6.8 (Pushouts): [hit.pushout](hit/pushout.hlean). Some of the "standard homotopy-theoretic constructions" have separate files, although not all of them have been defined explicitly yet
- 6.10 (Quotients): [hit.set_quotient](hit/set_quotient.hlean) (up to 6.10.3). We define integers differently, to make them compute, in the folder [types.int](types/int/int.md). 6.10.13 is in [types.int.hott](types/int/hott.hlean)
- 8.4 (Fiber sequences and the long exact sequence): Mostly in [homotopy.chain_complex](homotopy/chain_complex.hlean), [homotopy.LES_of_homotopy_groups](homotopy/LES_of_homotopy_groups.hlean). Definitions 8.4.1 and 8.4.2 in [types.pointed](types/pointed.hlean), Corollary 8.4.8 in [homotopy.homotopy_group](homotopy/homotopy_group.hlean).
- 8.5 (The Hopf fibration): [circle](homotopy/circle.hlean) (multiplication on the circle, Lemma 8.5.8), [join](homotopy/join.hlean) (join is associative, Lemma 8.5.9), [hopf](homotopy/hopf.hlean) (The Hopf construction, Lemmas 8.5.5 and 8.5.7), [complex_hopf](homotopy/complex_hopf.hlean) (the H-space structure on the circle and the complex Hopf fibration)
- 8.6 (The Freudenthal suspension theorem): [connectedness](homotopy/connectedness.hlean) (Lemma 8.6.1), [wedge](homotopy/wedge.hlean) (Wedge connectivity, Lemma 8.6.2). Corollary 8.6.14 is proven directly in [homotopy.freudenthal](homotopy/freudenthal.hlean), however, we don't prove it as a Corollary of Theorem 8.6.4, which is not proven. Stability of iterated suspensions is also in [homotopy.freudenthal](homotopy/freudenthal.hlean). The homotopy groups of spheres in this section are computed in [homotopy.sphere2](homotopy/sphere2.hlean).
- 8.8 (Whitehead’s theorem and Whitehead’s principle): 8.8.1 and 8.8.2 at the bottom of [types.trunc](types/trunc.hlean), 8.8.3 in [homotopy.homotopy_group](homotopy/homotopy_group.hlean).
- 10.1 (The category of sets): The category of sets is in [algebra.category.constructions.set](algebra/category/constructions/set.hlean). The proof that it is complete and cocomplete is in [algebra.category.limits.set](algebra/category/limits/set.hlean). Most other properties of the category of sets has not been formalized.
- 10.2 (Cardinal numbers): not formalized
- 10.3 (Ordinal numbers): not formalized
- 10.4 (Classical well-orderings): not formalized
- 10.5 (The cumulative hierarchy): not formalized, and probably not formalizable, because Lean lacks induction-recursion.
