Library HoTT.Categories.HomotopyPreCategory

Homotopy precategory of types

Require Import Category.Core.
Require Import HoTT.Basics HoTT.Truncations.Core.

Set Implicit Arguments.
Generalizable All Variables.

Local Open Scope path_scope.

Local Open Scope category_scope.

Quoting the HoTT Book:

Example. There is a precategory whose type of objects is U and with hom(X,Y) : ∥X → Y∥₀, and composition defined by induction on truncation from ordinary composition (Y → Z) → (X → Y) → (X → Z). We call this the homotopy precategory of types.

We don't want access to all of the internals of this category at top level.

Module HomotopyPreCategoryInternals.
  Section homotopy_precategory.
    Local Notation object := Type (only parsing).
    Local Notation morphism s d := (Trunc 0 (s → d)) (only parsing).

    Definition compose s d d' (m : morphism d d') (m' : morphism s d)
    : morphism s d'.
    Proof.
      revert m'; apply Trunc_rec; intro m'.
      revert m; apply Trunc_rec; intro m.
      apply tr.
      exact (m o m')%core.
    Defined.

    Definition identity x : morphism x x
      := tr idmap.

    Global Arguments compose [s d d'] m m' / .
    Global Arguments identity x / .
  End homotopy_precategory.
End HomotopyPreCategoryInternals.

The homotopy precategory of types

Definition homotopy_precategory : PreCategory.
Proof.
  refine (@Build_PreCategory
            Type
            _
            (@HomotopyPreCategoryInternals.identity)
            (@HomotopyPreCategoryInternals.compose)
            _
            _
            _
            _);
  simpl; intros;
  repeat match goal with
           | [ m : Trunc _ _ |- _ ]
             ⇒ revert m;
               apply Trunc_ind;
               [ intro;
                 match goal with
                   | [ |- IsHSet (?a = ?b :> ?T) ]
                     ⇒ generalize a b; intros;
                        let H := fresh in
                        assert (H : ∀ x y : T, IsHProp (x = y))
                 end;
                 typeclasses eauto
               | intro ]
         end;
  simpl;
  apply ap;
  exact idpath.
Defined.