146 lines
4.6 KiB
Text
146 lines
4.6 KiB
Text
/-
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Copyright (c) 2016 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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The definition of pointed types. This file is here to avoid circularities in the import graph
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-/
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prelude
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import init.trunc
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open eq equiv is_equiv is_trunc
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structure pointed [class] (A : Type) :=
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(point : A)
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structure pType :=
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(carrier : Type)
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(Point : carrier)
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notation `Type*` := pType
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namespace pointed
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attribute pType.carrier [coercion]
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variables {A : Type}
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definition pt [reducible] [unfold 2] [H : pointed A] := point A
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definition Point [reducible] [unfold 1] (A : Type*) := pType.Point A
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definition carrier [reducible] [unfold 1] (A : Type*) := pType.carrier A
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protected definition Mk [constructor] {A : Type} (a : A) := pType.mk A a
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protected definition MK [constructor] (A : Type) (a : A) := pType.mk A a
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protected definition mk' [constructor] (A : Type) [H : pointed A] : Type* :=
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pType.mk A (point A)
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definition pointed_carrier [instance] [constructor] (A : Type*) : pointed A :=
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pointed.mk (Point A)
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end pointed
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open pointed
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section
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universe variable u
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structure ptrunctype (n : ℕ₋₂) extends trunctype.{u} n, pType.{u}
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definition is_trunc_ptrunctype [instance] {n : ℕ₋₂} (X : ptrunctype n)
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: is_trunc n (ptrunctype.to_pType X) :=
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trunctype.struct X
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end
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notation n `-Type*` := ptrunctype n
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abbreviation pSet [parsing_only] := 0-Type*
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notation `Set*` := pSet
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namespace pointed
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protected definition ptrunctype.mk' [constructor] (n : ℕ₋₂)
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(A : Type) [pointed A] [is_trunc n A] : n-Type* :=
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ptrunctype.mk A _ pt
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protected definition pSet.mk [constructor] := @ptrunctype.mk (-1.+1)
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protected definition pSet.mk' [constructor] := ptrunctype.mk' (-1.+1)
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definition ptrunctype_of_trunctype [constructor] {n : ℕ₋₂} (A : n-Type) (a : A)
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: n-Type* :=
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ptrunctype.mk A _ a
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definition ptrunctype_of_pType [constructor] {n : ℕ₋₂} (A : Type*) (H : is_trunc n A)
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: n-Type* :=
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ptrunctype.mk A _ pt
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definition pSet_of_Set [constructor] (A : Set) (a : A) : Set* :=
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ptrunctype.mk A _ a
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definition pSet_of_pType [constructor] (A : Type*) (H : is_set A) : Set* :=
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ptrunctype.mk A _ pt
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attribute ptrunctype._trans_of_to_pType ptrunctype.to_pType ptrunctype.to_trunctype [unfold 2]
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-- Any contractible type is pointed
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definition pointed_of_is_contr [instance] [priority 800] [constructor]
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(A : Type) [H : is_contr A] : pointed A :=
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pointed.mk !center
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end pointed
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/- pointed maps -/
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structure ppi_gen {A : Type*} (P : A → Type) (x₀ : P pt) :=
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(to_fun : Π a : A, P a)
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(resp_pt : to_fun (Point A) = x₀)
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definition ppi {A : Type*} (P : A → Type*) : Type :=
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ppi_gen P pt
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-- We could try to define pmap as a special case of ppi
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-- definition pmap (A B : Type*) := @ppi A (λa, B)
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structure pmap (A B : Type*) :=
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(to_fun : A → B)
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(resp_pt : to_fun (Point A) = Point B)
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namespace pointed
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definition ppi.mk [constructor] [reducible] {A : Type*} {P : A → Type*} (f : Πa, P a)
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(p : f pt = pt) : ppi P :=
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ppi_gen.mk f p
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definition ppi.to_fun [unfold 3] [coercion] [reducible] {A : Type*} {P : A → Type*} (f : ppi P)
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(a : A) : P a :=
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ppi_gen.to_fun f a
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definition ppi.resp_pt [unfold 3] [reducible] {A : Type*} {P : A → Type*} (f : ppi P) :
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f pt = pt :=
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ppi_gen.resp_pt f
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abbreviation respect_pt [unfold 3] := @pmap.resp_pt
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notation `map₊` := pmap
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infix ` →* `:28 := pmap
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attribute pmap.to_fun ppi_gen.to_fun [coercion]
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-- notation `Π*` binders `, ` r:(scoped P, ppi _ P) := r
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-- definition pmap.mk [constructor] {A B : Type*} (f : A → B) (p : f pt = pt) : A →* B :=
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-- ppi.mk f p
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-- definition pmap.to_fun [coercion] [unfold 3] {A B : Type*} (f : A →* B) : A → B := f
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end pointed open pointed
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/- pointed homotopies -/
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definition phomotopy {A B : Type*} (f g : A →* B) : Type :=
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ppi_gen (λa, f a = g a) (respect_pt f ⬝ (respect_pt g)⁻¹)
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-- structure phomotopy {A B : Type*} (f g : A →* B) : Type :=
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-- (homotopy : f ~ g)
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-- (homotopy_pt : homotopy pt ⬝ respect_pt g = respect_pt f)
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namespace pointed
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variables {A B : Type*} {f g : A →* B}
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infix ` ~* `:50 := phomotopy
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definition phomotopy.mk [reducible] [constructor] (h : f ~ g)
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(p : h pt ⬝ respect_pt g = respect_pt f) : f ~* g :=
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ppi_gen.mk h (eq_con_inv_of_con_eq p)
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definition to_homotopy [coercion] [unfold 5] [reducible] (p : f ~* g) : Πa, f a = g a := p
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definition to_homotopy_pt [unfold 5] [reducible] (p : f ~* g) :
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p pt ⬝ respect_pt g = respect_pt f :=
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con_eq_of_eq_con_inv (ppi_gen.resp_pt p)
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end pointed
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