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generalize many results about pointed homotopies of nondependent maps to dependent maps

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This commit is contained in:
Floris van Doorn 2017-07-21 15:03:17 +01:00
parent 1a26d405ef
commit c8477d28ba
6 changed files with 164 additions and 93 deletions
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19
hott/init/funext.hlean
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@ -285,6 +285,25 @@ namespace eq
induction p using homotopy.rec_on, induction q, exact H
end
definition homotopy.rec_idp [recursor] {A : Type} {P : A → Type} {f : Πa, P a}
(Q : Π{g}, (f ~ g) → Type) (H : Q (homotopy.refl f)) {g : Π x, P x} (p : f ~ g) : Q p :=
homotopy.rec_on_idp p H
definition homotopy_rec_on_apd10 {A : Type} {P : A → Type} {f g : Πa, P a}
(Q : f ~ g → Type) (H : Π(q : f = g), Q (apd10 q)) (p : f = g) :
homotopy.rec_on (apd10 p) H = H p :=
begin
unfold [homotopy.rec_on],
refine ap (λp, p ▸ _) !adj ⬝ _,
refine !tr_compose⁻¹ ⬝ _,
apply apdt
end
definition homotopy_rec_idp_refl {A : Type} {P : A → Type} {f : Πa, P a}
(Q : Π{g}, f ~ g → Type) (H : Q homotopy.rfl) :
homotopy.rec_idp @Q H homotopy.rfl = H :=
!homotopy_rec_on_apd10
definition eq_of_homotopy_inv {f g : Π x, P x} (H : f ~ g)
: eq_of_homotopy (λx, (H x)⁻¹) = (eq_of_homotopy H)⁻¹ :=
begin

25
hott/init/pathover.hlean
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@ -82,6 +82,9 @@ namespace eq
definition change_path [unfold 9] (q : p = p') (r : b =[p] b₂) : b =[p'] b₂ :=
q ▸ r
definition change_path_idp [unfold_full] (r : b =[p] b₂) : change_path idp r = r :=
by reflexivity
-- infix ` ⬝ ` := concato
infix ` ⬝o `:72 := concato
infix ` ⬝op `:73 := concato_eq
@ -103,25 +106,25 @@ namespace eq
/- Some of the theorems analogous to theorems for = in init.path -/
definition cono_idpo (r : b =[p] b₂) : r ⬝o idpo =[con_idp p] r :=
pathover.rec_on r idpo
definition cono_idpo (r : b =[p] b₂) : r ⬝o idpo = r :=
by reflexivity
definition idpo_cono (r : b =[p] b₂) : idpo ⬝o r =[idp_con p] r :=
pathover.rec_on r idpo
by induction r; constructor
definition cono.assoc' (r : b =[p] b₂) (r₂ : b₂ =[p₂] b₃) (r₃ : b₃ =[p₃] b₄) :
r ⬝o (r₂ ⬝o r₃) =[!con.assoc'] (r ⬝o r₂) ⬝o r₃ :=
pathover.rec_on r₃ (pathover.rec_on r₂ (pathover.rec_on r idpo))
by induction r₃; constructor
definition cono.assoc (r : b =[p] b₂) (r₂ : b₂ =[p₂] b₃) (r₃ : b₃ =[p₃] b₄) :
(r ⬝o r₂) ⬝o r₃ =[!con.assoc] r ⬝o (r₂ ⬝o r₃) :=
pathover.rec_on r₃ (pathover.rec_on r₂ (pathover.rec_on r idpo))
by induction r₃; constructor
definition cono.right_inv (r : b =[p] b₂) : r ⬝o r⁻¹ᵒ =[!con.right_inv] idpo :=
pathover.rec_on r idpo
by induction r; constructor
definition cono.left_inv (r : b =[p] b₂) : r⁻¹ᵒ ⬝o r =[!con.left_inv] idpo :=
pathover.rec_on r idpo
by induction r; constructor
definition eq_of_pathover {a' a₂' : A'} (q : a' =[p] a₂') : a' = a₂' :=
by cases q;reflexivity
@ -159,6 +162,11 @@ namespace eq
eq_of_pathover_idp (pathover_of_eq (idpath x) p) = p :=
by induction p; reflexivity
variable (B)
definition idpo_concato_eq (r : b = b') : eq_of_pathover_idp (@idpo A B a b ⬝op r) = r :=
by induction r; reflexivity
variable {B}
-- definition pathover_idp (b : B a) (b' : B a) : b =[idpath a] b' ≃ b = b' :=
-- pathover_equiv_tr_eq idp b b'
@ -245,6 +253,9 @@ namespace eq
definition apd [unfold 6] (f : Πa, B a) (p : a = a₂) : f a =[p] f a₂ :=
by induction p; constructor
definition apd_idp [unfold_full] (f : Πa, B a) : apd f idp = @idpo A B a (f a) :=
by reflexivity
definition apo [unfold 12] {f : A → A'} (g : Πa, B a → B'' (f a)) (q : b =[p] b₂) :
g a b =[p] g a₂ b₂ :=
by induction q; constructor

25
hott/init/pointed.hlean
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@ -88,31 +88,26 @@ structure ppi {A : Type*} (P : A → Type) (x₀ : P pt) :=
(to_fun : Π a : A, P a)
(resp_pt : to_fun (Point A) = x₀)
definition ppi_const [constructor] {A : Type*} (P : A → Type*) : ppi P pt :=
ppi.mk (λa, pt) idp
definition pppi' [reducible] {A : Type*} (P : A → Type*) : Type :=
ppi P pt
definition ppi_const [constructor] {A : Type*} (P : A → Type*) : pppi' P :=
ppi.mk (λa, pt) idp
definition pppi [constructor] [reducible] {A : Type*} (P : A → Type*) : Type* :=
pointed.MK (pppi' P) (ppi_const P)
notation `Π*` binders `, ` r:(scoped P, pppi P) := r
-- We could try to define pmap as a special case of ppi
-- definition pmap' (A B : Type*) : Type := @pppi' A (λa, B)
-- todo: make this already pointed?
-- do we want to make this already pointed?
definition pmap [reducible] (A B : Type*) : Type := @pppi A (λa, B)
-- structure pmap (A B : Type*) :=
-- (to_fun : A → B)
-- (resp_pt : to_fun (Point A) = Point B)
namespace pointed
attribute ppi.to_fun [coercion]
notation `map₊` := pmap
infix ` →* `:28 := pmap
notation `Π*` binders `, ` r:(scoped P, pppi P) := r
namespace pointed
definition pppi.mk [constructor] [reducible] {A : Type*} {P : A → Type*} (f : Πa, P a)
(p : f pt = pt) : pppi P :=
@ -126,7 +121,7 @@ namespace pointed
(p : f (Point A) = Point B) : A →* B :=
pppi.mk f p
definition pmap.to_fun [unfold 3] [reducible] {A B : Type*} (f : A →* B) : A → B :=
abbreviation pmap.to_fun [unfold 3] [reducible] [coercion] {A B : Type*} (f : A →* B) : A → B :=
pppi.to_fun f
definition respect_pt [unfold 4] [reducible] {A : Type*} {P : A → Type} {p₀ : P pt}

47
hott/types/pointed.hlean
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@ -116,10 +116,10 @@ namespace pointed
/- categorical properties of pointed maps -/
definition pid [constructor] (A : Type*) : A →* A :=
definition pid [constructor] [refl] (A : Type*) : A →* A :=
pmap.mk id idp
definition pcompose [constructor] {A B C : Type*} (g : B →* C) (f : A →* B) : A →* C :=
definition pcompose [constructor] [trans] {A B C : Type*} (g : B →* C) (f : A →* B) : A →* C :=
pmap.mk (λa, g (f a)) (ap g (respect_pt f) ⬝ respect_pt g)
infixr ` ∘* `:60 := pcompose
@ -323,7 +323,7 @@ namespace pointed
protected definition phomotopy.refl [constructor] : k ~* k :=
phomotopy.mk homotopy.rfl !idp_con
variable {k}
protected definition phomotopy.rfl [constructor] [refl] : k ~* k :=
protected definition phomotopy.rfl [reducible] [constructor] [refl] : k ~* k :=
phomotopy.refl k
protected definition phomotopy.symm [constructor] [symm] (p : k ~* l) : l ~* k :=
@ -447,8 +447,7 @@ namespace pointed
definition eq_of_phomotopy (p : k ~* l) : k = l :=
to_inv (ppi_eq_equiv k l) p
definition eq_of_phomotopy_refl {X Y : Type*} (f : X →* Y) :
eq_of_phomotopy (phomotopy.refl f) = idpath f :=
definition eq_of_phomotopy_refl (k : ppi P p₀) : eq_of_phomotopy (phomotopy.refl k) = idpath k :=
begin
apply to_inv_eq_of_eq, reflexivity
end
@ -480,11 +479,17 @@ namespace pointed
induction t, exact phomotopy_of_eq_idp k ▸ q,
end
definition phomotopy_rec_idp' (Q : Π ⦃k' : ppi P p₀⦄, (k ~* k') → (k = k') → Type)
(q : Q phomotopy.rfl idp) ⦃k' : ppi P p₀⦄ (H : k ~* k') : Q H (eq_of_phomotopy H) :=
begin
induction H using phomotopy_rec_idp,
exact transport (Q phomotopy.rfl) !eq_of_phomotopy_refl⁻¹ q
end
attribute phomotopy.rec' [recursor]
definition phomotopy_rec_eq_phomotopy_of_eq {A B : Type*} {f g: A →* B}
{Q : (f ~* g) → Type} (p : f = g) (H : Π(q : f = g), Q (phomotopy_of_eq q)) :
phomotopy_rec_eq (phomotopy_of_eq p) H = H p :=
definition phomotopy_rec_eq_phomotopy_of_eq {Q : (k ~* l) → Type} (p : k = l)
(H : Π(q : k = l), Q (phomotopy_of_eq q)) : phomotopy_rec_eq (phomotopy_of_eq p) H = H p :=
begin
unfold phomotopy_rec_eq,
refine ap (λp, p ▸ _) !adj ⬝ _,
@ -492,11 +497,33 @@ namespace pointed
apply apdt
end
definition phomotopy_rec_idp_refl {A B : Type*} (f : A →* B)
{Q : Π{g}, (f ~* g) → Type} (H : Q (phomotopy.refl f)) :
definition phomotopy_rec_idp_refl {Q : Π{l}, (k ~* l) → Type} (H : Q (phomotopy.refl k)) :
phomotopy_rec_idp phomotopy.rfl H = H :=
!phomotopy_rec_eq_phomotopy_of_eq
definition phomotopy_rec_idp'_refl (Q : Π ⦃k' : ppi P p₀⦄, (k ~* k') → (k = k') → Type)
(q : Q phomotopy.rfl idp) :
phomotopy_rec_idp' Q q phomotopy.rfl = transport (Q phomotopy.rfl) !eq_of_phomotopy_refl⁻¹ q :=
!phomotopy_rec_idp_refl
/- maps out of or into contractible types -/
definition phomotopy_of_is_contr_cod [constructor] (k l : ppi P p₀) [Πa, is_contr (P a)] :
k ~* l :=
phomotopy.mk (λa, !eq_of_is_contr) !eq_of_is_contr
definition phomotopy_of_is_contr_cod_pmap [constructor] (f g : A →* B) [is_contr B] : f ~* g :=
phomotopy_of_is_contr_cod f g
definition phomotopy_of_is_contr_dom [constructor] (k l : ppi P p₀) [is_contr A] : k ~* l :=
begin
fapply phomotopy.mk,
{ intro a, exact eq_of_pathover_idp (change_path !is_prop.elim
(apd k !is_prop.elim ⬝op respect_pt k ⬝ (respect_pt l)⁻¹ ⬝o apd l !is_prop.elim)) },
rewrite [▸*, +is_prop_elim_self, +apd_idp, cono_idpo],
refine ap (λx, eq_of_pathover_idp (change_path x _)) !is_prop_elim_self ◾ idp ⬝ _,
xrewrite [change_path_idp, idpo_concato_eq, inv_con_cancel_right],
end
/- adjunction between (-)₊ : Type → Type* and pType.carrier : Type* → Type -/
definition pmap_equiv_left (A : Type) (B : Type*) : A₊ →* B ≃ (A → B) :=
begin

126
hott/types/pointed2.hlean
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@ -18,7 +18,7 @@ import algebra.homotopy_group eq2
open pointed eq unit is_trunc trunc nat group is_equiv equiv sigma function bool
namespace pointed
variables {A B C : Type*}
variables {A B C : Type*} {P : A → Type} {p₀ : P pt} {k k' l m n : ppi P p₀}
section psquare
/-
@ -134,14 +134,35 @@ namespace pointed
definition punit_pmap_phomotopy [constructor] {A : Type*} (f : punit →* A) :
f ~* pconst punit A :=
!phomotopy_of_is_contr_dom
definition punit_ppi [constructor] (P : punit → Type*) (p₀ : P ⋆) : ppi P p₀ :=
begin
fapply phomotopy.mk,
{ intro u, induction u, exact respect_pt f },
{ reflexivity }
fapply ppi.mk, intro u, induction u, exact p₀,
reflexivity
end
definition punit_ppi_phomotopy [constructor] {P : punit → Type*} {p₀ : P ⋆} (f : ppi P p₀) :
f ~* punit_ppi P p₀ :=
!phomotopy_of_is_contr_dom
definition is_contr_punit_ppi (P : punit → Type*) (p₀ : P ⋆) : is_contr (ppi P p₀) :=
is_contr.mk (punit_ppi P p₀) (λf, eq_of_phomotopy (punit_ppi_phomotopy f)⁻¹*)
definition is_contr_punit_pmap (A : Type*) : is_contr (punit →* A) :=
is_contr.mk (pconst punit A) (λf, eq_of_phomotopy (punit_pmap_phomotopy f)⁻¹*)
!is_contr_punit_ppi
-- definition phomotopy_eq_equiv (h₁ h₂ : k ~* l) :
-- (h₁ = h₂) ≃ Σ(p : to_homotopy h₁ ~ to_homotopy h₂),
-- whisker_right (respect_pt l) (p pt) ⬝ to_homotopy_pt h₂ = to_homotopy_pt h₁ :=
-- begin
-- refine !ppi_eq_equiv ⬝e !phomotopy.sigma_char ⬝e sigma_equiv_sigma_right _,
-- intro p,
-- end
/- Short term TODO: generalize to dependent maps (use ppi_eq_equiv?)
Long term TODO: use homotopies between pointed homotopies, not equalities
-/
definition phomotopy_eq_equiv {A B : Type*} {f g : A →* B} (h k : f ~* g) :
(h = k) ≃ Σ(p : to_homotopy h ~ to_homotopy k),
@ -173,12 +194,11 @@ namespace pointed
(q : square (to_homotopy_pt h) (to_homotopy_pt k) (whisker_right (respect_pt g) (p pt)) idp) : h = k :=
phomotopy_eq p (eq_of_square q)⁻¹
definition trans_refl {A B : Type*} {f g : A →* B} (p : f ~* g) : p ⬝* phomotopy.refl g = p :=
definition trans_refl (p : k ~* l) : p ⬝* phomotopy.rfl = p :=
begin
induction A with A a₀, induction B with B b₀,
induction f with f f₀, induction g with g g₀, induction p with p p₀,
esimp at *, induction g₀, induction p₀,
reflexivity
induction A with A a₀,
induction k with k k₀, induction l with l l₀, induction p with p p₀', esimp at * ⊢,
induction l₀, induction p₀', reflexivity,
end
definition eq_of_phomotopy_trans {X Y : Type*} {f g h : X →* Y} (p : f ~* g) (q : g ~* h) :
@ -188,77 +208,66 @@ namespace pointed
exact ap eq_of_phomotopy !trans_refl ⬝ whisker_left _ !eq_of_phomotopy_refl⁻¹
end
definition refl_trans {A B : Type*} {f g : A →* B} (p : f ~* g) : phomotopy.refl f ⬝* p = p :=
definition refl_trans (p : k ~* l) : phomotopy.rfl ⬝* p = p :=
begin
induction p using phomotopy_rec_idp,
induction A with A a₀, induction B with B b₀,
induction f with f f₀, esimp at *, induction f₀,
reflexivity
apply trans_refl
end
definition trans_assoc {A B : Type*} {f g h i : A →* B} (p : f ~* g) (q : g ~* h)
(r : h ~* i) : p ⬝* q ⬝* r = p ⬝* (q ⬝* r) :=
definition trans_assoc (p : k ~* l) (q : l ~* m) (r : m ~* n) : p ⬝* q ⬝* r = p ⬝* (q ⬝* r) :=
begin
induction r using phomotopy_rec_idp,
induction q using phomotopy_rec_idp,
induction p using phomotopy_rec_idp,
induction B with B b₀,
induction f with f f₀, esimp at *, induction f₀,
induction k with k k₀, induction k₀,
reflexivity
end
definition refl_symm {A B : Type*} (f : A →* B) : phomotopy.rfl⁻¹* = phomotopy.refl f :=
definition refl_symm : phomotopy.rfl⁻¹* = phomotopy.refl k :=
begin
induction B with B b₀,
induction f with f f₀, esimp at *, induction f₀,
induction k with k k₀, induction k₀,
reflexivity
end
definition symm_symm {A B : Type*} {f g : A →* B} (p : f ~* g) : p⁻¹*⁻¹* = p :=
phomotopy_eq (λa, !inv_inv)
definition symm_symm (p : k ~* l) : p⁻¹*⁻¹* = p :=
begin
induction p using phomotopy_rec_idp, induction f with f f₀, induction B with B b₀,
esimp at *, induction f₀, reflexivity
induction p using phomotopy_rec_idp, induction k with k k₀, induction k₀, reflexivity
end
definition trans_right_inv {A B : Type*} {f g : A →* B} (p : f ~* g) : p ⬝* p⁻¹* = phomotopy.rfl :=
definition trans_right_inv (p : k ~* l) : p ⬝* p⁻¹* = phomotopy.rfl :=
begin
induction p using phomotopy_rec_idp, exact !refl_trans ⬝ !refl_symm
end
definition trans_left_inv {A B : Type*} {f g : A →* B} (p : f ~* g) : p⁻¹* ⬝* p = phomotopy.rfl :=
definition trans_left_inv (p : k ~* l) : p⁻¹* ⬝* p = phomotopy.rfl :=
begin
induction p using phomotopy_rec_idp, exact !trans_refl ⬝ !refl_symm
end
definition trans2 {A B : Type*} {f g h : A →* B} {p p' : f ~* g} {q q' : g ~* h}
(r : p = p') (s : q = q') : p ⬝* q = p' ⬝* q' :=
definition trans2 {p p' : k ~* l} {q q' : l ~* m} (r : p = p') (s : q = q') : p ⬝* q = p' ⬝* q' :=
ap011 phomotopy.trans r s
definition pcompose3 {A B C : Type*} {g g' : B →* C} {f f' : A →* B}
{p p' : g ~* g'} {q q' : f ~* f'} (r : p = p') (s : q = q') : p ◾* q = p' ◾* q' :=
ap011 pcompose2 r s
definition symm2 {A B : Type*} {f g : A →* B} {p p' : f ~* g} (r : p = p') : p⁻¹* = p'⁻¹* :=
definition symm2 {p p' : k ~* l} (r : p = p') : p⁻¹* = p'⁻¹* :=
ap phomotopy.symm r
infixl ` ◾** `:80 := pointed.trans2
infixl ` ◽* `:81 := pointed.pcompose3
postfix `⁻²**`:(max+1) := pointed.symm2
definition trans_symm {A B : Type*} {f g h : A →* B} (p : f ~* g) (q : g ~* h) :
(p ⬝* q)⁻¹* = q⁻¹* ⬝* p⁻¹* :=
definition trans_symm (p : k ~* l) (q : l ~* m) : (p ⬝* q)⁻¹* = q⁻¹* ⬝* p⁻¹* :=
begin
induction p using phomotopy_rec_idp, induction q using phomotopy_rec_idp,
exact !trans_refl⁻²** ⬝ !trans_refl⁻¹ ⬝ idp ◾** !refl_symm⁻¹
end
definition phwhisker_left {A B : Type*} {f g h : A →* B} (p : f ~* g) {q q' : g ~* h}
(s : q = q') : p ⬝* q = p ⬝* q' :=
definition phwhisker_left (p : k ~* l) {q q' : l ~* m} (s : q = q') : p ⬝* q = p ⬝* q' :=
idp ◾** s
definition phwhisker_right {A B : Type*} {f g h : A →* B} {p p' : f ~* g} (q : g ~* h)
(r : p = p') : p ⬝* q = p' ⬝* q :=
definition phwhisker_right {p p' : k ~* l} (q : l ~* m) (r : p = p') : p ⬝* q = p' ⬝* q :=
r ◾** idp
definition pwhisker_left_refl {A B C : Type*} (g : B →* C) (f : A →* B) :
@ -325,36 +334,36 @@ namespace pointed
refine ap (pwhisker_right f) !refl_symm ⬝ !pwhisker_right_refl ⬝ !refl_symm⁻¹
end
definition trans_eq_of_eq_symm_trans {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
{r : f ~* h} (s : q = p⁻¹* ⬝* r) : p ⬝* q = r :=
definition trans_eq_of_eq_symm_trans {p : k ~* l} {q : l ~* m} {r : k ~* m} (s : q = p⁻¹* ⬝* r) :
p ⬝* q = r :=
idp ◾** s ⬝ !trans_assoc⁻¹ ⬝ trans_right_inv p ◾** idp ⬝ !refl_trans
definition eq_symm_trans_of_trans_eq {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
{r : f ~* h} (s : p ⬝* q = r) : q = p⁻¹* ⬝* r :=
definition eq_symm_trans_of_trans_eq {p : k ~* l} {q : l ~* m} {r : k ~* m} (s : p ⬝* q = r) :
q = p⁻¹* ⬝* r :=
!refl_trans⁻¹ ⬝ !trans_left_inv⁻¹ ◾** idp ⬝ !trans_assoc ⬝ idp ◾** s
definition trans_eq_of_eq_trans_symm {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
{r : f ~* h} (s : p = r ⬝* q⁻¹*) : p ⬝* q = r :=
definition trans_eq_of_eq_trans_symm {p : k ~* l} {q : l ~* m} {r : k ~* m} (s : p = r ⬝* q⁻¹*) :
p ⬝* q = r :=
s ◾** idp ⬝ !trans_assoc ⬝ idp ◾** trans_left_inv q ⬝ !trans_refl
definition eq_trans_symm_of_trans_eq {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
{r : f ~* h} (s : p ⬝* q = r) : p = r ⬝* q⁻¹* :=
definition eq_trans_symm_of_trans_eq {p : k ~* l} {q : l ~* m} {r : k ~* m} (s : p ⬝* q = r) :
p = r ⬝* q⁻¹* :=
!trans_refl⁻¹ ⬝ idp ◾** !trans_right_inv⁻¹ ⬝ !trans_assoc⁻¹ ⬝ s ◾** idp
definition eq_trans_of_symm_trans_eq {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
{r : f ~* h} (s : p⁻¹* ⬝* r = q) : r = p ⬝* q :=
definition eq_trans_of_symm_trans_eq {p : k ~* l} {q : l ~* m} {r : k ~* m} (s : p⁻¹* ⬝* r = q) :
r = p ⬝* q :=
!refl_trans⁻¹ ⬝ !trans_right_inv⁻¹ ◾** idp ⬝ !trans_assoc ⬝ idp ◾** s
definition symm_trans_eq_of_eq_trans {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
{r : f ~* h} (s : r = p ⬝* q) : p⁻¹* ⬝* r = q :=
definition symm_trans_eq_of_eq_trans {p : k ~* l} {q : l ~* m} {r : k ~* m} (s : r = p ⬝* q) :
p⁻¹* ⬝* r = q :=
idp ◾** s ⬝ !trans_assoc⁻¹ ⬝ trans_left_inv p ◾** idp ⬝ !refl_trans
definition eq_trans_of_trans_symm_eq {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
{r : f ~* h} (s : r ⬝* q⁻¹* = p) : r = p ⬝* q :=
definition eq_trans_of_trans_symm_eq {p : k ~* l} {q : l ~* m} {r : k ~* m} (s : r ⬝* q⁻¹* = p) :
r = p ⬝* q :=
!trans_refl⁻¹ ⬝ idp ◾** !trans_left_inv⁻¹ ⬝ !trans_assoc⁻¹ ⬝ s ◾** idp
definition trans_symm_eq_of_eq_trans {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
{r : f ~* h} (s : r = p ⬝* q) : r ⬝* q⁻¹* = p :=
definition trans_symm_eq_of_eq_trans {p : k ~* l} {q : l ~* m} {r : k ~* m} (s : r = p ⬝* q) :
r ⬝* q⁻¹* = p :=
s ◾** idp ⬝ !trans_assoc ⬝ idp ◾** trans_right_inv q ⬝ !trans_refl
section phsquare
@ -362,7 +371,7 @@ namespace pointed
Squares of pointed homotopies
-/
variables {f f' f₀₀ f₂₀ f₄₀ f₀₂ f₂₂ f₄₂ f₀₄ f₂₄ f₄₄ : A →* B}
variables {f f' f₀₀ f₂₀ f₄₀ f₀₂ f₂₂ f₄₂ f₀₄ f₂₄ f₄₄ : ppi P p₀}
{p₁₀ : f₀₀ ~* f₂₀} {p₃₀ : f₂₀ ~* f₄₀}
{p₀₁ : f₀₀ ~* f₀₂} {p₂₁ : f₂₀ ~* f₂₂} {p₄₁ : f₄₀ ~* f₄₂}
{p₁₂ : f₀₂ ~* f₂₂} {p₃₂ : f₂₂ ~* f₄₂}
@ -495,6 +504,13 @@ namespace pointed
!pwhisker_left_refl⁻¹ ◾** !pwhisker_right_refl⁻¹
end
end phsquare
section nondep_phsquare
variables {f f' f₀₀ f₂₀ f₀₂ f₂₂ : A →* B}
{p₁₀ : f₀₀ ~* f₂₀} {p₀₁ : f₀₀ ~* f₀₂} {p₂₁ : f₂₀ ~* f₂₂} {p₁₂ : f₀₂ ~* f₂₂}
definition pwhisker_left_phsquare (f : B →* C) (p : phsquare p₁₀ p₁₂ p₀₁ p₂₁) :
phsquare (pwhisker_left f p₁₀) (pwhisker_left f p₁₂)
(pwhisker_left f p₀₁) (pwhisker_left f p₂₁) :=
@ -505,9 +521,9 @@ namespace pointed
(pwhisker_right f p₀₁) (pwhisker_right f p₂₁) :=
!pwhisker_right_trans⁻¹ ⬝ ap (pwhisker_right f) p ⬝ !pwhisker_right_trans
end phsquare
end nondep_phsquare
definition phomotopy_of_eq_con {A B : Type*} {f g h : A →* B} (p : f = g) (q : g = h) :
definition phomotopy_of_eq_con (p : k = l) (q : l = m) :
phomotopy_of_eq (p ⬝ q) = phomotopy_of_eq p ⬝* phomotopy_of_eq q :=
begin induction q, induction p, exact !trans_refl⁻¹ end

3
hott/types/unit.hlean
View file

@ -32,6 +32,9 @@ namespace unit
notation `unit*` := punit
definition is_contr_punit [instance] : is_contr punit :=
is_contr_unit
definition unit_arrow_eq {X : Type} (f : unit → X) : (λx, f ⋆) = f :=
by apply eq_of_homotopy; intro u; induction u; reflexivity