feat(hott): minor changes
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8 changed files with 20 additions and 10 deletions
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@ -19,8 +19,8 @@ namespace eq
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definition homotopy_group [reducible] (n : ℕ) (A : Type*) : Type :=
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phomotopy_group n A
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notation `π*[`:95 n:0 `] `:0 := phomotopy_group n
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notation `π[`:95 n:0 `] `:0 := homotopy_group n
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notation `π*[`:95 n:0 `]`:0 := phomotopy_group n
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notation `π[`:95 n:0 `]`:0 := homotopy_group n
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definition group_homotopy_group [instance] [constructor] [reducible] (n : ℕ) (A : Type*)
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: group (π[succ n] A) :=
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@ -126,8 +126,8 @@ namespace eq
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definition homotopy_group_functor (n : ℕ) {A B : Type*} (f : A →* B) : π[n] A → π[n] B :=
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phomotopy_group_functor n f
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notation `π→*[`:95 n:0 `] `:0 := phomotopy_group_functor n
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notation `π→[`:95 n:0 `] `:0 := homotopy_group_functor n
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notation `π→*[`:95 n:0 `]`:0 := phomotopy_group_functor n
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notation `π→[`:95 n:0 `]`:0 := homotopy_group_functor n
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definition phomotopy_group_functor_phomotopy [constructor] (n : ℕ) {A B : Type*} {f g : A →* B}
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(p : f ~* g) : π→*[n] f ~* π→*[n] g :=
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@ -267,6 +267,6 @@ namespace eq
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exact ap tr !con_inv
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end
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notation `π→g[`:95 n:0 ` +1] `:0 f:95 := homotopy_group_homomorphism n f
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notation `π→g[`:95 n:0 ` +1]`:0 := homotopy_group_homomorphism n
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end eq
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@ -1,4 +1,4 @@
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/-
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/-
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Copyright (c) 2015 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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@ -76,7 +76,6 @@ We get the long exact sequence of homotopy groups by taking the set-truncation o
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import .chain_complex algebra.homotopy_group eq2
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open eq pointed sigma fiber equiv is_equiv sigma.ops is_trunc nat trunc algebra function sum
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/--------------
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PART 1
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--------------/
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@ -53,6 +53,15 @@ namespace is_trunc
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[H : is_conn_fun n f] (H2 : k ≤ n) : is_contr (π[k] (pfiber f)) :=
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@(trivial_homotopy_group_of_is_conn (pfiber f) H2) (H pt)
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theorem homotopy_group_trunc_of_le (A : Type*) (n k : ℕ) (H : k ≤ n)
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: π*[k] (ptrunc n A) ≃* π*[k] A :=
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begin
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refine !phomotopy_group_pequiv_loop_ptrunc ⬝e* _,
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refine loopn_pequiv_loopn _ (ptrunc_ptrunc_pequiv_left _ _) ⬝e* _,
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exact of_nat_le_of_nat H,
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exact !phomotopy_group_pequiv_loop_ptrunc⁻¹ᵉ*,
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end
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/- Corollaries of the LES of homotopy groups -/
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local attribute comm_group.to_group [coercion]
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local attribute is_equiv_tinverse [instance]
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@ -3,7 +3,7 @@ Copyright (c) 2016 Jakob von Raumer. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Jakob von Raumer, Ulrik Buchholtz
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The Wedge Sum of Two pType Types
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The Wedge Sum of Two Pointed Types
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-/
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import hit.pointed_pushout .connectedness types.unit
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@ -49,7 +49,7 @@ namespace eq
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definition con_idp [unfold_full] (p : x = y) : p ⬝ idp = p :=
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idp
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-- The identity path is a right unit.
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-- The identity path is a left unit.
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definition idp_con [unfold 4] (p : x = y) : idp ⬝ p = p :=
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by induction p; reflexivity
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@ -39,6 +39,8 @@ begin
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intro p, induction p, apply ap (mk i), apply !is_prop.elim
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end
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definition fin_eq := @eq_of_veq
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definition eq_of_veq_refl (i : fin n) : eq_of_veq (refl (val i)) = idp :=
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!is_prop.elim
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@ -496,7 +496,7 @@ end
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/- set difference -/
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definition diff (s t : set X) : set X := {x ∈ s | x ∉ t}
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infix `\`:70 := diff
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infix ` \ `:70 := diff
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theorem mem_diff {s t : set X} {x : X} (H1 : x ∈ s) (H2 : x ∉ t) : x ∈ s \ t :=
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and.intro H1 H2
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