feat(hott): minor changes

This commit is contained in:
Floris van Doorn 2016-07-13 09:39:16 +01:00
parent ddec6f77ee
commit 467001c0a9
8 changed files with 20 additions and 10 deletions

View file

@ -19,8 +19,8 @@ namespace eq
definition homotopy_group [reducible] (n : ) (A : Type*) : Type :=
phomotopy_group n A
notation `π*[`:95 n:0 `] `:0 := phomotopy_group n
notation `π[`:95 n:0 `] `:0 := homotopy_group n
notation `π*[`:95 n:0 `]`:0 := phomotopy_group n
notation `π[`:95 n:0 `]`:0 := homotopy_group n
definition group_homotopy_group [instance] [constructor] [reducible] (n : ) (A : Type*)
: group (π[succ n] A) :=
@ -126,8 +126,8 @@ namespace eq
definition homotopy_group_functor (n : ) {A B : Type*} (f : A →* B) : π[n] A → π[n] B :=
phomotopy_group_functor n f
notation `π→*[`:95 n:0 `] `:0 := phomotopy_group_functor n
notation `π→[`:95 n:0 `] `:0 := homotopy_group_functor n
notation `π→*[`:95 n:0 `]`:0 := phomotopy_group_functor n
notation `π→[`:95 n:0 `]`:0 := homotopy_group_functor n
definition phomotopy_group_functor_phomotopy [constructor] (n : ) {A B : Type*} {f g : A →* B}
(p : f ~* g) : π→*[n] f ~* π→*[n] g :=
@ -267,6 +267,6 @@ namespace eq
exact ap tr !con_inv
end
notation `π→g[`:95 n:0 ` +1] `:0 f:95 := homotopy_group_homomorphism n f
notation `π→g[`:95 n:0 ` +1]`:0 := homotopy_group_homomorphism n
end eq

View file

@ -1,4 +1,4 @@
/-
/-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.

View file

@ -76,7 +76,6 @@ We get the long exact sequence of homotopy groups by taking the set-truncation o
import .chain_complex algebra.homotopy_group eq2
open eq pointed sigma fiber equiv is_equiv sigma.ops is_trunc nat trunc algebra function sum
/--------------
PART 1
--------------/

View file

@ -53,6 +53,15 @@ namespace is_trunc
[H : is_conn_fun n f] (H2 : k ≤ n) : is_contr (π[k] (pfiber f)) :=
@(trivial_homotopy_group_of_is_conn (pfiber f) H2) (H pt)
theorem homotopy_group_trunc_of_le (A : Type*) (n k : ) (H : k ≤ n)
: π*[k] (ptrunc n A) ≃* π*[k] A :=
begin
refine !phomotopy_group_pequiv_loop_ptrunc ⬝e* _,
refine loopn_pequiv_loopn _ (ptrunc_ptrunc_pequiv_left _ _) ⬝e* _,
exact of_nat_le_of_nat H,
exact !phomotopy_group_pequiv_loop_ptrunc⁻¹ᵉ*,
end
/- Corollaries of the LES of homotopy groups -/
local attribute comm_group.to_group [coercion]
local attribute is_equiv_tinverse [instance]

View file

@ -3,7 +3,7 @@ Copyright (c) 2016 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Ulrik Buchholtz
The Wedge Sum of Two pType Types
The Wedge Sum of Two Pointed Types
-/
import hit.pointed_pushout .connectedness types.unit

View file

@ -49,7 +49,7 @@ namespace eq
definition con_idp [unfold_full] (p : x = y) : p ⬝ idp = p :=
idp
-- The identity path is a right unit.
-- The identity path is a left unit.
definition idp_con [unfold 4] (p : x = y) : idp ⬝ p = p :=
by induction p; reflexivity

View file

@ -39,6 +39,8 @@ begin
intro p, induction p, apply ap (mk i), apply !is_prop.elim
end
definition fin_eq := @eq_of_veq
definition eq_of_veq_refl (i : fin n) : eq_of_veq (refl (val i)) = idp :=
!is_prop.elim

View file

@ -496,7 +496,7 @@ end
/- set difference -/
definition diff (s t : set X) : set X := {x ∈ s | x ∉ t}
infix `\`:70 := diff
infix ` \ `:70 := diff
theorem mem_diff {s t : set X} {x : X} (H1 : x ∈ s) (H2 : x ∉ t) : x ∈ s \ t :=
and.intro H1 H2