From 6889eba5ea60526766cb7a29bb76e5143139b880 Mon Sep 17 00:00:00 2001
From: Steve Awodey <sawodey@gmail.com>
Date: Thu, 21 Jan 2016 14:33:54 -0500
Subject: [PATCH] added homotopy subdirectory and sample file

to formalize Chapter 8 of the book
---
 homotopy/sample.hlean | 194 ++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 194 insertions(+)
 create mode 100644 homotopy/sample.hlean

diff --git a/homotopy/sample.hlean b/homotopy/sample.hlean
new file mode 100644
index 0000000..f4949ae
--- /dev/null
+++ b/homotopy/sample.hlean
@@ -0,0 +1,194 @@
+/-
+Copyright (c) 2015 Ulrik Buchholtz. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Ulrik Buchholtz
+-/
+import types.trunc types.arrow_2 types.fiber .susp
+
+open eq is_trunc is_equiv nat equiv trunc function fiber
+
+namespace homotopy
+
+  definition is_conn [reducible] (n : trunc_index) (A : Type) : Type :=
+  is_contr (trunc n A)
+
+  definition is_conn_equiv_closed (n : trunc_index) {A B : Type}
+    : A ≃ B → is_conn n A → is_conn n B :=
+  begin
+    intros H C,
+    fapply @is_contr_equiv_closed (trunc n A) _,
+    apply trunc_equiv_trunc,
+    assumption
+  end
+
+  definition is_conn_map (n : trunc_index) {A B : Type} (f : A → B) : Type :=
+  Πb : B, is_conn n (fiber f b)
+
+  namespace is_conn_map
+  section
+    parameters {n : trunc_index} {A B : Type} {h : A → B}
+               (H : is_conn_map n h) (P : B → n -Type)
+
+    private definition helper : (Πa : A, P (h a)) → Πb : B, trunc n (fiber h b) → P b :=
+    λt b, trunc.rec (λx, point_eq x ▸ t (point x))
+
+    private definition g : (Πa : A, P (h a)) → (Πb : B, P b) :=
+    λt b, helper t b (@center (trunc n (fiber h b)) (H b))
+
+    -- induction principle for n-connected maps (Lemma 7.5.7)
+    definition rec : is_equiv (λs : Πb : B, P b, λa : A, s (h a)) :=
+    adjointify (λs a, s (h a)) g
+    begin
+      intro t, apply eq_of_homotopy, intro a, unfold g, unfold helper,
+      rewrite [@center_eq _ (H (h a)) (tr (fiber.mk a idp))],
+    end
+    begin
+      intro k, apply eq_of_homotopy, intro b, unfold g,
+      generalize (@center _ (H b)), apply trunc.rec, apply fiber.rec,
+      intros a p, induction p, reflexivity
+    end
+
+    definition elim : (Πa : A, P (h a)) → (Πb : B, P b) :=
+    @is_equiv.inv _ _ (λs a, s (h a)) rec
+
+    definition elim_β : Πf : (Πa : A, P (h a)), Πa : A, elim f (h a) = f a :=
+    λf, apd10 (@is_equiv.right_inv _ _ (λs a, s (h a)) rec f)
+
+  end
+
+  section
+    universe variables u v
+    parameters {n : trunc_index} {A : Type.{u}} {B : Type.{v}} {h : A → B}
+    parameter sec : ΠP : B → trunctype.{max u v} n,
+                    is_retraction (λs : (Πb : B, P b), λ a, s (h a))
+
+    private definition s := sec (λb, trunctype.mk' n (trunc n (fiber h b)))
+
+    include sec
+
+    -- the other half of Lemma 7.5.7
+    definition intro : is_conn_map n h :=
+    begin
+      intro b,
+      apply is_contr.mk (@is_retraction.sect _ _ _ s (λa, tr (fiber.mk a idp)) b),
+      apply trunc.rec, apply fiber.rec, intros a p,
+      apply transport
+               (λz : (Σy, h a = y), @sect _ _ _ s (λa, tr (mk a idp)) (sigma.pr1 z) =
+                                    tr (fiber.mk a (sigma.pr2 z)))
+               (@center_eq _ (is_contr_sigma_eq (h a)) (sigma.mk b p)),
+      exact apd10 (@right_inverse _ _ _ s (λa, tr (fiber.mk a idp))) a
+    end
+  end
+  end is_conn_map
+
+  -- Connectedness is related to maps to and from the unit type, first to
+  section
+    parameters (n : trunc_index) (A : Type)
+
+    definition is_conn_of_map_to_unit
+      : is_conn_map n (λx : A, unit.star) → is_conn n A :=
+    begin
+      intro H, unfold is_conn_map at H,
+      rewrite [-(ua (fiber.fiber_star_equiv A))],
+      exact (H unit.star)
+    end
+
+    -- now maps from unit
+    definition is_conn_of_map_from_unit (a₀ : A) (H : is_conn_map n (const unit a₀))
+      : is_conn n .+1 A :=
+    is_contr.mk (tr a₀)
+    begin
+      apply trunc.rec, intro a,
+      exact trunc.elim (λz : fiber (const unit a₀) a, ap tr (point_eq z))
+                            (@center _ (H a))
+    end
+
+    definition is_conn_map_from_unit (a₀ : A) [H : is_conn n .+1 A]
+      : is_conn_map n (const unit a₀) :=
+    begin
+      intro a,
+      apply is_conn_equiv_closed n (equiv.symm (fiber_const_equiv A a₀ a)),
+      apply @is_contr_equiv_closed _ _ (tr_eq_tr_equiv n a₀ a),
+    end
+
+  end
+
+  -- Lemma 7.5.2
+  definition minus_one_conn_of_surjective {A B : Type} (f : A → B)
+    : is_surjective f → is_conn_map -1 f :=
+  begin
+    intro H, intro b,
+    exact @is_contr_of_inhabited_hprop (∥fiber f b∥) (is_trunc_trunc -1 (fiber f b)) (H b),
+  end
+
+  definition is_surjection_of_minus_one_conn {A B : Type} (f : A → B)
+    : is_conn_map -1 f → is_surjective f :=
+  begin
+    intro H, intro b,
+    exact @center (∥fiber f b∥) (H b),
+  end
+
+  definition merely_of_minus_one_conn {A : Type} : is_conn -1 A → ∥A∥ :=
+  λH, @center (∥A∥) H
+
+  definition minus_one_conn_of_merely {A : Type} : ∥A∥ → is_conn -1 A :=
+  @is_contr_of_inhabited_hprop (∥A∥) (is_trunc_trunc -1 A)
+
+  section
+    open arrow
+
+    variables {f g : arrow}
+
+    -- Lemma 7.5.4
+    definition retract_of_conn_is_conn [instance] (r : arrow_hom f g) [H : is_retraction r]
+      (n : trunc_index) [K : is_conn_map n f] : is_conn_map n g :=
+    begin
+      intro b, unfold is_conn,
+      apply is_contr_retract (trunc_functor n (retraction_on_fiber r b)),
+      exact K (on_cod (arrow.is_retraction.sect r) b)
+    end
+
+  end
+
+  -- Corollary 7.5.5
+  definition is_conn_homotopy (n : trunc_index) {A B : Type} {f g : A → B}
+    (p : f ~ g) (H : is_conn_map n f) : is_conn_map n g :=
+  @retract_of_conn_is_conn _ _ (arrow.arrow_hom_of_homotopy p) (arrow.is_retraction_arrow_hom_of_homotopy p) n H 
+
+  -- all types are -2-connected
+  definition minus_two_conn [instance] (A : Type) : is_conn -2 A :=
+  _
+
+  -- Theorem 8.2.1
+  open susp
+
+  definition is_conn_susp [instance] (n : trunc_index) (A : Type)
+    [H : is_conn n A] : is_conn (n .+1) (susp A) :=
+  is_contr.mk (tr north)
+  begin
+    apply trunc.rec,
+    fapply susp.rec,
+    { reflexivity },
+    { exact (trunc.rec (λa, ap tr (merid a)) (center (trunc n A))) },
+    { intro a,
+      generalize (center (trunc n A)),
+      apply trunc.rec,
+      intro a',
+      apply pathover_of_tr_eq,
+      rewrite [transport_eq_Fr,idp_con],
+      revert H, induction n with [n, IH],
+      { intro H, apply is_hprop.elim },
+      { intros H,
+        change ap (@tr n .+2 (susp A)) (merid a) = ap tr (merid a'),
+        generalize a',
+        apply is_conn_map.elim
+              (is_conn_map_from_unit n A a)
+              (λx : A, trunctype.mk' n (ap (@tr n .+2 (susp A)) (merid a) = ap tr (merid x))),
+        intros,
+        change ap (@tr n .+2 (susp A)) (merid a) = ap tr (merid a),
+        reflexivity
+      }
+    }
+  end
+
+end homotopy