feat(hott/algebra) start to formalize the category of sets (seems to be pretty tough)

hott/algebra/category/Set.hlean

@@ -0,0 +1,71 @@
+-- Copyright (c) 2015 Jakob von Raumer. All rights reserved.
+-- Released under Apache 2.0 license as described in the file LICENSE.
+-- Authors: Jakob von Raumer
+-- Category of sets
+
+import .basic types.pi trunc
+
+open truncation sigma sigma.ops pi function eq morphism precategory
+open equiv
+
+namespace precategory
+
+  universe variable l
+
+  definition set_precategory : precategory.{l+1 l} (Σ (A : Type.{l}), is_hset A) :=
+  begin
+    fapply precategory.mk.{l+1 l},
+                  intros, apply (a.1 → a_1.1),
+                intros, apply trunc_pi, intros, apply b.2,
+              intros, intro x, exact (a_1 (a_2 x)),
+            intros, exact (λ (x : a.1), x),
+          intros, apply funext.path_pi, intro x, apply idp,
+        intros, apply funext.path_pi, intro x, apply idp,
+      intros, apply funext.path_pi, intro x, apply idp,
+  end
+
+end precategory
+
+namespace category
+
+  universe variable l
+  instance precategory.set_precategory.{l+1 l}
+
+  check @equiv
+  definition set_category_equiv_iso (a b : (Σ (A : Type.{l}), is_hset A))
+    : (a ≅ b) = (a.1 ≃ b.1) :=
+  /-begin
+    apply ua, fapply equiv.mk,
+      intro H,
+        apply (isomorphic.rec_on H), intros (H1, H2),
+        apply (is_iso.rec_on H2), intros (H3, H4, H5),
+        fapply equiv.mk,
+        apply (isomorphic.rec_on H), intros (H1, H2),
+        exact H1,
+      fapply is_equiv.adjointify, exact H3,
+          exact sorry,
+        exact sorry,
+  end-/ sorry
+
+  definition set_category : category.{l+1 l} (Σ (A : Type.{l}), is_hset A) :=
+  begin
+    assert (C : precategory.{l+1 l} (Σ (A : Type.{l}), is_hset A)),
+      apply precategory.set_precategory,
+    apply category.mk,
+    assert (p : (λ A B p, (set_category_equiv_iso A B) ▹ iso_of_path p) = (λ A B p, @equiv_path A.1 B.1 p)),
+    /-apply is_equiv.adjointify,
+        intros,
+        apply (isomorphic.rec_on a_1), intros (iso', is_iso'),
+        apply (is_iso.rec_on is_iso'), intros (f', f'sect, f'retr),
+        fapply sigma.path,
+          apply ua, fapply equiv.mk, exact iso',
+          fapply is_equiv.adjointify,
+              exact f',
+            intros, apply (f'retr ▹ _),
+          intros, apply (f'sect ▹ _),
+        apply (@is_hprop.elim),
+        apply is_trunc_is_hprop,
+      intros, -/
+  end
+
+end category