feat(hott/algebra) add structure for strict precategories

hott/algebra/precategory/functor.hlean

@@ -102,6 +102,7 @@ namespace functor
 
   set_option apply.class_instance false
   -- "functor C D" is equivalent to a certain sigma type
+  set_option unifier.max_steps 38500
   protected definition sigma_char :
     (Σ (to_fun_ob : C → D)
     (to_fun_hom : Π ⦃a b : C⦄, hom a b → hom (to_fun_ob a) (to_fun_ob b)),
@@ -227,27 +228,3 @@ namespace functor
   -- end
 
 end functor
-
-namespace category
-  open functor
-
-  --TODO: make this a structure
-  definition precat_strict_precat : precategory (Σ (C : Precategory), is_hset C) :=
-  precategory.mk (λ a b, functor a.1 b.1)
-     (λ a b, @functor.is_hset_functor a.1 b.1 b.2)
-     (λ a b c g f, functor.compose g f)
-     (λ a, functor.id)
-     (λ a b c d h g f, !functor.assoc)
-     (λ a b f, !functor.id_left)
-     (λ a b f, !functor.id_right)
-
-  definition Precat_of_strict_cats := precategory.Mk precat_strict_precat
-
-  namespace ops
-
-    abbreviation SPreCat := Precat_of_strict_cats
-    --attribute precat_strict_precat [instance]
-
-  end ops
-
-end category

hott/algebra/precategory/strict.hlean

@@ -0,0 +1,45 @@
+import .basic .functor
+
+open category is_trunc eq function sigma sigma.ops
+
+namespace category
+  structure strict_precategory[class] (ob : Type) extends precategory ob :=
+    (is_hset_ob : is_hset ob)
+
+  attribute strict_precategory.is_hset_ob [instance]
+
+  definition strict_precategory.mk' [reducible] (ob : Type) (C : precategory ob)
+    (H : is_hset ob) : strict_precategory ob :=
+  precategory.rec_on C strict_precategory.mk H
+
+  structure Strict_precategory : Type :=
+    (carrier : Type)
+    (struct : strict_precategory carrier)
+
+  attribute Strict_precategory.struct [instance] [coercion]
+
+  definition Strict_precategory.to_Precategory [coercion] [reducible]
+    (C : Strict_precategory) : Precategory :=
+  Precategory.mk (Strict_precategory.carrier C) C
+
+  open functor
+
+  definition precat_strict_precat : precategory Strict_precategory :=
+  precategory.mk (λ a b, functor a b)
+     (λ a b, @functor.is_hset_functor a b _)
+     (λ a b c g f, functor.compose g f)
+     (λ a, functor.id)
+     (λ a b c d h g f, !functor.assoc)
+     (λ a b f, !functor.id_left)
+     (λ a b f, !functor.id_right)
+
+  definition Precat_of_strict_precats := precategory.Mk precat_strict_precat
+
+  namespace ops
+
+    abbreviation SPreCat := Precat_of_strict_precats
+    --attribute precat_strict_precat [instance]
+
+  end ops
+
+end category