feat(hott) add type functors, maybe they should be changed to be un-funexted.

hott/types/type_functor.hlean

@@ -0,0 +1,56 @@
+/-
+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
+
+Pointed and unpointed type functor, and adjoint pairs.
+More or less ported from Evan Cavallo's HoTT-Agda homotopy library.
+-/
+
+import types.pointed
+
+open equiv function pointed Pointed
+
+structure type_functor : Type :=
+  (fun_ty : Type → Type)
+  (fun_arr : Π {A B}, (A → B) → (fun_ty A → fun_ty B))
+  (respect_id : Π {A}, fun_arr (@id A) = id)
+  (respect_comp : Π {A B C} (g : B → C) (f : A → B),
+    fun_arr (g ∘ f) = (fun_arr g) ∘ (fun_arr f))
+
+attribute [coercion] type_functor.fun_ty
+
+section type_adjoint
+open type_functor
+
+structure type_adjoint (F G : type_functor) : Type :=
+  (η : Π X, X → G (F X))
+  (ε : Π U, F (G U) → U)
+  (ηnat : Π X Y (h : X → Y), η Y ∘ h = fun_arr G (fun_arr F h) ∘ η X)
+  (εnat : Π U V (k : U → V), ε V ∘ fun_arr F (fun_arr G k) = k ∘ ε U)
+  (εF_Fη : Π X, ε (F X) ∘ fun_arr F (η X) = id)
+  (Gε_ηG : Π U, fun_arr G (ε U) ∘ η (G U) = id)
+
+end type_adjoint
+
+structure Type_functor : Type :=
+  (fun_ty : Type* → Type*)
+  (fun_arr : Π {A B}, (A →* B) → (fun_ty A →* fun_ty B))
+  (respect_id : Π {A}, fun_arr (pid A) = pid (fun_ty A))
+  (respect_comp : Π {A B C} (g : B →* C) (f : A →* B),
+    fun_arr (g ∘* f) = fun_arr g ∘* fun_arr f)
+
+attribute [coercion] Type_functor.fun_ty
+
+section Type_adjoint
+open Type_functor
+
+structure Type_adjoint (F G : Type_functor) : Type :=
+  (η : Π (X : Type*), X →* G (F X))
+  (ε : Π (U : Type*), F (G U) →* U)
+  (ηnat : Π {X Y} (h : X →* Y), η Y ∘* h = (fun_arr G (fun_arr F h)) ∘* η X)
+  (εnat : Π {U V} (k : U →* V), ε V ∘* (fun_arr F (fun_arr G k)) = k ∘* ε U)
+  (εF_Fη : Π {X}, ε (F X) ∘* (fun_arr F (η X)) = !pid)
+  (Gε_ηG : Π {U}, (fun_arr G (ε U)) ∘* η (G U) = !pid)
+
+end Type_adjoint