From d5bbdfe9627eabfa72b9a4b8d1770b88682b7955 Mon Sep 17 00:00:00 2001
From: Michael Zhang <mail@mzhang.io>
Date: Thu, 25 Apr 2024 14:35:17 +0000
Subject: [PATCH] auto gitdoc commit

---
 src/HottBook/Chapter2.lagda.md |  2 +-
 src/HottBook/Chapter5.lagda.md | 33 ++++++++++++++++++++++++
 src/HottBook/Chapter6.lagda.md | 46 ++++++++++++++++++++++++++--------
 src/HottBook/Chapter8.lagda.md | 19 ++++++++++++++
 src/HottBook/Chapter9.lagda.md | 13 ++++++++++
 5 files changed, 102 insertions(+), 11 deletions(-)
 create mode 100644 src/HottBook/Chapter5.lagda.md
 create mode 100644 src/HottBook/Chapter8.lagda.md
 create mode 100644 src/HottBook/Chapter9.lagda.md

diff --git a/src/HottBook/Chapter2.lagda.md b/src/HottBook/Chapter2.lagda.md
index 00c8dec..530ec63 100644
--- a/src/HottBook/Chapter2.lagda.md
+++ b/src/HottBook/Chapter2.lagda.md
@@ -369,7 +369,7 @@ code (suc x) (suc y) = code x y
 
 _For all $m, n : N$ we have $(m = n) \simeq \texttt{code}(m, n)$._
 
-```
+```text
 theorem2∙13∙1 : (m n : ℕ) → (m ≡ n) ≃ code m n
 theorem2∙13∙1 m n = encode m n , equiv
   where
diff --git a/src/HottBook/Chapter5.lagda.md b/src/HottBook/Chapter5.lagda.md
new file mode 100644
index 0000000..d65a49c
--- /dev/null
+++ b/src/HottBook/Chapter5.lagda.md
@@ -0,0 +1,33 @@
+```
+module HottBook.Chapter5 where
+
+open import Agda.Primitive
+open import HottBook.Chapter1
+```
+
+## 5.1 Introduction to inductive types
+
+### Theorem 5.1.1
+
+```
+-- theorem5∙1∙1 : {E : ℕ → Set}
+--   → (f g : (x : ℕ) → E x)
+--   → (z : f 0 ≡ g 0)
+--   → (s : 
+--   → f ≡ g
+```
+
+### Definition 5.4.1
+
+```
+ℕAlg : {l : Level} → Set (lsuc l)
+ℕAlg {l} = Σ (Set l) (λ C → C × (C → C))
+```
+
+### Definition 5.4.2
+
+```
+ℕHom : {l : Level} → ℕAlg {l} → ℕAlg {l} → Set l
+ℕHom alg1@(C , (cz , cs)) alg2@(D , (dz , ds)) = 
+  Σ (C → D) λ h → (h cz ≡ dz) × ((c : C) → h (cs c) ≡ ds (h c))
+```
diff --git a/src/HottBook/Chapter6.lagda.md b/src/HottBook/Chapter6.lagda.md
index 09a505f..ef3a243 100644
--- a/src/HottBook/Chapter6.lagda.md
+++ b/src/HottBook/Chapter6.lagda.md
@@ -1,16 +1,18 @@
 ```
 module HottBook.Chapter6 where
 
-open import Relation.Binary.PropositionalEquality
-
+open import HottBook.Chapter1
 open import HottBook.Chapter2
 ```
 
 ### Definition 6.2.2 (Dependent paths)
 
 ```
-dep-path : {A : Type} (P : A → Type) {x y : A} (p : x ≡ y)
-  → (u : P x) → (v : P y) → Type
+dep-path : {A : Set} 
+  → (P : A → Set) 
+  → {x y : A} 
+  → (p : x ≡ y)
+  → (u : P x) → (v : P y) → Set
 dep-path P p u v = transport P p u ≡ v
 ```
 
@@ -18,17 +20,41 @@ Circle definition
 
 ```
 postulate
-  𝕊¹ : Type
-  base : 𝕊¹
+  S¹ : Set
+  base : S¹
   loop : base ≡ base
-  𝕊¹-elim : (P : 𝕊¹ → Type) → (p-base : P base)
+  S¹-elim : (P : S¹ → Set)
+    → (p-base : P base)
     → (p-loop : dep-path P loop p-base p-base)
-    → (x : 𝕊¹) → P x
+    → (x : S¹) → P x
 ```
 
 ### Lemma 6.2.5
 
+```text
+lemma6∙2∙5 : {A : Set}
+  → (a : A)
+  → (p : a ≡ a)
+  → S¹ → A
+lemma6∙2∙5 {A} a p circ = S¹-elim P p-base p-loop circ
+  where
+    P : S¹ → Set
+    P _ = A
+
+    p-base = a
+
+    p-loop : transport P loop a ≡ a
+    p-loop =
+      let wtf = {! lemma2∙3∙8   !} in
+      {!   !}
 ```
-lemma625 : {A : Type} (a : A) → (p : a ≡ a) → 𝕊¹ → A
-lemma625 {A} a p circ = 𝕊¹-elim (λ _ → A) a ? circ
+
+## 6.3 The interval
+
 ```
+postulate
+  I : Set
+  0I : I
+  1I : I
+  seg : 0I ≡ 1I
+```
\ No newline at end of file
diff --git a/src/HottBook/Chapter8.lagda.md b/src/HottBook/Chapter8.lagda.md
new file mode 100644
index 0000000..2a8dead
--- /dev/null
+++ b/src/HottBook/Chapter8.lagda.md
@@ -0,0 +1,19 @@
+```
+module HottBook.Chapter8 where
+
+open import Agda.Primitive
+open import HottBook.Chapter6
+```
+
+## 8.1 π₁(S¹)
+
+### Definition 8.1.1
+
+```
+data ℤ : Set where
+```
+
+```
+code : {l : Level} → S¹ → Set l
+code {l} = S¹-elim (λ _ → Set l) ℤ (ua succ)
+```
\ No newline at end of file
diff --git a/src/HottBook/Chapter9.lagda.md b/src/HottBook/Chapter9.lagda.md
new file mode 100644
index 0000000..adb71c8
--- /dev/null
+++ b/src/HottBook/Chapter9.lagda.md
@@ -0,0 +1,13 @@
+```
+module HottBook.Chapter9 where
+
+open import Agda.Primitive
+```
+
+## 9.1 Categories and precategories
+
+```
+record precategory {l : Level} : Set (lsuc l) where
+  field
+    A : Set l
+```
\ No newline at end of file