Put cat, hom, string dia together

trees/tt-0001.tree

@@ -10,54 +10,6 @@
 
 \date{2024-05-28}
 
-\p{These are my notes on:}
+\transclude{tt-0005}
 
-\ul{
-  \li{Category theory}
-  \li{Topos theory}
-  \li{Type theory}
-  \li{Sheaf theory}
-  \li{Differential sheaves}
-  \li{SDG (Synthetic Differential Geometry)}
-}
-
-\p{The primary reference for these notes are:}
-
-\ul{
-  \li{\citek{kostecki2011introduction}} for a clean introduction from category theory to topos theory
-  \li{\citek{kostecki2009differential}} for its introduction to SDG
-  \li{\citek{mallios2015differential}} for its introduction to Differential sheaves
-  \li{\citek{rosiak2022sheaf}} for its examples of sheaves
-  \li{\citek{zhang2021type}} for a friendly introduction to type theory using the language of category theory
-  \li{\citek{chen2016infinitely}} for various preliminaries on category theory
-  \li{\citek{fauser2004grade}} for the use of Kuperberg graphical calculi over commutative diagrams
-}
-
-\p{Scattered notes:}
-
-\scope{
-  % \put\transclude/toc{true}
-  % \put\transclude/numbered{true}
-  % \put\transclude/metadata{true}
-  % \put\transclude/expanded{true}
-  \query{
-      \query/and{
-        \query/tag{draft}
-        \query/tag{tt}
-        % \query/or{
-        %   \query/taxon{definition}
-        %   \query/taxon{theorem}
-        %   \query/taxon{lemma}
-        %   \query/taxon{construction}
-        %   \query/taxon{observation}
-        %   \query/taxon{convention}
-        %   \query/taxon{corollary}
-        %   \query/taxon{axiom}
-        %   \query/taxon{example}
-        %   \query/taxon{exercise}
-        %   \query/taxon{proof}
-        %   \query/taxon{remark}
-        % }
-      }
-  }
-}
+\transclude{tt-0006}

trees/tt-0002.tree

@@ -1,7 +1,6 @@
 \import{tt-macros}
 % clifford hopf spin tt math draft
 \tag{tt}
-\tag{draft}
 
 % definition theorem lemma construction observation
 % convention corollary axiom example exercise proof
@@ -12,17 +11,22 @@
 \p{
 	A \newvocab{category} #{\C} consists of:
 	\ul{
-		\li{\newvocab{objects}: #{X, Y, \dots}, whose collection is denoted #{\Ob(\C)}}
-		\li{\newvocab{arrows} or \newvocab{morphisms}: #{f, g, \dots}, whose collection is denoted #{\Arr(\C)} or #{\Mor(\C)}}
-    \li{a pair of operations: for each arrow #{f},
+		\li{\newvocab{objects}: #{X, Y, \dots \in \Ob(\C)}}
+		\li{\newvocab{arrows} or \newvocab{morphisms}: #{f, g, \dots \in \Arr(\C)} or #{\Mor(\C)}}
+  }
+
+  (where #{\Ob(\C)} and #{\Arr(\C)} are [class](https://en.wikipedia.org/wiki/Class_(set_theory))es, i.e. a collection of sets), and
+
+  \ul{
+
+  \li{a pair of operations: \newvocab{codomain} #{\cod} and \newvocab{domain} #{\dom} for each arrow #{f}}
     \ul{
-      \li{\vocab{codomain} #{\cod : \Arr(\C) \to \Ob(\C), f \mapsto X},}
-      \li{\vocab{domain} #{\dom : \Arr(\C) \to \Ob(\C), f \mapsto Y},}
-    }
-    denoted by #{f : X \to Y} or #{\arr{X}{f}{Y}},
+      \li{ #{\cod : \Arr(\C) \to \Ob(\C), f \mapsto X},}
+      \li{ #{\dom : \Arr(\C) \to \Ob(\C), f \mapsto Y},}
     }
   }
-  such that,
+
+  denoted by #{f : X \to Y} or #{\arr{X}{f}{Y}}, satisfying,
 
   \ol{
     \li{(associativity of composition) there exists a \newvocab{composite arrow} of any pair of arrows #{f} and #{g}, denoted by #{g \circ f}, makes the diagrams 

trees/tt-0003.tree

@@ -1,14 +1,17 @@
 \import{tt-macros}
 % clifford hopf spin tt math draft
 \tag{tt}
-\tag{draft}
-\parent{tt-0002}
+% \parent{tt-0002}
 
 % definition theorem lemma construction observation
 % convention corollary axiom example exercise proof
 % discussion remark notation
 \taxon{notation}
 
-\p{In most literatures (e.g. \cite{chen2016infinitely}), objects in #{\C} are denoted like #{X, Y \in \Ob(\C)}, the set of these arrows are denoted by #{\Hom_\C(X, Y)}, thus an arrow from #{X} to #{Y} is #{f \in \Hom_\C(X, Y)}. }
+\title{Hom-class}
 
-\p{\cite{zhang2021type} simply writes the above as #{X \in \C} and #{f \in \C(X, Y)}, respectively, which is quite friendly, as long as one doesn't use the set theory mindset.}
+\p{In most literatures (e.g. \cite{chen2016infinitely}), for objects #{X, Y \in \Ob(\C)}, #{\Hom_\C(X, Y)} is called the hom-class between #{X} and #{Y} (not [hom-set](https://en.wikipedia.org/wiki/Morphism#Hom-set) as the collection of morphisms is not neccessarily a set), thus an arrow from #{X} to #{Y} is written as #{f \in \Hom_\C(X, Y)}.}
+
+\p{"#{\Hom}" are the first few letters of the word "homomorphism", since a morphism in category theory is a generalization of [homomorphism](https://en.wikipedia.org/wiki/Homomorphism) between algebraic structures.}
+
+\p{\cite{zhang2021type} simply writes the above as #{X \in \C} and #{f \in \C(X, Y)}, respectively, which is quite friendly (at least the latter), as long as one doesn't use the set theory mindset. }

trees/tt-0004.tree

@@ -1,19 +1,20 @@
 \import{tt-macros}
 % clifford hopf spin tt math draft
-\tag{draft}
 \tag{tt}
 
 % definition theorem lemma construction observation
 % convention corollary axiom example exercise proof
 % discussion remark notation
 \taxon{notation}
 
+\title{string diagrams}
+
 \p{
-  Later, when we have learned about functors and natural transformations, we will see that:
+  Later, when we have learned about functors and natural transformations, we will see that, in \vocab{string diagram}\citek{marsden2014category}:
 
   \ol{
     \li{
-      A \vocab{category} #{\C} is represented as a colored region in \vocab{string diagram}\citek{marsden2014category}:
+      A \vocab{category} #{\C} is represented as a colored region:
 
       \tikz{
         \begin{tikzpicture}[scale=0.5]

trees/tt-0005.tree

@@ -0,0 +1,16 @@
+\import{tt-macros}
+% clifford hopf spin tt math draft
+\tag{tt}
+
+% definition theorem lemma construction observation
+% convention corollary axiom example exercise proof
+% discussion remark notation
+% \taxon{}
+
+\title{Categories}
+
+\transclude{tt-0002}
+
+\transclude{tt-0003}
+
+\transclude{tt-0004}

trees/tt-0006.tree

@@ -0,0 +1,62 @@
+\import{tt-macros}
+% clifford hopf spin tt math draft
+\tag{tt}
+
+% definition theorem lemma construction observation
+% convention corollary axiom example exercise proof
+% discussion remark notation
+% \taxon{}
+
+\title{Appendix}
+
+\p{These are my notes on:}
+
+\ul{
+  \li{Category theory}
+  \li{Topos theory}
+  \li{Type theory}
+  \li{Sheaf theory}
+  \li{Differential sheaves}
+  \li{SDG (Synthetic Differential Geometry)}
+}
+
+\p{The primary reference for these notes are:}
+
+\ul{
+  \li{\citek{kostecki2011introduction}} for a clean introduction from category theory to topos theory
+  \li{\citek{kostecki2009differential}} for its introduction to SDG
+  \li{\citek{mallios2015differential}} for its introduction to Differential sheaves
+  \li{\citek{rosiak2022sheaf}} for its examples of sheaves
+  \li{\citek{zhang2021type}} for a friendly introduction to type theory using the language of category theory
+  \li{\citek{chen2016infinitely}} for various preliminaries on category theory
+  \li{\citek{fauser2004grade}} for the use of Kuperberg graphical calculi over commutative diagrams
+}
+
+\p{Scattered notes:}
+
+\scope{
+  % \put\transclude/toc{true}
+  % \put\transclude/numbered{true}
+  % \put\transclude/metadata{true}
+  % \put\transclude/expanded{true}
+  \query{
+      \query/and{
+        \query/tag{draft}
+        \query/tag{tt}
+        % \query/or{
+        %   \query/taxon{definition}
+        %   \query/taxon{theorem}
+        %   \query/taxon{lemma}
+        %   \query/taxon{construction}
+        %   \query/taxon{observation}
+        %   \query/taxon{convention}
+        %   \query/taxon{corollary}
+        %   \query/taxon{axiom}
+        %   \query/taxon{example}
+        %   \query/taxon{exercise}
+        %   \query/taxon{proof}
+        %   \query/taxon{remark}
+        % }
+      }
+  }
+}