Divide special functors

Author: utensil

trees/tt-0016.tree

@@ -5,38 +5,7 @@
 % definition theorem lemma construction observation
 % convention corollary axiom example exercise proof
 % discussion remark notation
-\taxon{example}\refnotet{special functors}{eq. 54}{kostecki2011introduction}{
-
-\p{
-The \newvocab{identity functor} #{\id_{\C}: \C \to \C} (denoted also by #{ 1_{\C}: \C \to \C}), defined by #{\id_{\C}(X)=X} and #{\id_{\C}(f)=f} for every #{X \in \Ob(\C)} and every #{f \in \Arr(\C)}.
-}
-
-\p{
-The \newvocab{constant functor} #{\Delta_O: \C \to \D} which assigns a fixed #{O \in \mathrm{Ob}(\D)} to any object of #{\C} and #{\id_O}, the identity arrow on #{O}, to any arrows from #{\C} :
-
-\tikz{
-\begin{tikzcd}
-	X &&&& O \\
-	& {} && {} \\
-	Y &&&& O \\
-	\C &&&& \D
-	\arrow["f"', from=1-1, to=3-1]
-	\arrow["{\id_O}", from=1-5, to=3-5]
-	\arrow["{\Delta_O}", Rightarrow, maps to, from=2-2, to=2-4]
-	\arrow["{\Delta_O}", Rightarrow, from=4-1, to=4-5]
-\end{tikzcd}
+\taxon{example}\refnotet{other special functors}{3.1, example 10, 11, 4.6}{kostecki2011introduction}{
+\p{Some other special functors are introduced in later sections in context, e.g. \vocabk{hom-functor}{tt-001S}, \vocabk{Yoneda embedding functors}{tt-002T}, \vocabk{object functor}{tt-003H}.
 }
-
-with compositions and identities preserved in a trivial way.}
-
-\p{The \newvocab{forgetful functor}, which \em{forgets} some part of structure, however arrows, compositions and identities are preserved.
-}
-
-\p{
-Let #{\C} be a subcategory of #{\D}. The \newvocab{inclusion functor}, denoted #{\hookrightarrow}, sends objects and arrows of #{\D} into themselves in category #{\D}.}
-
-\p{
-The \newvocab{diagonal functor} #{\Delta: \C \to \C \times \C, \Delta(X)=(X, X)} and #{\Delta(f)=(f, f)} for #{f: X \to X^{\prime}}.
-}
-
 }

trees/tt-0027.tree

@@ -13,7 +13,7 @@
 Let #{\C^{\J}} be a [functor category](tt-001F), where #{\J} is a [small category](tt-000A). 
 }
 
-\p{Let #{\Delta_O} be a [constant functor](tt-0016), which assigns the same object #{O} in #{\C} to any object #{J} in #{\J}.
+\p{Let #{\Delta_O} be a [constant functor](tt-003Q), which assigns the same object #{O} in #{\C} to any object #{J} in #{\J}.
 }
 
 \p{Let #{K \in \Ob(\J)} and let #{j \in \operatorname{Arr}(\J)} such that #{j: J \to K}. Let #{\fF} be any functor in #{\C^{\J}}, i.e. it's a [diagram](tt-0025) in #{\C} of shape #{\J}.

trees/tt-0028.tree

@@ -34,7 +34,7 @@ of arrows in #{\C} such that for all arrows #{J \to J'} in #{\J}, the triangle
 commutes.
 }
 
-\p{The family of arrows are components of a \vocabk{natural transformation}{tt-001E} #{\pi: \Delta_V \to \fD}, i.e. from the \vocabk{constant functor}{tt-0016} ( which assigns the same object #{V} to any object #{J_i} in #{\J}) to diagram functor #{\fD}.
+\p{The family of arrows are components of a \vocabk{natural transformation}{tt-001E} #{\pi: \Delta_V \to \fD}, i.e. from the \vocabk{constant functor}{tt-003Q} ( which assigns the same object #{V} to any object #{J_i} in #{\J}) to diagram functor #{\fD}.
 }
 
 \p{For simplicity, we refer to a cone by "a cone #{(V, \pi)} on #{\fD}".

trees/tt-003N.tree

@@ -12,4 +12,14 @@
 
 \title{special functors}
 
+\transclude{tt-003P}
+
+\transclude{tt-003Q}
+
+\transclude{tt-003R}
+
+\transclude{tt-003S}
+
+\transclude{tt-003T}
+
 \transclude{tt-0016}

trees/tt-003P.tree

@@ -0,0 +1,12 @@
+\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
+\refdeft{identity functor}{3.1, example 1}{kostecki2011introduction}{
+\p{
+The \newvocab{identity functor} #{\id_{\C}: \C \to \C} (denoted also by #{ 1_{\C}: \C \to \C}), defined by #{\id_{\C}(X)=X} and #{\id_{\C}(f)=f} for every #{X \in \Ob(\C)} and every #{f \in \Arr(\C)}.
+}
+}

trees/tt-003Q.tree

@@ -0,0 +1,27 @@
+\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
+\refdeft{constant functor}{3.1, example 2}{kostecki2011introduction}{
+\p{
+The \newvocab{constant functor} #{\Delta_O: \C \to \D} which assigns a fixed #{O \in \mathrm{Ob}(\D)} to any object of #{\C} and #{\id_O}, the identity arrow on #{O}, to any arrows from #{\C} :
+
+\tikz{
+\begin{tikzcd}
+	X &&&& O \\
+	& {} && {} \\
+	Y &&&& O \\
+	\C &&&& \D
+	\arrow["f"', from=1-1, to=3-1]
+	\arrow["{\id_O}", from=1-5, to=3-5]
+	\arrow["{\Delta_O}", Rightarrow, maps to, from=2-2, to=2-4]
+	\arrow["{\Delta_O}", Rightarrow, from=4-1, to=4-5]
+\end{tikzcd}
+}
+
+with compositions and identities preserved in a trivial way.}
+
+}

trees/tt-003R.tree

@@ -0,0 +1,11 @@
+\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
+\refdeft{forgetful functor}{3.1, example 3}{kostecki2011introduction}{
+\p{The \newvocab{forgetful functor}, which \em{forgets} some part of structure, however arrows, compositions and identities are preserved.
+}
+}

trees/tt-003S.tree

@@ -0,0 +1,11 @@
+\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
+\refdeft{inclusion functor}{3.1, example 4}{kostecki2011introduction}{
+\p{
+Let #{\C} be a \vocabk{subcategory}{tt-002Y} of #{\D}. The \newvocab{inclusion functor}, denoted #{\hookrightarrow}, sends objects and arrows of #{\D} into themselves in category #{\D}.}
+}

trees/tt-003T.tree

@@ -0,0 +1,12 @@
+\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
+\refdeft{diagonal functor}{3.1, example 6}{kostecki2011introduction}{
+\p{
+The \newvocab{diagonal functor} #{\Delta: \C \to \C \times \C, \Delta(X)=(X, X)} and #{\Delta(f)=(f, f)} for #{f: X \to X^{\prime}}.
+}
+}