Polish "limit via representation"

tex/diagrams.tex

@@ -9,6 +9,7 @@
 \usetikzlibrary{decorations.pathreplacing}
 \usetikzlibrary{decorations.pathmorphing}
 \usetikzlibrary{decorations.markings}
+\usetikzlibrary{nfold}
 
 \tikzset{
   desc/.style={sloped, fill=white,inner sep=2pt},

trees/tt-000X.tree

@@ -45,7 +45,7 @@
 \li{coequalizer:
 \tikz{
 \begin{tikzcd}
-    & \bullet \\
+    & \anyobj \\
     \\
     & E \\
     \bullet && \bullet
@@ -62,7 +62,7 @@
 \li{pushout (fiber coproduct):
 \tikz{
 \begin{tikzcd}
-    \bullet \\
+    \anyobj \\
     \\
     && {X +_{O} Y} && Y \\
     \\

trees/tt-0013.tree

@@ -13,6 +13,8 @@
 
 \transclude{tt-0015}
 
+\transclude{tt-0041}
+
 \transclude{tt-001D}
 
 % \transclude{tt-0016}

trees/tt-001O.tree

@@ -12,6 +12,8 @@
 A \vocab{set-valued} functor #{\fF:  \C \to \Set} is called \newvocab{covariantly representable} if for some #{X \in \C},
 
 ##{\tau: \fF \iso \C(X,-)}
+
+where #{\iso} denotes a \vocabk{natural isomorphism}{tt-001H}.
 }
 
 \p{Conversely, a set-valued functor #{\fG : \C^{op} \to \Set} is called \newvocab{contravariantly representable} if for some #{X \in \C},

trees/tt-003Y.tree

@@ -12,13 +12,47 @@
 
 \taxon{lemma}\refnotet{limit via representation}{6.1.1}{leinster2016basic}{
 \p{
-Let #{\J} be a small category, #{\C} a category, and #{\fD: \J \to \C} a diagram. Then there is a one-to-one correspondence between limit cones on #{\fD} and representations of the \vocabk{cone functor}{tt-0040}
+Let #{\J} be a small category, #{\C} a category, and #{\fD: \J \to \C} a diagram. Then there is a one-to-one correspondence between
 
-##{\Cone(-, \fD): \C^{op} \to \Set}
+\ul{
+  \li{limit cones on #{\fD}}
+  \li{representations of the natural transformation \vocabk{Cone}{tt-0040}}
+}
+
+with the representing objects being the limit objects (i.e. the vertices) of #{\fD}.
 
-with the representing objects of #{\Cone(-, \fD)} being the limit objects (that is, the vertices of the limit cones) of #{\fD}.
 }
 
-\p{Briefly put: a \vocabk{limit}{tt-002A} of #{\fD} is a \vocabk{representation}{tt-001O} of #{[\J, \C] (\Delta_{-}, \fD)}.
+\p{Briefly put: a \vocabk{limit}{tt-002A} #{(V, \pi)} of #{\fD} is a \vocabk{representation}{tt-001O} of #{[\J, \C] (\Delta_{-}, \fD)}.
+}
+
+\p{Schematically,
+\tikz{
+\begin{tikzcd}
+	&&& \anyobj \\
+	\\
+	&&&& V \\
+	J &&&&& \bullet \\
+	\\
+	{J'} &&&&& \bullet \\
+	\J &&& \C
+	\arrow[dashed, from=1-4, to=3-5]
+	\arrow["{\pi_J}", from=3-5, to=4-6]
+	\arrow["{\pi_{J'}}"'{pos=0.2}, shorten >=6pt, from=3-5, to=6-6]
+	\arrow[""{name=0, anchor=center, inner sep=0}, from=4-1, to=6-1]
+	\arrow[""{name=1, anchor=center, inner sep=0}, from=4-6, to=6-6]
+	\arrow[""{name=2, anchor=center, inner sep=0}, "{\Delta_{\anyobj}}"{pos=0.8}, shorten <=12pt, Rightarrow, maps to, from=0, to=1-4]
+	\arrow["{\Delta_V}"{pos=0.8}, shorten <=13pt, Rightarrow, maps to, from=0, to=3-5]
+	\arrow[""{name=3, anchor=center, inner sep=0}, "\fD"', shorten <=17pt, Rightarrow, maps to, from=0, to=1]
+	\arrow["{[\J, \C] (\Delta_{-}, \fD)}"{pos=0.8}, shorten <=5pt, shorten >=5pt, Rightarrow, scaling nfold=3, maps to, from=2, to=3]
+\end{tikzcd}
+}
+}
+
+\p{It implies that
+
+##{\Cone(\anyobj, \fD) \iso \C\left(\anyobj, \lim\limits_{\longleftarrow \J} \fD\right)}
+
+for any #{\anyobj \in \C}.
 }
 }

trees/tt-0040.tree

@@ -5,7 +5,7 @@
 % definition theorem lemma construction observation
 % convention corollary axiom example exercise proof
 % discussion remark notation
-\refdeft{cone functor}{eq. 6.1}{leinster2016basic}{
+\refdeft{cone as a natural transformation}{eq. 6.1}{leinster2016basic}{
 \p{
 Now, given a diagram #{\fD: \J \rightarrow \C} and an object #{V \in \C}, a \vocabk{cone}{tt-0028} on #{\fD} with vertex #{V} is simply a natural transformation from the \vocabk{diagonal functor}{tt-003T} #{\Delta_V} to the diagram #{\fD}.
 
@@ -25,7 +25,7 @@ Now, given a diagram #{\fD: \J \rightarrow \C} and an object #{V \in \C}, a \voc
 \Cone(V, \fD)=[\J, \C] (\Delta_V, \fD) .
 }}
 
-\p{Thus, #{\Cone(V, \fD)} is functorial in #{V} (contravariantly) and #{\fD} (covariantly).
+\p{Thus, #{\Cone(V, \fD)} is \vocabk{functorial in}{tt-0041} #{V} (contravariantly) and #{\fD} (covariantly).
 }
 
 }

trees/tt-0041.tree

@@ -0,0 +1,29 @@
+\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{functorial in}{sec. 4.1}{leinster2016basic}{
+\p{
+For some expression #{E(X)} containing #{X}, when we say #{E(X)} is (covariant) \newvocab{functorial in} #{X}, we mean that there exists a functor #{\fF} such that
+
+\tikz{
+\begin{tikzcd}
+	X &&& {E(X)} \\
+	\\
+	{X'} &&& {E(X')}
+	\arrow[""{name=0, anchor=center, inner sep=0}, "f"', from=1-1, to=3-1]
+	\arrow[""{name=1, anchor=center, inner sep=0}, from=1-4, to=3-4]
+	\arrow["\fF", shorten <=20pt, shorten >=20pt, Rightarrow, maps to, from=0, to=1]
+\end{tikzcd}
+}
+
+for every #{f : X \to X'}.
+}
+
+\p{Dually, we use the term \newvocab{contravariantly functorial in}.
+}
+
+}

trees/tt-macros.tree

@@ -55,7 +55,7 @@
 \def\initobj{\mathrm{0}}
 
 \def\uniqobj{\mathrm{*}}
-\def\anyobj{\mathrm{Z}}
+\def\anyobj{\mathrm{-}}
 
 \def\obj{\operatorname{obj}}
 \def\Hom{\operatorname{Hom}}