resolving notes & some color checking

P1-Polynomials.tex

@@ -5349,17 +5349,17 @@ \section{Dependent dynamical systems}\label{sec.poly.dyn_sys.depend_sys}
 \[
 \begin{tikzpicture}[scale=.5]
   \draw[step=1cm,gray,very thin] (-3,-3) grid (4,4);
-	\draw[->, red ] (-2.5,3.5) -- (-1.5, 3.5);
-	\draw[->, red ] (-2.5,3.5) -- (-1.5, 2.5);
-	\draw[->, red ] (-2.5,3.5) -- (-2.5, 2.5);
-	\draw[->, blue] (1.5, 0.5) -- (0.5, 0.5);
-	\draw[->, blue] (1.5, 0.5) -- (2.5, 0.5);
-	\draw[->, blue] (1.5, 0.5) -- (1.5, 1.5);
-	\draw[->, blue] (1.5, 0.5) -- (1.5,-0.5);
-	\draw[->, blue] (1.5, 0.5) -- (0.5, 1.5);
-	\draw[->, blue] (1.5, 0.5) -- (0.5,-0.5);
-	\draw[->, blue] (1.5, 0.5) -- (2.5, 1.5);
-	\draw[->, blue] (1.5, 0.5) -- (2.5,-0.5);
+	\draw[->] (-2.5,3.5) -- (-1.5, 3.5);
+	\draw[->] (-2.5,3.5) -- (-1.5, 2.5);
+	\draw[->] (-2.5,3.5) -- (-2.5, 2.5);
+	\draw[->] (1.5, 0.5) -- (0.5, 0.5);
+	\draw[->] (1.5, 0.5) -- (2.5, 0.5);
+	\draw[->] (1.5, 0.5) -- (1.5, 1.5);
+	\draw[->] (1.5, 0.5) -- (1.5,-0.5);
+	\draw[->] (1.5, 0.5) -- (0.5, 1.5);
+	\draw[->] (1.5, 0.5) -- (0.5,-0.5);
+	\draw[->] (1.5, 0.5) -- (2.5, 1.5);
+	\draw[->] (1.5, 0.5) -- (2.5,-0.5);
 \end{tikzpicture}
 \]
 In this picture, $\ord{n}\coloneqq\7$, and the $4$ directions at position $(1,7)$ and $9$ directions at position $(5,4)$ are shown (recall that remaining still is an option in either case).
@@ -6199,10 +6199,10 @@ \subsection{Wrapping juxtaposed dynamical systems together}
 \begin{tikzpicture}
 	\node (m1) {\faMotorcycle};
 	\node[above=-.15 of m1] (e1) {\faEye};
-	\node[draw, thick, blue!10, fit = (m1) (e1)] {};
+	\node[draw, thick, fit = (m1) (e1)] {};
 	\node[below right=0 and 1 of m1] (m2) {\scalebox{-1}[1]{\faMotorcycle}};
 	\node[above=-.15 of m2] (e2) {\faEye};
-	\node[draw, thick, blue!10, fit = (m2) (e2)] {};
+	\node[draw, thick, fit = (m2) (e2)] {};
 \end{tikzpicture}
 \]
 The ordered pair comprising the position of each interface indicates the location of the corresponding system, while the range of possible directions indicate the locations that the system could observe, relative to the location of the system itself.
@@ -7063,7 +7063,7 @@ \subsection{More examples of general interaction}
 	\node[bb={1}{1}, fit= (y1) (y2)] (outer) {};
 	\draw[->, shorten >= -4mm] (y1_in1) -- (outer_in1) node[left=4.5mm, font=\tiny] (R){Force};
 	\draw[->, shorten >= -4mm] (y2_out1) -- (outer_out1) node[right=4.5mm, font=\tiny] {Force};
-	\node[starburst, draw, minimum width=2cm, minimum height=1.5cm,red,fill=orange,line width=1.5pt] at ($(L)!.5!(R)$)
+	\node[starburst, draw, minimum width=2cm, minimum height=1.5cm,line width=1.5pt,fill=blue!10] at ($(L)!.5!(R)$)
 {Snap!};
 \end{tikzpicture}
 \]
@@ -7143,7 +7143,7 @@ \subsection{More examples of general interaction}
 	\draw (s2'_out1) to node[above, fill=none, font=\tiny] {$W$} (c'_in1);
 	\draw (s1'_out1) to +(5pt,0) node[fill=none] {$\bullet$};
 \end{scope}
-	\node[starburst, draw, minimum width=2cm, minimum height=2cm,align=center,fill=white, font=\small,line width=1.5pt] at ($(c.east)!.5!(s2'.west)$)
+	\node[starburst, draw, minimum width=2cm, minimum height=2cm,align=center,font=\small,line width=1.5pt] at ($(c.east)!.5!(s2'.west)$)
 {Change\\supplier!};
 \end{tikzpicture}
 \]
@@ -7179,7 +7179,7 @@ \subsection{More examples of general interaction}
 	\draw[->] (A'_out1) -- (B'_in1);
 \end{scope}
 %
-	\node[starburst, draw, minimum width=2cm, minimum height=2cm,fill=blue!50,line width=1.5pt, align=center, font=\upshape] at ($(B)!.5!(A')-(0,.6cm)$)
+	\node[starburst, draw, minimum width=2cm, minimum height=2cm,line width=1.5pt, align=center, font=\upshape] at ($(B)!.5!(A')-(0,.6cm)$)
 {Attach!};
 \end{tikzpicture}
 \end{equation*}
@@ -8021,7 +8021,7 @@ \section{Epi-mono factorization of lenses}
 Hint: You may use the following facts.
 \begin{enumerate}
     \item A function that is both a monomorphism and an epimorphism in $\smset$ is an isomorphism.
-    \item A lens is an isomorphism if and only if the on-positions function is an isomorphism and every on-directions function is an isomorphism.
+    \item A lens is an isomorphism if and only if the on-positions function is an isomorphism and every on-directions function is an isomorphism. \qedhere
 \end{enumerate}
 \begin{solution}
 Let $f \colon p \to q$ be a lens in $\poly$ that is both a monomorphism and an epimorphism.

P2-Comonoids.tex

@@ -26,7 +26,8 @@ \chapter{The composition product}\label{ch.comon.comp}
 \index{polynomial functor, substitution of polynomials|see{polynomial functor!composition of polynomials}}
 \index{polynomial functor!composition of polynomials}
 
-It turns out that this operation, which we'll see soon is a monoidal product, has a lot to do with time.
+It turns out that this operation, which we will soon see is a monoidal product, has a lot to do with time.
+\index{time!composition product and}
 There is a strong sense---made precise in \cref{prop.poly_closed_comp}---in which the polynomial $p\circ q$ represents ``starting at a position $i$ in $p$, choosing a direction in $p[i]$, landing at a position $j$ in $q$, choosing a direction in $q[j]$, and then landing... somewhere.''
 This is exactly what we need to run through multiple steps of a dynamical system, the very thing we didn't know how to do in \cref{ex.do_nothing}.
 We'll continue that story in \cref{subsec.comon.comp.def.dyn_sys}.
@@ -2062,10 +2063,9 @@ \subsection{Dynamical systems and the composition product} \label{subsec.comon.c
 
 \index{time!composition product and}
 
-This is what we meant in the introduction when we said that the composition product has to do with time.\dnote{Where did we say this? I didn't find it. Please add $\backslash$ index$\{$time!composition product and$\}$ wherever it is.} %copy this: \index{time!composition product and}
 \index{dynamical system!trees}\index{interface}
 
-This is what we meant in the introduction when we said that the substitution product has to do with time.
+This is what we meant at the start of this chapter when we said that the substitution product has to do with time.
 It takes a specification $\varphi$ for how a state system and an interface can interact back-and-forth---or, indeed, any interaction pattern between wrapper interfaces---and extends it to a multistep model $\varphi\tripow{n}$ that simulates $n$ successive interaction cycles over time, accounting for all possible external directions that the interface could encounter.
 Alternatively, we can think of $\varphi\tripow{n}$ as ``speeding up'' the original dynamical system $\varphi$ by a factor of $n$, as it runs $n$ steps in one---as long as whatever's connected to its new interface $p\tripow{n}$ can keep up with its pace and feed it $n$ directions of $p$ at a time!
 The lens $\varphi$ tells us how the machine can run, but it is $\tri$ that makes the clock tick.
@@ -3372,8 +3372,8 @@ \subsection{Interaction with limits on the left} \label{subsec.comon.comp.prop.l
 But this is equivalent to the following gadget (to visualize this equivalence, imagine leaving the positions box for $p$, the arrow $\varphi^r$, and the polyboxes for $r$ untouched, while dragging the polyboxes for $q$ leftward to the directions box for $p$, merging all the data from $q$ and the arrows $\varphi^q$ and $\varphi^\sharp$ into a single on-directions arrow and directions box):
 \[
 \begin{tikzpicture}[polybox, mapstos]
-    \node[poly, dom, "$\lchom{q}{p}$" left] (l) {$(j,\varphi^\sharp)$\at$i$};
-    \node[poly, cod, "$r$" right, right=of l, yshift=-0.5ex] (r) {$c$\at$k$};
+    \node[poly, dom, "$\lchom{q}{p}$" left] (l) {$(j,\varphi^\sharp)$\at$i\vphantom{k}$};
+    \node[poly, cod, "$r$" right, right=of l, yshift=-0.5ex] (r) {$c\vphantom{(j,\varphi^\sharp)}$\at$k$};
     \draw (l_pos) -- node[below] {$\varphi^r$} (r_pos);
     \draw (r_dir) -- node[above] {} (l_dir);
 \end{tikzpicture}
@@ -6780,10 +6780,6 @@ \subsubsection{Retrofunctors to monoids}
 \end{solution}
 \end{exercise}
 
-
-
-\dnote{What is this ``displaced content"?}
-%** displaced content here
 \begin{exercise}[Monoid actions]\label{exc.monoid_action}
 Recall from \cref{ex.monoid_action} that every monoid action $\alpha\colon S\times M\to S$, where $S$ is a set and $(M,e,*)$ is a monoid, gives rise to a category carried by $S\yon^M$.
 Show that the projection $S\yon^M\to\yon^M$ is a retrofunctor.