resolve blanks

P1-Polynomials.tex

@@ -4270,7 +4270,7 @@ \section{Summary and further reading}
 In this chapter, we introduced the category $\poly$, whose objects are polynomial functors and whose morphisms are the natural transformations between them.
 We call these natural transformations \emph{dependent lenses}, or \emph{lenses} for short.
 We also proved our first categorical property of $\poly$: that it hs all small coproducts.
-In the next chapter, we will review the coproduct construction before turning to other operations on our polynomials.
+%In the next chapter, we will review the coproduct construction before turning to other operations on our polynomials.
 
 The main result of this chapter was a concrete characterization of our dependent lenses between polynomial functors.
 A dependent lens $f\colon p\to q$ is characterized by its
@@ -4283,10 +4283,10 @@ \section{Summary and further reading}
 This perspective is exhibited by our corolla and polybox pictures for lenses.
 We studied examples of lenses between special polynomials: in particular, lenses between monomials are known as \emph{bimorphic lenses} in functional programming literature.
 
-Finally, we unwound our interpretation of natural transformations between polynomials as dependent lenses with on-positions and on-directions functions to describe what happens to these functions when lenses compose.
+We then unwound our interpretation of natural transformations between polynomials as dependent lenses with on-positions and on-directions functions to describe what happens to these functions when lenses compose.
 This gave us an accessible way to interpret commutative diagrams in $\poly$ that is particularly convenient to express using polyboxes.
 
-Once $\poly$ was defined, we considered various properties it has, e.g.\ that it has all products and coproducts, and that these distribute: $\prod\sum\to\sum\prod$.
+Finally, we considered various categorical properties of $\poly$, e.g.\ that it has all products and coproducts, and that these distribute: $\prod\sum\to\sum\prod$.
 \begin{align*}
 	\sum_{a\in A}p_a&\coloneqq\sum_{(a,i)\in\sum_{a\in A}p_a(1)}\yon^{p_a[i]}&
 	\prod_{a\in A}p_a&\coloneqq\sum_{i\in\prod_{a\in A}p_a(1)}\yon^{\sum_{a\in A}p_a[i a]}
@@ -4799,8 +4799,8 @@ \section{Dependent dynamical systems}\label{sec.poly.dyn_sys.depend_sys}
   \begin{equation*}
     \begin{tikzpicture}[polybox, mapstos]
       \node[poly, dom] (S) {$t$\at$s\vphantom{i}$};
-        \node[left=0pt of s_pos] {$S$};
-        \node[left=0pt of s_dir] {$S$};
+        \node[left=0pt of S_pos] {$S$};
+        \node[left=0pt of S_dir] {$S$};
 
       \node[poly, cod, right=of S, "$p$" right] (p) {$a\vphantom{t}$\at$i$};
 
@@ -6561,20 +6561,20 @@ \subsection{More examples of general interaction}
 It could be dictated by a given vertex $v_0\in V$ in the sense that its state completely determines the neighbor function $V\to\2^V$; this would be expressed by saying that $N'_-$ factors as $\2^V\to\2^{\{v_0\}}\cong\2\To{I_0}(\2^V)^V$ for some $I_0$.
 \end{example}
 
-\begin{exercise}
-We can change \cref{ex.cell_auto_vote_interaction} slightly by replacing the wrapper interface $\yon$ with some other interface.
-\begin{enumerate}
-	\item First change it to $A\yon$ for some set $A$ of your choice, and update \eqref{eqn.polymap_misc9237} so that the system outputs some aspect of the current state configuration of all the vertices $S\in\2^V$.
-	\item What would it mean to change \eqref{eqn.polymap_misc9237} to a map $\bigotimes_{v\in V}p_v\to\yon^A$ for some $A$?
-\qedhere
-\end{enumerate}
-\begin{solution}
-\begin{enumerate}
-    \item **
-    \item **
-\end{enumerate}
-\end{solution}
-\end{exercise}
+%\begin{exercise}
+%We can change \cref{ex.cell_auto_vote_interaction} slightly by replacing the wrapper interface $\yon$ with some other interface.
+%\begin{enumerate}
+%	\item First change it to $A\yon$ for some set $A$ of your choice, and update \eqref{eqn.polymap_misc9237} so that the system outputs some aspect of the current state configuration of all the vertices $S\in\2^V$.
+%	\item What would it mean to change \eqref{eqn.polymap_misc9237} to a map $\bigotimes_{v\in V}p_v\to\yon^A$ for some $A$?
+%\qedhere
+%\end{enumerate}
+%\begin{solution}
+%\begin{enumerate}
+%    \item **
+%    \item **
+%\end{enumerate}
+%\end{solution}
+%\end{exercise}
 
 Here are some more examples of using dependent dynamical systems to model changing wiring diagrams.
 

P2-Comonoids.tex

@@ -7233,9 +7233,9 @@ \subsubsection{As discrete opfibrations}
 \begin{proposition}
 The category of $\cat{C}$-coalgebras is isomorphic to the category $\Cat{dopf}(\cat{C})$ of discrete opfibrations over $\cat{C}$.
 \end{proposition}
-\begin{proof}
-**
-\end{proof}
+%\begin{proof}
+%**
+%\end{proof}
 
 
 \subsubsection{As copresheaves}
@@ -7383,19 +7383,19 @@ \subsubsection{As copresheaves}
 Given a dynamical system $S\yon^S\to p$, we extend it to a cofunctor $\varphi\colon S\yon^S\cof\cofree{p}$. By \cref{prop.tfae_dopf,prop.ds_dopf}, we can consider it as a discrete opfibration over $\cofree{p}$. By \cref{exc.elts_free_grph} the category $\elts \varphi$ is again free on a graph. It is this graph that we usually draw when depicting the dynamical system, e.g.\ in \eqref{eqn.dyn_sys_misc573}.
 
 
-\begin{exercise}
-Give an example of a dynamical system on $p\coloneqq\yon^\2+\yon$ for which the corresponding copresheaf has in its image a set with at least two elements.
-\begin{solution}
-**
-\end{solution}
-\end{exercise}
+%\begin{exercise}
+%Give an example of a dynamical system on $p\coloneqq\yon^\2+\yon$ for which the corresponding copresheaf has in its image a set with at least two elements.
+%\begin{solution}
+%**
+%\end{solution}
+%\end{exercise}
 
-\begin{exercise}
-Given a cofunctor $F\colon S\yon^S\cof\yon$, what does the corresponding copresheaf look like?
-\begin{solution}
-**
-\end{solution}
-\end{exercise}
+%\begin{exercise}
+%Given a cofunctor $F\colon S\yon^S\cof\yon$, what does the corresponding copresheaf look like?
+%\begin{solution}
+%**
+%\end{solution}
+%\end{exercise}
 
 To summarize, we have four equivalent notions:
 \begin{enumerate}[label=(\arabic*)]