Fix typos

Author: amuricys

book/C0-Preface.tex

@@ -13,14 +13,14 @@ \chapter{Preface}\label{chapter.0}
 
 Categorical systems theory is an emerging field of mathematics which seeks to apply the methods of category theory to general systems theory. General systems theory is the study of systems --- ways things can be and change, and models thereof --- in full generality. The difficulty is that there doesn't seem to be a single core idea of what it means to be a ``system''. Different people have, for different purposes, come up with a vast array of different modeling techniques and definitions that could be called ``systems''. There is often little the same in the precise content of these definitions, though there are still strong, if informal, analogies to be made accross these different fields. This makes coming up with a mathematical theory of general systems tantalizing but difficult: what, after all, is a system in general?
 
-Category theory has been describe as the mathematics of formal analogy making. It allows us to make analogies between fields by focusing not on content of the objects of those fields, but by the ways that the objects of those fields relate to one another. Categorical systems theory applies this idea to general systems theory, avoiding the issue of not having a contentful definition of system by instead focusing on the ways that systems interact with eachother and their environment.
+Category theory has been described as the mathematics of formal analogy making. It allows us to make analogies between fields by focusing not on content of the objects of those fields, but by the ways that the objects of those fields relate to one another. Categorical systems theory applies this idea to general systems theory, avoiding the issue of not having a contentful definition of system by instead focusing on the ways that systems interact with each other and their environment.
 
 These are the main ideas of categorical systems theory:
 \begin{enumerate}
 \item Any system interacts with its environment through an \emph{interface}, which can be described separately from the system itself.
 \item All interactions of a system with its environment take place through its interface, so that from the point of view of the environment, all we need to know about a system is what is going on at the interface.
 \item Systems interact with other systems through their respective interfaces. So, to understand complex systems in terms of their component subsystems, we need to understand the ways that interfaces can be connected. We call these ways that interfaces can be connected \emph{composition patterns}.
-\item Given a composition pattern describing how some interface are to be connected, and some systems with those interfaces, we should have a \emph{composite} system which consists of those subsystems interacting according to the composition pattern. The ability to form composite systems of interacting component systems is called \emph{modularity}, and is a well known boon in the design of complex systems.
+\item Given a composition pattern describing how some interfaces are to be connected, and some systems with those interfaces, we should have a \emph{composite} system which consists of those subsystems interacting according to the composition pattern. The ability to form composite systems of interacting component systems is called \emph{modularity}, and is a well known boon in the design of complex systems.
 \end{enumerate}
 
 In a sense, the definitions of categorical systems theory are all about \emph{modularity} --- how can systems be composed of subsystems. On the other hand, the theorems of categorical systems theory often take the form of \emph{compositionality} results. These say that certain facts and features of composite systems can be understood or calculated in terms of their component systems and the composition pattern.
@@ -35,9 +35,9 @@ \chapter{Preface}\label{chapter.0}
    \item What does it mean to be a system? Does it have a notion of states, or of behaviors? Or is it a diagram describing the way some primitive parts are organized?
    \item What should the interface of a system be?
    \item How can interfaces be connected in composition patterns?
-  \item How are systems composed through composition patterns between their interfaces.
+  \item How are systems composed through composition patterns between their interfaces?
   \item What is a map between systems, and how does it affect their interfaces?
-  \item When can maps between systems be composed along the same composition patterns as the systems.
+  \item When can maps between systems be composed along the same composition patterns as the systems?
   \end{itemize}
   \end{informal}
 
@@ -91,12 +91,12 @@ \chapter{Preface}\label{chapter.0}
   \item Can the kinds of input a
     system takes in depend on what it's putting out, and how do they depend on it?
   \item What sorts of changes are possible in a given state?
-  \item What does it mean for states to change.
+  \item What does it mean for states to change?
   \item How should the way the state changes vary with the input?
   \end{itemize}
 \end{informal}
 
-We will make this definition fully formal in \cref{Chapter.3}, after introducing enough category theory to state it. Once we have made the definition of systems theory formal, we can make the definition of system. But what is interesting about dynamical systems is how they \emph{behave}.
+We will make this definition fully formal in \cref{Chapter.3}, after introducing enough category theory to state it. Once we have made the definition of systems theory formal, we can make the definition of system formal. But what is interesting about dynamical systems is how they \emph{behave}.
 \begin{informal}\label{inf:behavior}
   A \emph{behavior} of a dynamical system is a particular way its states can
   change according to its dynamics.
@@ -123,7 +123,7 @@ \chapter{Preface}\label{chapter.0}
     it with the first one and then with the second one.
   \item Suppose that we have a pair of
     wiring patterns and compatible charts between them. If we
-    take a bunch of behaviors whose charts are compatable according to the first
+    take a bunch of behaviors whose charts are compatible according to the first
     wiring pattern, wire them together, and then compose with a behavior of the
     second chart, we get the same thing as if we compose them all with behaviors
     of the first chart, noted that they were compatible with the second wiring

book/C1-.tex

@@ -86,7 +86,7 @@ \section{Introduction}\label{sec.chap1_intro}
   \item Can the kinds of input a
     system takes in depend on what it's putting out, and how do they depend on it?
   \item What sorts of changes are possible in a given state?
-  \item What does it mean for states to change. 
+  \item What does it mean for states to change? 
   \item How should the way the state changes vary with the input?
   \end{itemize}
 \end{informal}
@@ -113,7 +113,7 @@ \section{Introduction}\label{sec.chap1_intro}
 
 The dynamical systems we will see in this book are \emph{open} in the sense that
 they take in inputs from their environment and expose outputs back to their
-environment. Because of this, our systems can interact with eachother. One
+environment. Because of this, our systems can interact with each other. One
 system can take what the other system outputs as part of its input, and the
 other can take what the first outputs as part of its input. For example, when we
 have a conversation, I take what I hear from you and use it to change how I
@@ -387,7 +387,7 @@ \subsection{Deterministic systems}\label{sec.deterministic_system}
   \draw[label] node [right=2pt of Outer_out1] {$\Set{DinerState}$};
 \end{tikzpicture}
 \end{equation}
-If we want to, we can peak into the clock with display and see that it is itself
+If we want to, we can peek into the clock with display and see that it is itself
 made out of a clock wired to a display:
 \begin{equation}\label{eqn.diner_system_box}
 \begin{tikzpicture}[oriented WD, every fit/.style={inner xsep=\bbx, inner ysep=\bby}, bbx = 1cm, bby =.3cm, bb min width=1cm, bb port length=4pt, bb port sep=1, baseline=(Outer.center)]
@@ -553,7 +553,7 @@ \subsection{Deterministic systems}\label{sec.deterministic_system}
 \end{example}
 
 \begin{example}\label{ex.moore_machine}
-  Not only is the term \emph{Moore machine} is used for the mathematical notion of
+  Not only is the term \emph{Moore machine} used for the mathematical notion of
   deterministic system we've just presented, but it is also used for actual,
   real-life circuits which are designed on that principle.
 
@@ -1471,7 +1471,7 @@ \subsection{Deterministic and differential systems as lenses}
 resulting node. When we wire together systems presented as transition diagrams,
 the dynamics then involve reading the input labels of all inner systems, moving
 along all the arrows with those labels, and then outputing the labels at each
-state, possible into the input of another system.
+state, possibly into the input of another system.
 
 \begin{exercise}\label{ex.wiring_transition_diagrams}
 Here are two systems, $\Sys{S_1}$ and $\Sys{S_2}$ presented in terms of
@@ -1986,7 +1986,7 @@ \subsection{Wiring diagrams as lenses in categories of arities}\label{sec:wiring
  Wiring diagrams are designed to
 express the flow of variables through the system; how they are to be copied from
 one port to another, how they are to be shuffled about, and (though we haven't
-had need for this yet) how they are to be deleted or forgotton.
+had need for this yet) how they are to be deleted or forgotten.
 
 
 
@@ -1999,7 +1999,7 @@ \subsection{Wiring diagrams as lenses in categories of arities}\label{sec:wiring
 
 \begin{definition}\label{defn:category.of.arities}
   The category $\arity$ of arities is the free cartesian category generated by
-  a single object $\XX$. That is, $\arity$ constains an object $\XX$, called
+  a single object $\XX$. That is, $\arity$ contains an object $\XX$, called
   the \emph{generic object}, and for any finite set $I$, there is an $I$-fold
   power $\XX^I$ of $\XX$. The only maps are those that can be defined from the
   product structure by pairing and projection.
@@ -2112,7 +2112,7 @@ \subsection{Wiring diagrams as lenses in categories of arities}\label{sec:wiring
       \XX^{\ord{2}}$ given by $\fun{swap} : \ord{2} \to \ord{2}$ defined by $0
       \mapsto 1$ and $1 \mapsto 0$.
     \item What map corresponds to the map $1 : \ord{1} \to \ord{2}$ picking out
-      $1 \in \ord{2} = \{1, 2\}$? What about $2 : \ord{1} \to \ord{2}$. 
+      $1 \in \ord{2} = \{1, 2\}$? What about $2 : \ord{1} \to \ord{2}$? 
     \item Convince yourself that \emph{any} map $\XX^I \to \XX^J$ you can express with the
       universal property of products can be expressed by choosing an appropriate
       $f : J \to I$.
@@ -2506,7 +2506,7 @@ \subsection{Wiring diagrams as lenses in categories of arities}\label{sec:wiring
 
 \begin{exercise}
   We blew through that isomorphism $\arity_{\cat{T}} \cong (\Cat{Fin} \downarrow
-  \cat{T})\op$ quickly, but its not entirely trivial. The category $\Cat{Fin} \downarrow
+  \cat{T})\op$ quickly, but it's not entirely trivial. The category $\Cat{Fin} \downarrow
   \cat{T}$ has objects functions $\tau : I \to \cat{T}$ where $I$ is a finite
   set, and a morphism is a commuting triangle like this:
   \[

book/C3-.tex

@@ -433,7 +433,7 @@ \subsubsection{Steady states in the deterministic systems theory}
 While the steady states of a wired together system can be calculated from those
 of its components, this is not true for steady-looking trajectories.
 Intuitively, this is because the internal systems can be exposing changing
-outputs between eachother even while the eventual external output remains
+outputs between each other even while the eventual external output remains
 unchanged.
 \end{remark}
 
@@ -1743,7 +1743,7 @@ \subsection{The double category of arenas in the deterministic systems theory}
   \end{equation}
 \end{proposition}
 \begin{proof}
-  This is a simple matter of checking the definitions against eachother. The
+  This is a simple matter of checking the definitions against each other. The
   defining equations of \cref{def.double_category_of_arenas_discrete} specialize
   to the defining equations of \cref{def.behavior_discrete}.
 \end{proof}
@@ -2628,7 +2628,7 @@ \section{Theories of Dynamical Systems}\label{sec.doctrines}
   \item Can the kinds of input a
     system takes in depend on what it's putting out, and how do they depend on it?
   \item What sorts of changes are possible in a given state?
-  \item What does it mean for states to change. 
+  \item What does it mean for states to change? 
   \item How should the way the state changes vary with the input?
   \end{enumerate}
 \end{informal}

book/C4-.tex

@@ -32,7 +32,7 @@ \section{Introduction}
   \item Can the kinds of input a
     system takes in depend on what it's putting out, and how do they depend on it?
   \item What sorts of changes are possible in a given state?
-  \item What does it mean for states to change. 
+  \item What does it mean for states to change? 
   \item How should the way the state changes vary with the input?
   \end{enumerate}
 \end{informal}
@@ -1970,7 +1970,7 @@ \subsection{Definition}
   \item Can the kinds of input a
     system takes in depend on what it's putting out, and how do they depend on it?
   \item What sorts of changes are possible in a given state?
-  \item What does it mean for states to change. 
+  \item What does it mean for states to change? 
   \item How should the way the state changes vary with the input?
   \end{enumerate}
 \end{informal}

book/C5-.tex

@@ -43,7 +43,7 @@ \section{Introduction}
 compositionality theorem concerning any sort of behavior in any systems theory.
 
 But the behaviors of the component systems must be compatible with
-eachother: if a system $\Sys{S_1}$ has its parameters set by the exposed
+each other: if a system $\Sys{S_1}$ has its parameters set by the exposed
 variables of a system $\Sys{S_2}$, then a behavior $\phi_1$ of $\Sys{S_1}$ will be
 compatible with a behavior $\phi_2$ of $\Sys{S_2}$ when $\phi_2$ is a behavior
 for the parameters charted by the variables exposed by $\phi_1$.
@@ -1478,7 +1478,7 @@ \subsection{How behaviors of systems wire together: representable doubly indexed
     it with the first one and then with the second one.
   \item (Functorial Interchange) This asks us to suppose that we have a pair of
     wiring patterns and compatible charts between them (a square in $\Cat{Arena}$). The law then says that if we
-    take a bunch of behaviors whose charts are compatable according to the first
+    take a bunch of behaviors whose charts are compatible according to the first
     wiring pattern, wire them together, and then compose with a behavior of the
     second chart, we get the same thing as if we compose them all with behaviors
     of the first chart, noted that they were compatible with the second wiring

book/C6-.tex

@@ -29,9 +29,9 @@ \section{Introduction}
    \item What does it mean to be a system? Does it have a notion of states, or of behaviors? Or is it a diagram describing the way some primitive parts are organized?
    \item What should the interface of a system be?
    \item How can interfaces be connected in composition patterns?
-  \item How are systems composed through composition patterns between their interfaces.
+  \item How are systems composed through composition patterns between their interfaces?
   \item What is a map between systems, and how does it affect their interfaces?
-  \item When can maps between systems be composed along the same composition patterns as the systems.
+  \item When can maps between systems be composed along the same composition patterns as the systems?
   \end{itemize}
   \end{informal}
 
@@ -192,7 +192,7 @@ \section{The Behavioral Approach to Systems Theory}\label{sec:behavioral.approac
 that the parameters of the original differential system
 $\Sys{LK}$ are considered as exposed variables of state in the behavioral
 approach. This is because the behavioral approach composes systems by setting
-exposed variables equal to eachother, so the parameters must be considered as
+exposed variables equal to each other, so the parameters must be considered as
 exposed variables so that they can be set equal to other variables.
 \end{remark}
 
@@ -2208,7 +2208,7 @@ \subsection{Port-plugging systems theories: Labelled graphs}
  The category $\Cat{Graph}$ of graphs is the category of presheaves on the category $0 \rightrightarrows 1$ consisting of two objects $0$ and $1$ and two arrows $s$ and $t$ from $0$ to $1$.
 \end{proposition}
 \begin{proof}
-This is a matter of checking definitions against eachother. A presheaf $G$ on that small category would consists of two sets $G(0)$ and $G(1)$ together with two functions $G(s), G(t) : G(1) \rightrightarrows G(0)$ --- precisely a graph. Furthermore, a natural transformation between these presheaves will be a graph map.
+This is a matter of checking definitions against each other. A presheaf $G$ on that small category would consists of two sets $G(0)$ and $G(1)$ together with two functions $G(s), G(t) : G(1) \rightrightarrows G(0)$ --- precisely a graph. Furthermore, a natural transformation between these presheaves will be a graph map.
 \end{proof}
 
 As a corollary, we note that the category of graphs has all limits and colimits, and that they may be calculated in the category of sets. That is, the (co)limit of a diagram of graphs has as nodes the (co)limit of the diagram of sets of nodes, and similarly for its edges. In particular, the category of graphs is has all finite colimits.