change document class to scrbook

Author: benediktahrens

tex/CT4P.tex

@@ -1,4 +1,4 @@
-\documentclass[a4paper,10pt]{scrartcl}
+\documentclass[a4paper,11pt, oneside,titlepage=false]{scrbook}
 \usepackage[utf8]{inputenc}
 
 \usepackage{tikz-cd}
@@ -284,9 +284,9 @@
 \newpage
 
 
-\section{Introduction}
-
+\chapter{Introduction}
 
+\section{About Category Theory}
 \emph{Category theory} is a mathematical area of endeavour and language developed to reconcile and unify mathematical phenomena from different disciplines.
 It was developed from the 1940s on, in particular by Samuel Eilenberg and Saunders Mac Lane.
 Category theory studies objects by studying the way they interact with other objects of the same kind.
@@ -300,7 +300,7 @@ \section{Introduction}
 We also study infinite datatypes (see \cref{sec:coinductive}).
 
 
-\subsection{Learning Material on Category Theory}
+\section{Learning Material on Category Theory}
 \label{sec:material}
 
 
@@ -317,16 +317,19 @@ \subsection{Learning Material on Category Theory}
 \item The rather substantial textbook by Barr and Wells \cite{barr-wells} (available for free online) covers a lot more than we are going to discuss in these notes.
 
 \item The Catsters \cite{catsters} provide a lecture series on category theory on YouTube.
+  
+\item A list of resources on category theory is maintained at \url{https://www.logicmatters.net/categories/}.
 \end{itemize}
 
 Throughout these notes, pointers to further sources, such as textbooks and research articles, are given.
 
-\subsection{Notations}
+\chapter{Brief Summary of Logical Foundations}
 \label{sec:notation}
 
-A list of notations which we use throughout these lectures notes.
+
 \begin{itemize}
 \item If $X$ is a set and $x$ is an element of $X$, we write $x \in X$.
+\item If $X$ and $Y$ are sets
 \item If $X$ is a set and $P$ and $Q$ are properties dependent over the elements of $X$, we write $P\implies Q$ to express that if $P(x)$ holds for an element $x\in X$, then also $Q(x)$ should hold for the element $x$. Moreover, we write $P \iff Q$ if $P\implies Q$ and $Q\implies P$.
 \item If $X$ is a set and $P$ is a property dependent over the elements of $X$, we write:
 \begin{itemize}
@@ -343,7 +346,7 @@ \subsection{Notations}
 
 
 
-\section{Categories}
+\chapter{Categories}
 \label{sec:categories}
 
 \begin{reading*}
@@ -357,7 +360,7 @@ \section{Categories}
 \end{reading*}
 
 
-\subsection{Categories and Examples}
+\section{Categories and Examples}
 
 \begin{dfn}\label{dfn:category}
   A \textbf{category} $\CC$ consists of the following data:
@@ -798,7 +801,9 @@ \subsection{Categories and Examples}
   Define a category $\REL$ of sets and binary relations. Recall that given sets $X$ and $Y$ a binary relation $R$ is a subset of the cartesian product $X \times Y$ of the sets.
 \end{exer}
 
-\subsection{Isomorphisms}
+\chapter{Special Morphisms in a Category}
+
+\section{Isomorphisms}
 \label{sec:isos}
 
 \begin{reading*}
@@ -872,7 +877,7 @@ \subsection{Isomorphisms}
 Can you characterize/describe the isomorphisms in $\POS(\NN,\leq)$, $\SKELFINSET$ and $\MAT$?
 \end{exer}
 
-\subsection{Sections and Retractions}
+\section{Sections and Retractions}
 \label{sec:sections}
 
 
@@ -902,7 +907,7 @@ \subsection{Sections and Retractions}
 \end{exer}
 
 
-\subsection{Monomorphisms and Epimorphisms}
+\section{Monomorphisms and Epimorphisms}
 \label{sec:mono-epi}
 
 \begin{reading*}
@@ -1002,7 +1007,7 @@ \subsection{Monomorphisms and Epimorphisms}
 \]
 \end{exer}
 
-\section{Universal Properties}\label{sec:universal}
+\chapter{Universal Properties}\label{sec:universal}
 
 In category theory, we study objects of a category by studying the ``interactions'' they have with other objects in the category.
 What does this mean?
@@ -1025,7 +1030,7 @@ \section{Universal Properties}\label{sec:universal}
 
 \end{reading*}
 
-\subsection{Initial Objects}
+\section{Initial Objects}
 \label{sec:initial-objects}
 
 We say that an object of a category $\CC$ is \emph{initial} if it has a unique morphism to any object in the category:
@@ -1139,7 +1144,7 @@ \subsection{Initial Objects}
   we will see this in \cref{sec:initial-algs}.
 \end{rem}
 
-\subsection{Terminal Objects}
+\section{Terminal Objects}
 \label{sec:terminal-objects}
 
 
@@ -1206,7 +1211,7 @@ \subsection{Terminal Objects}
   Identify a terminal object in the category $\REL$ and show that it is terminal.
 \end{exer}
 
-\subsection{(Binary) Products}
+\section{(Binary) Products}
 \label{sec:products}
 
 
@@ -1367,7 +1372,7 @@ \subsection{(Binary) Products}
 \end{exer}
 
 
-\subsection{(Binary) Coproducts}
+\section{(Binary) Coproducts}
 \label{sec:coproducts}
 
 \begin{dfn}
@@ -1553,7 +1558,7 @@ \subsection{(Binary) Coproducts}
 
 
 
-\section{Functors}\label{sec:functors}
+\chapter{Functors}\label{sec:functors}
 An important aspect in computer programming is the transformation of data. For example, if you have a data type $X$, then one can consider also the data type $\List(X)$ of lists with values in $X$. If one thinks of the objects in a category to be data types, then we can ask even more. If $f:X\to Y$ is a function (between the data types), then this also induces  a function from the $X$-valued lists to the $Y$-valued lists as follows:
 \begin{align}\label{eqn:function_on_list}
   \List(f) : \List(X)&\to \List(Y)
@@ -1798,7 +1803,7 @@ \subsection{Categories as Objects of a Category?}
 
 
 
-\section{Natural Transformations}
+\chapter{Natural Transformations}
 \label{sec:nat-trans}
 
 \begin{dfn} Let $F,G: \CC\to\DD$ be functors. A \textbf{natural transformation} $\alpha$ from $F$ to $G$ consists of the following data:
@@ -2000,7 +2005,7 @@ \subsection{Equivalence of Categories}
 \input{data}
 
 
-\section{Conclusions and Further Reading}
+\chapter{Conclusions and Further Reading}
 
 We have given a brief, and necessarily incomplete, introduction to category theory.
 The choice of topics was guided by the applications to programming we considered.

tex/adjunctions.tex

@@ -1,4 +1,4 @@
-\section{Adjunctions}
+\chapter{Adjunctions}
 \label{sec:adjunctions}
 
 \begin{dfn} A pair $(F,G)$ of functors $F : \CC\to\DD, G:\DD\to\CC$ is called an \textbf{adjoint pair} if for every objects $X\in\Ob{\CC}$ and $Y\in\Ob{\DD}$, there exists a bijection 

tex/contravariant.tex

@@ -1,4 +1,4 @@
-\section{Contravariant Functors}
+\chapter{Contravariant Functors}
 
 A variation on functors are \textbf{contravariant functors}.
 A contravariant functor consists of a map on objects, just like a functor.

tex/data.tex

@@ -1,8 +1,8 @@
 
-\section{Inductive Datatypes and Initial Algebras}
+\chapter{Inductive Datatypes and Initial Algebras}
 \label{sec:initial-algs}
 
-\subsection{Examples}
+\section{Examples}
 \label{sec:examples}
 
 
@@ -110,7 +110,7 @@ \subsection{Examples}
   Construct a suitable category $\Cat{L}$ and show that ($\mathsf{NatList}, \nil, \cons$) is initial in that category.
 \end{exer}
 
-\subsection{Datatypes as Initial Algebras}
+\section{Datatypes as Initial Algebras}
 \label{sec:datatypes-as-initial}
 
 
@@ -557,7 +557,7 @@ \section{Fusion Property}\label{sec:fusion}
   A guide to further literature on recursion operators is given in \cite[\S6]{DBLP:journals/jfp/Hutton99}.
 \end{reading*}
 
-\section{Terminal Coalgebras and Coinductive Datatypes}\label{sec:coinductive}
+\chapter{Terminal Coalgebras and Coinductive Datatypes}\label{sec:coinductive}
 
 
 \begin{reading*}

tex/forget-free.tex

@@ -1,5 +1,5 @@
 
-\section{Forgetful and Free Functors}
+\chapter{Forgetful and Free Functors}
 A lot of (mathematical) structures are defined as some other kind of mathematical structure, but where extra structure is added. An example of this is the following:\\
 Recall that a monoid is a set $M$ together with a binary operation $m:M\to M\to M$ which is associative and such that there is an identity element $e$ (see \cref{monoidcategory}). In particular, any monoid has an underlying set and any morphism of monoids has an underlying function (between those sets). So forgetting the binary operation and identity element defines a functor from $\MON$ to $\SET$ which is called a \textit{forgetful functor}:
 \begin{exa}\label{example:forgetful_montoset} The \textbf{forgetful functor from $\MON$ to $\SET$} is the functor specified by the following data:

tex/monads.tex

@@ -1,5 +1,5 @@
 
-\section{Monads and Effects}
+\chapter{Monads and Effects}
 \label{sec:monads}
 
 In this section, we study the definition of monads as well as instances of monads useful in functional programming.

tex/solutions.tex

@@ -1,4 +1,4 @@
-\section{Solutions}
+\chapter{Solutions to Selected Exercises}
 \label{sec:solutions}
 
 \begin{solution}[\cref{exer:post_antisymmetry}]\label{sol:post_antisymmetry}