Add section on Functors (#32)

* Add section on Functors

* Fix spelling error

report/background/categorytheory.tex

@@ -16,6 +16,19 @@ \subsection{Categories}
 \todo{introduce hom-set}
 
 \subsection{Functors}
+\theoremstyle{definition}\newtheorem*{covfunctordef}{Functor}
+
+Similar to the arrows we have within categories, we can transform categories to other categories. This is the notion of a functor.
+\begin{covfunctordef}
+  Given two categories \cat{C} and \cat{D}, a (covariant) functor $\functor{F}: \cat{C} \to \cat{D}$ is two related functions, an \emph{object function} and an \emph{arrow function} (both written as \functor{F}). The object function $\functor{F}: \objs{C} \to \objs{D}$ maps an object $a$ in \cat{C} to $\functorobj{F}{a}$ in \cat{D}. Given an arrow $\morph{f}: \obj{a} \to \obj{b}$ in \cat{C}, the arrow function maps \morph{f} to $\functormorph{F}{f}: \functorobj{F}{a} \to \functorobj{F}{b}$. The functor must obey the following properties: \todo{Should I say the arrow is in the cat or the morphs of the cat}
+  \begin{itemize}
+    \item It must preserve the identity morphism, i.e. $\functormorph{F}{\id{a}} = \id{\functorobj{F}{a}}$.
+    \item It must also preserve composition, i.e. $\functormorph{F}{\left(\morph{g} \circ \morph{f}\right)} = \functormorph{F}{\morph{g}} \circ \functormorph{F}{\morph{f}}$.
+  \end{itemize}
+\end{covfunctordef}
+
+Intuitively, it should be obvious that categories and functors themselves form a category, \commoncatname{Cat}.\\
+
 \subsection{Natural transformations}
 \subsection{Adjunctions}
 \todo{Describe why adjunctions are such a key character in this paper}

report/background/utils.sty

@@ -8,6 +8,12 @@
 \newcommand{\domain}[1]{\ensuremath{\mathrm{dom}\,\morph{#1}}}
 \newcommand{\codomain}[1]{\ensuremath{\mathrm{cod}\,\morph{#1}}}
 \newcommand{\id}[1]{\ensuremath{\mathrm{id}_{\obj{#1}}}}
+\newcommand{\commoncatname}[1]{\ensuremath{\mathrm{#1}}}
+
+% Functor
+\newcommand{\functor}[1]{\ensuremath{\mathnormal{#1}}}
+\newcommand{\functorobj}[2]{\ensuremath{\functor{#1} \obj{#2}}}
+\newcommand{\functormorph}[2]{\ensuremath{\functor{#1} \morph{#2}}}
 
 % Pointed set
 \newcommand{\pset}[2]{\ensuremath{\left(#1,\,#2\right)}}