TFG.tex

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%% FONTS: libertine+biolinum+stix
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% =====================
% = Datos importantes =
% =====================
% hay que rellenar estos datos y luego
% ir a \begin{document}

\title{Teoría de Categorías y Programación Funcional}
\author{Diego Pedraza López}
\date{\today}
\newcommand{\tutores}[1]{\newcommand{\guardatutores}{#1}}
\tutores{Prof. Tutor José A. Alonso Jiménez}

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% ======================
% = Páginas de títulos =
% ======================
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  \begin{center}
  2018
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  \vspace*{1.8cm}
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  \begin{minipage}{8.45cm}
      Memoria presentada como parte de los requisitos para la obtención del título de
      Grado en Matemáticas por la Universidad de Sevilla.

      \vspace*{7.5mm}

      Tutorizada por
      % \vspace*{5mm}
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% ======================================
% = Color de la Universidad de Sevilla =
% ======================================
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% =========
% = Otros =
% =========
\usepackage[]{tabularx}
\usepackage[]{enumitem}
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% ==========================
% = Matemáticas y teoremas =
% ==========================
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% ============================
% = Composición de la página =
% ============================

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% ==============================
% = Composición de los títulos =
% ==============================

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% =======================
% = Cabeceras de página =
% =======================
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\usepackage{makeidx}
\makeindex
\usepackage{csquotes} % comillas españolas

% =============================
% = El documento empieza aquí =
% =============================
\begin{document}

\begin{titlepage}
  \centering
  {\scshape\Large Trabajo Fin de Grado\par}
  \vspace{1.5cm}
  {\huge\bfseries Teoría de Categorías y Programación Funcional\par}
  \vspace{0.025cm}
  \rule{0.4\textwidth}{0.4pt}
  \vspace{1cm}

  {\Large\itshape Diego Pedraza López\par}
  \vspace{1cm}
  tutorizado por\par
  Prof.~Tutor~José A. Alonso Jiménez\par
  \vspace{1cm}
  %\includegraphics[width=0.6\textwidth]{bookworm}
  \vspace{7cm}

  {\scshape\LARGE Universidad de Sevilla \par}
  \vfill
\end{titlepage}

\maketitle

\begin{dedication}
Dedicado a Mila.  
\end{dedication}
\frontmatter

\tableofcontents

\mainmatter

\chapter*{Sumario}
\addcontentsline{toc}{chapter}{Sumario}
\markright{Sumario}
En esencia, la teoría de categorías es el estudio de la composición.
Una categoría es una colección de objetos y morfismos entre ellos de manera que la composición tenga sentido.
Este tipo de estructura ressulta ser muy común en la mayoría de los campos de las matemáticas.
Es más, tiene un fuerte vínculo con la lógica y la teoría de tipos a través de las categorías cartesianamente cerradas.
En programación funcional, algunos diseños como las mónadas son originarias de la teoría de categorías.
Para poder hablar de estos temas, primero tendremos que entender las construcciones comunes que pueden ser definidas sobre una categoría o incluso entre categorías.
Exploraremos el campo de la teoría de categorías y sus conexiones con otros campos de la matemáticas usando programación funcional, específicamente Haskell.

\section*{Abstract}
\begin{otherlanguage}{english}
    In essence, category theory is the study of the composition.
    A category is a collection of objects and morphism between them where composition of morphism makes sense.
    This kind of structure happens to be very common in most fields of mathematics.
    Furthermore, it has strong links with logic and type theory through cartesian closed categories.
    In functional programming, some design patterns like monads originate from category theory.
    In order to discuss this topics, first we'll have to understand common constructions that can be defined over a category or even between categories. 
    We will explore the realm of category theory and its connections with other fields of mathematics using functional programming, specifically Haskell.
\end{otherlanguage}

\include{introduccion}
\include{fundamentos}
\include{yoneda}
\include{cartesianas}
\include{monoides}
\include{monadas}
\include{f-algebras}
\include{epilogo}

\backmatter

\chapter*{Bibliografía}
\markright{Bibliografía}
\addcontentsline{toc}{chapter}{Bibliografía}
%\nocite{milewski}
%\bibliographystyle{acm}
%\bibliography{bibliografia}

\begin{itemize}
  \setlength\itemsep{1em}
  \item \textsc{Awodey, Steve}, \emph{Category Theory}, Oxford University Press, 2006.
  \item \textsc{Buurlage, Jan-Willem}, \emph{Lecture Notes on Categories and Haskell}, 2017. Disponible en \url{https://jwbuurlage.github.io/}.
  \item \textsc{Lambek, Joachim} y \textsc{Scott, Philip J.}, \emph{Introduction to Higher Order Categorical Logic}, Cambridge University Press, 1986.
  \item \textsc{Lipovača, Miran}, \emph{Learn You a Haskell for Great Good!: A Beginner's Guide}, No Starch Press (2011). Disponible en \url{http://learnyouahaskell.com/}.
  \item \textsc{Mac Lane, Saunder}, \emph{Categories for the Working Mathematician}, Springer, 2ª edición, 1998.
  \item \textsc{Milewski, Bartosz}, \emph{Category Theory for Programmers}, 2018. Disponible en \url{https://github.com/hmemcpy/milewski-ctfp-pdf}.
  \item \textsc{Pierce, Benjamin C.}, \emph{A taste of category theory for computer scientist}, Carnegie Mellon University, 1988.
  \item \textsc{Riehl, Emily}, \emph{Categorical homotopy theory}, Cambridge University Press, 2014.
  \item \textsc{Riehl, Emily}, \emph{Category Theory in Context}, Cambridge University Press, 2016.
  \item \textsc{Thompson, Simon}, \emph{Type theory \& functional programming}, Addison-Wesley, 1991.
\end{itemize}

\printindex

\end{document}