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Export macros to catprog.sty, work on F-algebras section

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Leon Vatthauer 2024-03-27 10:35:12 +01:00
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6
tex/.vscode/ltex.dictionary.en-US.txt vendored
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@ -37,3 +37,9 @@ monoids
n-ary n-ary
Cocartesian Cocartesian
Yoneda Yoneda
endofunctor
F-Coalgebras
Corecursion
Coinduction
Colimits
Colimit

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tex/.vscode/settings.json vendored
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@ -27,6 +27,8 @@
"ltex.latex.commands": { "ltex.latex.commands": {
"\\customlabel{}": "ignore", "\\customlabel{}": "ignore",
"\\setminted[]{}": "ignore", "\\setminted[]{}": "ignore",
"\\setmintedinline[]{}": "ignore" "\\setmintedinline[]{}": "ignore",
"\\setmathfont[]{}": "ignore",
"\\setmathfont{}": "ignore"
} }
} }

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tex/catprog.sty Normal file
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@ -0,0 +1,221 @@
%
% Standard macros to typeset papers on category theory and semantics
%
% Unless \catname is defined, make it bold.
% So, call \providecommand{\catname}{\mathcal}
% or \reprovidecommand{\catname}{\mathcal}
% before calling this file if you prefer calligraphic names for categories.
%
% Same applies to other commands.
%
% COPIED FROM https://gitlab.cs.fau.de/i8/TexCommon/ AND THEN ADJUSTED.
\RequirePackage{bm} % Needed for defbbname
\RequirePackage{mathtools} % Needed for coloneqq
\providecommand{\catname}{\mathsf}
%\providecommand{\mndname}{\mathbbb}
\providecommand{\clsname}{\mathscr}
\providecommand{\oname}[1]{\operatorname{\mathsf{#1}}}
%% Defining category names like \BA, \BB, etc
\def\defcatname#1{\expandafter\def\csname B#1\endcsname{\catname{#1}}}
\def\defcatnames#1{\ifx#1\defcatnames\else\defcatname#1\expandafter\defcatnames\fi}
\defcatnames ABCDEFGHIJKLMNOPQRSTUVWXYZ\defcatnames
%% Defining \CA, \CB, etc
\def\defclsname#1{\expandafter\def\csname C#1\endcsname{\clsname{#1}}}
\def\defclsnames#1{\ifx#1\defclsnames\else\defclsname#1\expandafter\defclsnames\fi}
\defclsnames ABCDEFGHIJKLMNOPQRSTUVWXYZ\defclsnames
%% Defining \BBA, \BBB, etc
\def\defbbname#1{\expandafter\def\csname BB#1\endcsname{{\bm{\mathsf{#1}}}}}
\def\defbbnames#1{\ifx#1\defbbnames\else\defbbname#1\expandafter\defbbnames\fi}
\defbbnames ABCDEFGHIJKLMNOPQRSTUVWXYZ\defbbnames
%% Defining \BMA, \BMB, etc
\def\defbbname#1{\expandafter\def\csname BM#1\endcsname{{\bm{\mathsf{#1}}}}}
\def\defbbnames#1{\ifx#1\defbbnames\else\defbbname#1\expandafter\defbbnames\fi}
\defbbnames ABCDEFGHIJKLMNOPQRSTUVWXYZ\defbbnames
%% Some standard categories
\def\Set{\ensuremath{\catname{Set}}}
\def\Par{\ensuremath{\catname{Par}}}
\def\Rel{\ensuremath{\catname{Rel}}}
\def\Cpo{\ensuremath{\catname{Cpo}}}
\def\Pos{\ensuremath{\catname{Pos}}}
\def\Mon{\ensuremath{\catname{Mon}}}
\def\Gra{\ensuremath{\catname{Gra}}}
\providecommand{\Alg}[1]{\ensuremath{\catname{Alg}(#1)}}
%% Objects of category
\providecommand{\obj}[1]{\ensuremath{\vert #1 \vert}}
%% Dual category
\providecommand{\dual}[1]{\ensuremath{#1^{op}}}
% Misc
%% Commutation of diagrams
\providecommand{\comm}{\circlearrowleft}
%% Banana brackets for catamorphisms
\RequirePackage{stmaryrd}
\providecommand{\cata}[1]{\llparenthesis #1 \rrparenthesis}
\providecommand{\eps}{{\operatorname\epsilon}}
\providecommand{\amp}{\mathbin{\&}}
\providecommand{\argument}{\operatorname{-\!-}}
\providecommand{\altargument}{\underline{\;\;}}
\providecommand{\wave}[1]{\widetilde{#1}} % Overline wave
\providecommand{\ul}{\underline} % Underline
\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
\providecommand{\mplus}{{\scriptscriptstyle\bf+}} % Small '+' for indexing
\providecommand{\smin}{\smallsetminus}
\providecommand{\ass}{\mathrel{\coloneqq}}
\providecommand{\bnf}{\mathrel{\Coloneqq}}
%% Some standard functors
\providecommand{\PSet}{{\mathcal P}} % Powerset
\providecommand{\FSet}{{\mathcal P}_{\omega}} % Finite powerset
\providecommand{\CSet}{{\mathcal P}_{\omega_1}} % Countable powerset
\providecommand{\NESet}{{\mathcal P}^{\mplus}} % Non-empty powerset
\providecommand{\Id}{\operatorname{Id}}
%% General categorical notation
\providecommand{\Hom}{\mathsf{Hom}}
\providecommand{\id}{\mathsf{id}}
\providecommand{\op}{\mathsf{op}}
\providecommand{\comp}{\mathbin{\circ}}
\providecommand{\iso}{\mathbin{\cong}}
\providecommand{\tensor}{\mathbin{\otimes}}
\providecommand{\unit}{\star}
\providecommand{\bang}{\operatorname!} % Initial/final map
%% Various arrows
\providecommand{\from}{\leftarrow}
\providecommand{\ito}{\hookrightarrow} % Injection
\providecommand{\ifrom}{\hookleftarrow}
\providecommand{\pto}{\mathrel{\rightharpoonup}} % Partial function
\providecommand{\pfrom}{\mathrel{\leftarpoonup}} %
\providecommand{\tto}{\mathrel{\Rightarrow}} % Double arrow
\providecommand{\tfrom}{\mathrel{\Leftarrow}} %
\providecommand{\mto}{\mapsto}
\providecommand{\xto}[1]{\,\xrightarrow{#1}\,}
\providecommand{\xfrom}[1]{\,\xleftarrow{\;#1}\,}
\providecommand{\To}{\mathrel{\Rightarrow}} % Double arrow
\providecommand{\From}{\mathrel{\Leftarrow}}
\providecommand{\dar}{\kern-.75pt\operatorname{\downarrow}}
\providecommand{\uar}{\kern-.75pt\operatorname{\uparrow}}
\providecommand{\Dar}{\kern-.75pt\operatorname{\Downarrow}}
\providecommand{\Uar}{\kern-.75pt\operatorname{\Uparrow}}
%% Logic
\providecommand{\True}{\top}
\providecommand{\False}{\bot}
\providecommand{\bigor}{\bigvee}
\providecommand{\bigand}{\bigwedge}
\providecommand{\impl}{\Rightarrow}
\providecommand{\equ}{\Longleftrightarrow}
\providecommand{\entails}{\vdash}
%% Order
\providecommand{\appr}{\sqsubseteq}
\providecommand{\join}{\sqcup}
\providecommand{\meet}{\sqcap}
\providecommand{\bigjoin}{\bigsqcup}
\providecommand{\bigmeet}{\bigsqcap}
%% Products
\providecommand{\fst}{\oname{fst}}
\providecommand{\snd}{\oname{snd}}
\providecommand{\pr}{\oname{pr}}
\providecommand{\brks}[1]{\langle #1\rangle}
\providecommand{\Brks}[1]{\bigl\langle #1\bigr\rangle}
%% Coproducts
\providecommand{\inl}{\oname{inl}}
\providecommand{\inr}{\oname{inr}}
\providecommand{\inj}{\oname{in}}
\DeclareSymbolFont{Symbols}{OMS}{cmsy}{m}{n}
\DeclareMathSymbol{\iobj}{\mathord}{Symbols}{"3B}
%\DeclareRobustCommand{\iobj}{\emptyset}
%% CCC
\providecommand{\curry}{\oname{curry}}
\providecommand{\uncurry}{\oname{uncurry}}
\providecommand{\ev}{\oname{ev}}
% Semantic brackets
\RequirePackage{stmaryrd}
\providecommand{\lsem}{\llbracket}
\providecommand{\rsem}{\rrbracket}
\providecommand{\sem}[1]{\lsem #1 \rsem}
\providecommand{\Lsem}{\bigl\llbracket}
\providecommand{\Rsem}{\bigr\rrbracket}
\providecommand{\Sem}[1]{\Lsem #1 \Rsem}
% Typographic
\providecommand{\comma}{,\operatorname{}\linebreak[1]} % possibly line-breaking comma
\providecommand{\dash}{\nobreakdash-\hspace{0pt}} % non-line-breaking hyphen
\providecommand{\erule}{\rule{0pt}{0pt}} % Empty object whose emptiness
% is not detected by LaTeX
\providecommand{\by}[1]{\text{/\!\!/~#1}} % Comments in equations
\providecommand{\pacman}[1]{} % Hide a piece of text
\newcommand{\undefine}[1]{\let #1\relax} % Make a command undefined
\providecommand{\noqed}{\def\qed{}} % Undefine the QED symbol
% -1 superscript for the inversion operator
\providecommand{\mone}{{\text{\kern.5pt\rmfamily-}\mathsf{\kern-.5pt1}}}
\makeatletter
\@ifpackageloaded{enumitem}{}{
\RequirePackage[loadonly]{enumitem} % without [loadonly]
} % conflicts with Beamer
\makeatother
% Condensed list environments
\newlist{citemize}{itemize}{1}
\setlist[citemize]{label=\labelitemi,wide}
%leftmargin=0cm,itemindent=.7cm,labelwidth=\itemindent,labelsep=-.3cm,align=left}
\newlist{cenumerate}{enumerate}{1}
\setlist[cenumerate,1]{label=\arabic*.~,ref={\arabic*},wide}
%leftmargin=0cm,itemindent=.7cm,labelwidth=\itemindent,labelsep=-.3cm,align=left}
%\newenvironment{citemize}{\begin{itemize}[leftmargin=0cm,itemindent=.7cm,labelwidth=\itemindent,labelsep=-.3cm,align=left]}{\end{itemize}}
%\newenvironment{cenumerate}{\begin{enumerate}[leftmargin=0cm,itemindent=.7cm,labelwidth=\itemindent,labelsep=-.3cm,align=left]}{\end{enumerate}}
%% A macro for defining mixfix operators
\makeatletter
\def\mfix#1{\oname{#1}\@ifnextchar\bgroup\@mfix{}} % processing odd arguments
\def\@mfix#1{#1\@ifnextchar\bgroup\mfix{}} % processing even arguments
\makeatother
% %% Instances if mfix
% \providecommand{\ift}[3]{\mfix{if}{\mathbin{}#1}{then}{\mathbin{}#2}{else}{\mathbin{}#3}}
% \providecommand{\case}[3]{\mfix{case}{\mathbin{}#1}{of}{#2}{\kern-1pt;}{\mathbin{}#3}}

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tex/main.tex
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@ -6,9 +6,15 @@
\usepackage[ngerman, english]{babel} \usepackage[ngerman, english]{babel}
\babeltags{german=ngerman} \babeltags{german=ngerman}
\usepackage{anyfontsize} \usepackage{anyfontsize}
\usepackage{unicode-math} \usepackage{mathtools}
\usepackage[warnings-off={mathtools-colon,mathtools-overbracket}]{unicode-math}
\setmathfont[math-style=ISO,version=normal]{Latin Modern Math}
\setmathfont[version=bold,math-style=ISO,FakeBold = 3]{Latin Modern Math}
\setmathfont[range={\mathcal,\mathbfcal},StylisticSet=1]{XITS Math}
\setmathfont[range=\mathscr]{XITS Math}
\usepackage{amsmath} \usepackage{amsmath}
\usepackage{mathrsfs} \usepackage{stmaryrd}
% \usepackage{mathrsfs}
\usepackage{booktabs} \usepackage{booktabs}
\usepackage[scale=.85]{noto-mono} % TODO find better unicode mono font \usepackage[scale=.85]{noto-mono} % TODO find better unicode mono font
\usepackage[final]{hyperref} \usepackage[final]{hyperref}
@ -18,7 +24,7 @@
\renewcommand{\subsectionmark}[1]{\markright{\thesubsection\ #1}} \renewcommand{\subsectionmark}[1]{\markright{\thesubsection\ #1}}
\pagestyle{scrheadings} \pagestyle{scrheadings}
\newcounter{resumeenum} % for resuming enumerated lists, https://tex.stackexchange.com/a/1702 \newcounter{resumeenum} % for resuming enumerated lists, https://tex.stackexchange.com/a/1702
\usepackage{catprog}
%%%% %%%%
%%%% Metadata %%%% Metadata
@ -129,14 +135,8 @@
\makeatother \makeatother
%%% %%%
%%% Notation %%% Unicode substitutions
%% Categories \def\circlearrowleft{↺}
\newcommand{\C}{\ensuremath{\mathscr{C}}}
\newcommand{\D}{\ensuremath{\mathscr{D}}}
%% Objects of category
\newcommand{\obj}[1]{\ensuremath{\vert #1 \vert}}
%% Dual category
\newcommand{\dual}[1]{\ensuremath{#1^{op}}}
%%% %%%
%%%% %%%%

40
tex/sections/01_introduction.tex
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@ -1,11 +1,11 @@
\chapter{Introduction} \chapter{Introduction}\label{chp:introduction}
This is a summary of the course ``Algebra des Programmierens'' taught by Prof.\ Dr.\ Stefan Milius in the winter term 2023/2024 at the FAU~\footnote{Friedrich-Alexander-Universität Erlangen-Nürnberg}. This is a summary of the course ``Algebra des Programmierens'' taught by Prof.\ Dr.\ Stefan Milius in the winter term 2023/2024 at the FAU~\footnote{Friedrich-Alexander-Universität Erlangen-Nürnberg}.
The course is based on~\cite{poll1999algebra} with~\cite{adamek1990abstract} as a reference for category theory. The course is based on~\cite{poll1999algebra} with~\cite{adamek1990abstract} as a reference for category theory.
Goal of the course is to develop a mathematical theory for semantics of data types and their accompanying proof principles. The chosen environment is the field of category theory. Goal of the course is to develop a mathematical theory for semantics of data types and their accompanying proof principles. The chosen environment is the field of category theory.
\section{Functions} \section{Functions}
A function $f : X \rightarrow Y$ is a mapping from the set $X$ (the domain of $f$) to the set $Y$ (the codomain of $f$). A function $f : X \to Y$ is a mapping from the set $X$ (the domain of $f$) to the set $Y$ (the codomain of $f$).
More concretely $f$ is a relation $f \subseteq X \times Y$ which is More concretely $f$ is a relation $f \subseteq X \times Y$ which is
\begin{itemize} \begin{itemize}
\item \emph{left-total}, i.e.\ for all $x \in X$ exists some $y \in Y$ such that $(x,y) \in f$; \item \emph{left-total}, i.e.\ for all $x \in X$ exists some $y \in Y$ such that $(x,y) \in f$;
@ -14,24 +14,24 @@ More concretely $f$ is a relation $f \subseteq X \times Y$ which is
Often, one is also interested in the symmetrical properties, a function is called Often, one is also interested in the symmetrical properties, a function is called
\begin{itemize} \begin{itemize}
\item \emph{injective} or \emph{left-unique} if for every $x,x' \in X$ the implication $f(x) = f(x') \rightarrow x = x'$ holds; \item \emph{injective} or \emph{left-unique} if for every $x,x' \in X$ the implication $f(x) = f(x') \to x = x'$ holds;
\item \emph{surjective} or \emph{right-total} if for every $y \in Y$ there exists an $x \in X$ such that $f(x) = y$; \item \emph{surjective} or \emph{right-total} if for every $y \in Y$ there exists an $x \in X$ such that $f(x) = y$;
\item \emph{bijective} if it is injective and surjective. \item \emph{bijective} if it is injective and surjective.
\end{itemize} \end{itemize}
\begin{example} \begin{example}
\begin{enumerate} \begin{enumerate}
\item The identity function $id_A : A \rightarrow A$, $id_A(x) = x$ \item The identity function $id_A : A \to A$, $id_A(x) = x$
\item The constant function $b! : A \rightarrow B$ for $b \in B$ defined by $b!(x) = b$ \item The constant function $b! : A \to B$ for $b \in B$ defined by $b!(x) = b$
\item The inclusion function $i_A : A \rightarrow B$ for $A \subseteq B$ defined by $i_A(x) = x$ \item The inclusion function $i_A : A \to B$ for $A \subseteq B$ defined by $i_A(x) = x$
\item Constants $b : 1 \rightarrow B$, where $1 := {*}$. The function $b$ is in bijection with the set $B$. \item Constants $b : 1 \to B$, where $1 := {*}$. The function $b$ is in bijection with the set $B$.
\item Composition of function $f : A \rightarrow B, g : B \rightarrow C$ called $g \circ f : A \rightarrow C$ defined by $(g \circ f)(x) = g(f(x))$. \item Composition of function $f : A \to B, g : B \to C$ called $g \circ f : A \to C$ defined by $(g \circ f)(x) = g(f(x))$.
\item The empty function $¡ : \emptyset \rightarrow B$ \item The empty function $¡ : \emptyset \to B$
\item The singleton function $! : A \rightarrow 1$ \item The singleton function $! : A \to 1$
\end{enumerate} \end{enumerate}
\end{example} \end{example}
\section{Data Types} \section{Data Types}\label{sec:datatypes}
Programs work with data that should ideally be organized in a useful manner. Programs work with data that should ideally be organized in a useful manner.
A useful representation for data in functional programming is by means of \emph{algebraic data types}. A useful representation for data in functional programming is by means of \emph{algebraic data types}.
@ -54,7 +54,7 @@ Such data types (parametric or non-parametric) usually come with a principle for
Equivalently, every function defined by recursion can be defined via a \emph{fold}-function which satisfies an identity and fusion law, which replace the induction principle. Let us now consider two examples of data types and illustrate this. Equivalently, every function defined by recursion can be defined via a \emph{fold}-function which satisfies an identity and fusion law, which replace the induction principle. Let us now consider two examples of data types and illustrate this.
\subsection{Natural Numbers} \subsection{Natural Numbers}
The type of natural numbers comes with a fold function $foldn : C \rightarrow (Nat \rightarrow C) \rightarrow Nat \rightarrow C$ for every $C$, defined by The type of natural numbers comes with a fold function $foldn : C \to (Nat \to C) \to Nat \to C$ for every $C$, defined by
\begin{alignat*}{2} \begin{alignat*}{2}
& foldn\;c\;h\;zero & & = c \\ & foldn\;c\;h\;zero & & = c \\
& foldn\;c\;h\;(suc\;n) & & = h\;(foldn\;c\;h\;n) & foldn\;c\;h\;(suc\;n) & & = h\;(foldn\;c\;h\;n)
@ -63,11 +63,11 @@ The type of natural numbers comes with a fold function $foldn : C \rightarrow (N
\begin{example} Let us now consider some functions defined in terms of $foldn$. \begin{example} Let us now consider some functions defined in terms of $foldn$.
\begin{itemize} \begin{itemize}
\item $iszero : Nat \rightarrow Bool$ is defined by \item $iszero : Nat \to Bool$ is defined by
\[iszero = foldn\;true\;false!\] \[iszero = foldn\;true\;false!\]
\item $plus : Nat \rightarrow Nat \rightarrow Nat$ is defined by \item $plus : Nat \to Nat \to Nat$ is defined by
\[plus = foldn\;id\;(succ \circ eval) \] \[plus = foldn\;id\;(succ \circ eval) \]
where $eval : (A \rightarrow B) \rightarrow A \rightarrow B$ is defined by where $eval : (A \to B) \to A \to B$ is defined by
\[eval\;f\;a = f\;a\] \[eval\;f\;a = f\;a\]
\end{itemize} \end{itemize}
@ -78,7 +78,7 @@ The type of natural numbers comes with a fold function $foldn : C \rightarrow (N
\begin{enumerate} \begin{enumerate}
\item \customlabel{law:natident}{\textbf{Identity}}: $foldn\;zero\;succ = id_{Nat}$ \item \customlabel{law:natident}{\textbf{Identity}}: $foldn\;zero\;succ = id_{Nat}$
\item \customlabel{law:natfusion}{\textbf{Fusion}}: for all $c : C$, $h, h' : Nat \item \customlabel{law:natfusion}{\textbf{Fusion}}: for all $c : C$, $h, h' : Nat
\rightarrow C$ and $k : C \rightarrow C'$ with $kc = c'$ and $kh = h'k$ follows $k \circ foldn\;c\;h = foldn\;c'\;h'$, or diagrammatically: \to C$ and $k : C \to C'$ with $kc = c'$ and $kh = h'k$ follows $k \circ foldn\;c\;h = foldn\;c'\;h'$, or diagrammatically:
% https://q.uiver.app/#q=WzAsNSxbMiwwLCJDIl0sWzQsMCwiQyJdLFsyLDIsIkMnIl0sWzQsMiwiQyciXSxbMCwwLCIxIl0sWzQsMCwiYyJdLFswLDIsImsiXSxbNCwyLCJjJyIsMl0sWzAsMSwiaCIsMl0sWzIsMywiaCciLDJdLFsxLDMsImsiLDFdXQ== % https://q.uiver.app/#q=WzAsNSxbMiwwLCJDIl0sWzQsMCwiQyJdLFsyLDIsIkMnIl0sWzQsMiwiQyciXSxbMCwwLCIxIl0sWzQsMCwiYyJdLFswLDIsImsiXSxbNCwyLCJjJyIsMl0sWzAsMSwiaCIsMl0sWzIsMywiaCciLDJdLFsxLDMsImsiLDFdXQ==
\[ \[
\begin{tikzcd}[ampersand replacement=\&] \begin{tikzcd}[ampersand replacement=\&]
@ -139,7 +139,7 @@ The type of natural numbers comes with a fold function $foldn : C \rightarrow (N
\begin{example} \begin{example}
The identity and fusion laws can in turn be used to prove the following induction principle: The identity and fusion laws can in turn be used to prove the following induction principle:
For any predicate $p : Nat \rightarrow Bool$, For any predicate $p : Nat \to Bool$,
\begin{enumerate} \begin{enumerate}
\item $p\;zero = true$ and \item $p\;zero = true$ and
\item $p \circ succ = p$ \item $p \circ succ = p$
@ -182,7 +182,7 @@ The type of natural numbers comes with a fold function $foldn : C \rightarrow (N
trivially commutes. trivially commutes.
\end{example} \end{example}
\subsection{Lists}\label{subsec:lists} \subsection{Lists}\label{subsec:lists}
We will now look at the $List$ type and examine it for similar properties. Let us start with the fold function $foldr : C \rightarrow (A \rightarrow C \rightarrow C) \rightarrow List\;A \rightarrow C$, which is defined by We will now look at the $List$ type and examine it for similar properties. Let us start with the fold function $foldr : C \to (A \to C \to C) \to List\;A \to C$, which is defined by
\begin{alignat*}{2} \begin{alignat*}{2}
& foldr\;c\;h\;nil & & = c \\ & foldr\;c\;h\;nil & & = c \\
& foldr\;c\;h\;(cons\;x\;xs) & & = h\;a\;(foldr\;c\;h\;xs) & foldr\;c\;h\;(cons\;x\;xs) & & = h\;a\;(foldr\;c\;h\;xs)
@ -191,9 +191,9 @@ We will now look at the $List$ type and examine it for similar properties. Let u
\begin{example} \begin{example}
Again, let us define some functions using $foldr$. Again, let us define some functions using $foldr$.
\begin{itemize} \begin{itemize}
\item $length : List\;A \rightarrow Nat$ is defined by \item $length : List\;A \to Nat$ is defined by
\[length = foldr\;zero\;(succ !)\] \[length = foldr\;zero\;(succ !)\]
\item For $f : A \rightarrow B$ we can define $List$-mapping function $List\;f : List\;A \rightarrow List\;B$ by \item For $f : A \to B$ we can define $List$-mapping function $List\;f : List\;A \to List\;B$ by
\[List\;f = foldr\;nil\;(cons \circ f)\] \[List\;f = foldr\;nil\;(cons \circ f)\]
\end{itemize} \end{itemize}
\end{example} \end{example}

520
tex/sections/02_categories.tex
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@ -1,29 +1,30 @@
% chktex-file 1
\chapter{Category Theory} \chapter{Category Theory}
Categories consist of objects and morphisms between those objects, that can be composed in a coherent way. Categories consist of objects and morphisms between those objects, that can be composed in a coherent way.
This yields a framework for abstraction of many mathematical concepts that enables us to reason on a very abstract level. This yields a framework for abstraction of many mathematical concepts that enables us to reason on a very abstract level.
\begin{definition}[Category] A category \C consists of \begin{definition}[Category] A category $\CC$ consists of
\begin{itemize} \begin{itemize}
\item a class of objects denoted $\obj{\C}$, \item a class of objects denoted $\obj{\CC}$,
\item for every pair of objects $A,B \in \obj{\C}$ a set of morphisms $\C(A,B)$ called the hom-set, \item for every pair of objects $A,B \in \obj{\CC}$ a set of morphisms $\CC(A,B)$ called the hom-set,
\item a morphism $id_A : A \rightarrow A$ for every $A \in \obj{\C}$ \item a morphism $id_A : A \to A$ for every $A \in \obj{\CC}$
\item a composition operator $(-) \circ (-) : \C(B,C) \rightarrow \C(A,B) \rightarrow \C(A,C)$ for every $A,B,C \in \obj{\C}$ \item a composition operator $(\argument) \comp (\argument) : \CC(B,C) \to \CC(A,B) \to \CC(A,C)$ for every $A,B,C \in \obj{\CC}$
\end{itemize} \end{itemize}
additionally the composition must be associative and $f \circ id_A = f = id_B \circ f$ for any $f : A \rightarrow B$. additionally the composition must be associative and $f \comp id_A = f = id_B \comp f$ for any $f : A \to B$.
\end{definition} \end{definition}
\begin{example} Some standard examples of categories and their objects and morphisms include: \begin{example} Some standard examples of categories and their objects and morphisms include:
\begin{center} \begin{center}
\begin{tabular}{l l l} \begin{tabular}{l l l}
Category & Objects & Morphisms \\\midrule Category & Objects & Morphisms \\\midrule
\emph{Set} & Sets & Functions \\ \Set & Sets & Functions \\
\emph{Par} & Sets & Partial functions \\ \Par & Sets & Partial functions \\
\emph{Rel} & Sets & Binary relations \\ \Rel & Sets & Binary relations \\
\emph{Gra} & Directed Graphs & Graph homomorphisms \\ \Gra & Directed Graphs & Graph homomorphisms \\
\emph{Pos} & Partially ordered sets & Monotone mappings \\ \Pos & Partially ordered sets & Monotone mappings \\
\emph{Mon} & Monoids & Monoid homomorphisms \\ \Mon & Monoids & Monoid homomorphisms \\
Monoid $(M, \cdot, e)$ & A single object $*$ & $x : * \rightarrow *$ for every $x \in M$ \\ Monoid $(M, \cdot, e)$ & A single object $*$ & $x : * \to *$ for every $x \in M$ \\
Poset $(X, \leq)$ & Elements of $X$ & $x \leq y \iff \exists! f : x \rightarrow y$ Poset $(X, \leq)$ & Elements of $X$ & $x \leq y \iff \exists! f : x \to y$
\end{tabular} \end{tabular}
\end{center} \end{center}
\end{example} \end{example}
@ -35,17 +36,17 @@ In this chapter we will characterize (finite) products and coproducts, as well a
\begin{definition}[Initial and Terminal Objects] The following is the categorical abstraction of ``empty set'' and ``singleton set'' respectively. \begin{definition}[Initial and Terminal Objects] The following is the categorical abstraction of ``empty set'' and ``singleton set'' respectively.
\begin{enumerate} \begin{enumerate}
\item An object $0 \in \obj{\C}$ is called initial if for every $B \in \obj{C}$ there is a unique morphism $¡ : 0 \rightarrow B$. \item An object $0 \in \obj{\BC}$ is called initial if for every $B \in \obj{C}$ there is a unique morphism $¡ : 0 \to B$.
\item An object $1 \in \obj{\C}$ is called terminal if for every $A \in \obj{C}$ there is a unique morphism $! : A \rightarrow 1$. \item An object $1 \in \obj{\CC}$ is called terminal if for every $A \in \obj{C}$ there is a unique morphism $! : A \to 1$.
\end{enumerate} \end{enumerate}
\end{definition} \end{definition}
\begin{example} Oftentimes the initial object is an empty structure and the terminal object a singleton structure, some examples are: \begin{example} Oftentimes the initial object is an empty structure and the terminal object a singleton structure, some examples are:
\begin{center} \begin{center}
\begin{tabular}{l l l} \begin{tabular}{l l l}
Category & Initial Object & Terminal Object \\\midrule Category & Initial Object & Terminal Object \\\midrule
\emph{Set} & $\emptyset$ & ${*}$ \\ \Set & $\emptyset$ & ${*}$ \\
\emph{Pos} & $\emptyset$ & ${*}$ \\ \Pos & $\emptyset$ & ${*}$ \\
\emph{Gra} & Empty graph & Singleton graph \\ \Gra & Empty graph & Singleton graph \\
Poset $(X, \leq)$ & $\bot \in X$ such that $\forall x \in X. \bot \leq x$ & $\top \in X$ such that $\forall x \in X. x \leq \top$ Poset $(X, \leq)$ & $\bot \in X$ such that $\forall x \in X. \bot \leq x$ & $\top \in X$ such that $\forall x \in X. x \leq \top$
\end{tabular} \end{tabular}
\end{center} \end{center}
@ -56,18 +57,18 @@ In this chapter we will characterize (finite) products and coproducts, as well a
Now let us characterize special morphisms. Now let us characterize special morphisms.
\begin{definition}[Special Morphisms] \begin{definition}[Special Morphisms]
Let $f : A \rightarrow B$ be a morphism. $f$ is called Let $f : A \to B$ be a morphism. $f$ is called
\begin{itemize} \begin{itemize}
\item an \emph{isomorphism} (\emph{iso}), if there exists an inverse $f^{-1} : B \rightarrow A$ such that $f \circ g = id_B$ and $g \circ f = id_A$; \item an \emph{isomorphism} (\emph{iso}), if there exists an inverse $f^{-1} : B \to A$ such that $f \comp g = id_B$ and $g \comp f = id_A$;
\item a \emph{monomorphism} (\emph{mono}), if for all $g, h : C \rightarrow A$ the implication $f \circ g = f \circ h \Rightarrow g = h$ holds; \item a \emph{monomorphism} (\emph{mono}), if for all $g, h : C \to A$ the implication $f \comp g = f \comp h \Rightarrow g = h$ holds;
\item an \emph{epimorphism} (\emph{epi}), if for all $g, h : B \rightarrow C$ the implication $g \circ f = h \circ f \Rightarrow g = h$ holds. \item an \emph{epimorphism} (\emph{epi}), if for all $g, h : B \to C$ the implication $g \comp f = h \comp f \Rightarrow g = h$ holds.
\end{itemize} \end{itemize}
\end{definition} \end{definition}
\begin{example} Let us consider what these notions instantiate to in concrete categories. \begin{example} Let us consider what these notions instantiate to in concrete categories.
\begin{center} \begin{center}
\begin{tabular}{l l l l} \begin{tabular}{l l l l}
Category & Monomorphisms & Epimorphisms & Isomorphisms \\\midrule Category & Monomorphisms & Epimorphisms & Isomorphisms \\\midrule
\emph{Set} & injective functions & surjective functions & bijective functions \\ \Set & injective functions & surjective functions & bijective functions \\
\emph{Pos, Gra} & injective morphisms & surjective morphisms & bijective morphisms \\ \emph{Pos, Gra} & injective morphisms & surjective morphisms & bijective morphisms \\
Poset $(X, \leq)$ & all & all & all \\ Poset $(X, \leq)$ & all & all & all \\
Monoid $(M, \cdot, e)$ & left cancellative $a \in M$ & right cancellative $a \in M$ & invertible $a \in M$ Monoid $(M, \cdot, e)$ & left cancellative $a \in M$ & right cancellative $a \in M$ & invertible $a \in M$
@ -80,19 +81,19 @@ Now let us characterize special morphisms.
\begin{proof} \begin{proof}
Let $f$ be an isomorphism. Let $f$ be an isomorphism.
\begin{itemize} \begin{itemize}
\item $f \circ g = f \circ h$ implies $g = f^{-1} \circ f \circ g = f^{-1} \circ f \circ h = h$, thus $f$ is a monomorphism. \item $f \comp g = f \comp h$ implies $g = f^{-1} \comp f \comp g = f^{-1} \comp f \comp h = h$, thus $f$ is a monomorphism.
\item $g \circ f = h \circ f$ implies $g = g \circ f \circ f^{-1} = h \circ f \circ f^{-1} = h$, thus $f$ is an epimorphism. \item $g \comp f = h \comp f$ implies $g = g \comp f \comp f^{-1} = h \comp f \comp f^{-1} = h$, thus $f$ is an epimorphism.
\end{itemize} \end{itemize}
\end{proof} \end{proof}
\begin{proposition}\label{prop:monosplitting} If $f \circ m$ is a monomorphism then $m$ is also a monomorphism. \begin{proposition}\label{prop:monosplitting} If $f \comp m$ is a monomorphism then $m$ is also a monomorphism.
\end{proposition} \end{proposition}
\begin{proof} \begin{proof}
Let $m \circ g = m \circ h$. To show that $g = h$ it suffices to show that $f \circ m \circ g = f \circ m \circ h$, which indeed follows by assumption. Let $m \comp g = m \comp h$. To show that $g = h$ it suffices to show that $f \comp m \comp g = f \comp m \comp h$, which indeed follows by assumption.
\end{proof} \end{proof}
\begin{proposition}\label{prop:episplitting} If $e \circ f$ is an epimorphism then $e$ is also an epimorphism. \begin{proposition}\label{prop:episplitting} If $e \comp f$ is an epimorphism then $e$ is also an epimorphism.
\end{proposition} \end{proposition}
\begin{proof} \begin{proof}
Let $g \circ e = h \circ e$. To show that $g = h$ it suffices to show that $g \circ e \circ f = h \circ e \circ f$, which again follows by assumption. Let $g \comp e = h \comp e$. To show that $g = h$ it suffices to show that $g \comp e \comp f = h \comp e \comp f$, which again follows by assumption.
\end{proof} \end{proof}
Categorical structures like initial objects are usually not uniquely identified, there might be multiple initial objects in a category. However, all initial objects in a category are isomorphic, we call this ``unique up to isomorphism''. Categorical structures like initial objects are usually not uniquely identified, there might be multiple initial objects in a category. However, all initial objects in a category are isomorphic, we call this ``unique up to isomorphism''.
@ -100,7 +101,7 @@ Categorical structures like initial objects are usually not uniquely identified,
\begin{proposition}\label{prop:init_up_to} Initial objects are unique up to isomorphism. \begin{proposition}\label{prop:init_up_to} Initial objects are unique up to isomorphism.
\end{proposition} \end{proposition}
\begin{proof} \begin{proof}
Let $0, 0' \in \obj{\C}$ be two initial objects of $\C$ with the unique morphisms $¡_A : 0 \rightarrow A$ and ${¡'}_A : 0' \rightarrow A$. Let $0, 0' \in \obj{\CC}$ be two initial objects of $\CC$ with the unique morphisms $¡_A : 0 \to A$ and ${¡'}_A : 0' \to A$.
The isomorphism is: The isomorphism is:
% https://q.uiver.app/#q=WzAsMixbMCwwLCIwIl0sWzIsMCwiMCciXSxbMSwwLCJ7wqEnfV8wIiwyLHsiY3VydmUiOi0yfV0sWzAsMSwiwqFfezAnfSIsMCx7ImN1cnZlIjotMn1dXQ== % https://q.uiver.app/#q=WzAsMixbMCwwLCIwIl0sWzIsMCwiMCciXSxbMSwwLCJ7wqEnfV8wIiwyLHsiY3VydmUiOi0yfV0sWzAsMSwiwqFfezAnfSIsMCx7ImN1cnZlIjotMn1dXQ==
\[ \[
@ -110,12 +111,12 @@ Categorical structures like initial objects are usually not uniquely identified,
\arrow["{¡_{0'}}", curve={height=-12pt}, from=1-1, to=1-3] \arrow["{¡_{0'}}", curve={height=-12pt}, from=1-1, to=1-3]
\end{tikzcd} \end{tikzcd}
\] \]
Note that by uniqueness $¡_{0'} \circ {¡'}_0 = {¡'}_{0'} = id_{0'}$ and ${¡'}_0 \circ ¡_{0'} = ¡_0 = id_0$. Note that by uniqueness $¡_{0'} \comp {¡'}_0 = {¡'}_{0'} = id_{0'}$ and ${¡'}_0 \comp ¡_{0'} = ¡_0 = id_0$.
\end{proof} \end{proof}
\begin{proposition}\label{prop:term_up_to} Terminal objects are unique up to isomorphism. \begin{proposition}\label{prop:term_up_to} Terminal objects are unique up to isomorphism.
\end{proposition} \end{proposition}
\begin{proof} \begin{proof}
Let $1, 1' \in \obj{\C}$ be two terminal objects of $\C$ with the unique morphisms $!_A : A \rightarrow 1$ and ${!'}_A : A \rightarrow 1'$. Let $1, 1' \in \obj{\CC}$ be two terminal objects of $\CC$ with the unique morphisms $!_A : A \to 1$ and ${!'}_A : A \to 1'$.
The isomorphism is: The isomorphism is:
% https://q.uiver.app/#q=WzAsMixbMCwwLCIxIl0sWzIsMCwiMSciXSxbMCwxLCJ7ISd9XzEiLDAseyJjdXJ2ZSI6LTJ9XSxbMSwwLCIhX3sxJ30iLDAseyJjdXJ2ZSI6LTJ9XV0= % https://q.uiver.app/#q=WzAsMixbMCwwLCIxIl0sWzIsMCwiMSciXSxbMCwxLCJ7ISd9XzEiLDAseyJjdXJ2ZSI6LTJ9XSxbMSwwLCIhX3sxJ30iLDAseyJjdXJ2ZSI6LTJ9XV0=
\[ \[
@ -125,7 +126,7 @@ Categorical structures like initial objects are usually not uniquely identified,
\arrow["{!_{1'}}", curve={height=-12pt}, from=1-3, to=1-1] \arrow["{!_{1'}}", curve={height=-12pt}, from=1-3, to=1-1]
\end{tikzcd} \end{tikzcd}
\] \]
Note that by uniqueness ${!'}_1 \circ !_{1'} = {!'}_{1'} = id_{1'}$ and $!_{1'} \circ {!'}_1 = !_1 = id_1$. Note that by uniqueness ${!'}_1 \comp !_{1'} = {!'}_{1'} = id_{1'}$ and $!_{1'} \comp {!'}_1 = !_1 = id_1$.
\end{proof} \end{proof}
\section{Duality} \section{Duality}
@ -134,17 +135,17 @@ It seems that we should somehow be able to construct one proof from the other, s
This is actually the case, we can for example say that \autoref{prop:episplitting} follows from \autoref{prop:monosplitting} by \emph{duality}. This is actually the case, we can for example say that \autoref{prop:episplitting} follows from \autoref{prop:monosplitting} by \emph{duality}.
\begin{definition}[Dual Category] \begin{definition}[Dual Category]
Every category $\C$ has a \emph{dual category} $\dual{\C}$ defined by Every category $\CC$ has a \emph{dual category} $\dual{\CC}$ defined by
\begin{itemize} \begin{itemize}
\item $\obj{\dual{\C}} = \obj{\C}$ \item $\obj{\dual{\CC}} = \obj{\CC}$
\item $\dual{\C}(A,B) = \C(B,A)$ \item $\dual{\CC}(A,B) = \CC(B,A)$
\end{itemize} \end{itemize}
\end{definition} \end{definition}
\begin{example} Examples are: \begin{example} Examples are:
\begin{enumerate} \begin{enumerate}
\item In a poset the order relation gets reversed. \item In a poset the order relation gets reversed.
\item \emph{\dual{Rel}} is isomorphic to \emph{Rel}, since subsets of $A \times B$ are in bijection with subsets of $B \times A$ \item \emph{\dual{Rel}} is isomorphic to $\Rel$, since subsets of $A \times B$ are in bijection with subsets of $B \times A$
\item $\dual{(\dual{\C})} = \C$ \item $\dual{(\dual{\CC})} = \CC$
\end{enumerate} \end{enumerate}
\end{example} \end{example}
@ -163,7 +164,7 @@ This yields a proof principle ``by duality'', where every theorem yields another
\section{Products and Coproducts} \section{Products and Coproducts}
The categorical abstraction of Cartesian products is: The categorical abstraction of Cartesian products is:
\begin{definition}[Product] \begin{definition}[Product]
The \emph{product} of two objects $A, B \in \obj{\C}$ is an object that we call $A \times B$ together with morphisms $\pi_1 : A \times B \rightarrow A$ and $\pi_2 : A \times B \rightarrow B$ (the projections), where the following property holds: The \emph{product} of two objects $A, B \in \obj{\CC}$ is an object that we call $A \times B$ together with morphisms $\pi_1 : A \times B \to A$ and $\pi_2 : A \times B \to B$ (the projections), where the following property holds:
% https://q.uiver.app/#q=WzAsNCxbMiwyLCJBIFxcdGltZXMgQiJdLFs0LDIsIkIiXSxbMCwyLCJBIl0sWzIsMCwiQyJdLFswLDIsIlxccGlfMSJdLFswLDEsIlxccGlfMiIsMl0sWzMsMiwiZiIsMl0sWzMsMSwiZyJdLFszLDAsIlxcZXhpc3RzIVxcbGFuZ2xlIGYsIGcgXFxyYW5nbGUiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0= % https://q.uiver.app/#q=WzAsNCxbMiwyLCJBIFxcdGltZXMgQiJdLFs0LDIsIkIiXSxbMCwyLCJBIl0sWzIsMCwiQyJdLFswLDIsIlxccGlfMSJdLFswLDEsIlxccGlfMiIsMl0sWzMsMiwiZiIsMl0sWzMsMSwiZyJdLFszLDAsIlxcZXhpc3RzIVxcbGFuZ2xlIGYsIGcgXFxyYW5nbGUiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=
\[ \[
\begin{tikzcd} \begin{tikzcd}
@ -180,15 +181,15 @@ The categorical abstraction of Cartesian products is:
\end{definition} \end{definition}
\begin{example} Some examples include: \begin{example} Some examples include:
\begin{enumerate} \begin{enumerate}
\item \emph{Set}: The product of two sets $A$ and $B$ is the Cartesian product $A \times B = \{ (a,b)\;\vert\;a \in A, b \in B\}$. \item $\Set$: The product of two sets $A$ and $B$ is the Cartesian product $A \times B = \{ (a,b)\;\vert\;a \in A, b \in B\}$.
\item \emph{Gra}: The product of two graphs has as vertices the Cartesian product of the vertices of both graphs and an edge $(v_1, u_1) \rightarrow (v_2, u_2)$ iff there exists edges $v_1 \rightarrow v_2$ and $u_1 \rightarrow u_2$. \item $\Gra$: The product of two graphs has as vertices the Cartesian product of the vertices of both graphs and an edge $(v_1, u_1) \to (v_2, u_2)$ iff there exists edges $v_1 \to v_2$ and $u_1 \to u_2$.
\item \emph{Pos}: Given two posets $(A, \leq), (B, \leq)$, the product is the Cartesian product of $A$ and $B$ where $(a,b) \leq (a', b') \iff a \leq a' \land b \leq b'$. \item $\Pos$: Given two posets $(A, \leq), (B, \leq)$, the product is the Cartesian product of $A$ and $B$ where $(a,b) \leq (a', b') \iff a \leq a' \land b \leq b'$.
\item Let $(X, \leq)$ be a poset, the product of $a, b \in X$ is the greatest lower bound of $a$ and $b$. \item Let $(X, \leq)$ be a poset, the product of $a, b \in X$ is the greatest lower bound of $a$ and $b$.
\end{enumerate} \end{enumerate}
\end{example} \end{example}
Dual to products are: Dual to products are:
\begin{definition}[Coproduct] \begin{definition}[Coproduct]
The \emph{coproduct} of two objects $A, B \in \obj{\C}$ is an object that we call $A + B$ together with morphisms $i_1 : A \rightarrow A + B$ and $i_2 : B \rightarrow A + B$ (the injections), where the following property holds: The \emph{coproduct} of two objects $A, B \in \obj{\CC}$ is an object that we call $A + B$ together with morphisms $i_1 : A \to A + B$ and $i_2 : B \to A + B$ (the injections), where the following property holds:
% https://q.uiver.app/#q=WzAsNCxbMiwwLCJBK0IiXSxbMCwwLCJBIl0sWzQsMCwiQiJdLFsyLDIsIkMiXSxbMSwwLCJpXzEiXSxbMiwwLCJpXzIiLDJdLFsxLDMsImYiLDJdLFsyLDMsImciXSxbMCwzLCJcXGV4aXN0cyFbZixnXSIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ== % https://q.uiver.app/#q=WzAsNCxbMiwwLCJBK0IiXSxbMCwwLCJBIl0sWzQsMCwiQiJdLFsyLDIsIkMiXSxbMSwwLCJpXzEiXSxbMiwwLCJpXzIiLDJdLFsxLDMsImYiLDJdLFsyLDMsImciXSxbMCwzLCJcXGV4aXN0cyFbZixnXSIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==
\[ \[
\begin{tikzcd} \begin{tikzcd}
@ -205,11 +206,11 @@ Dual to products are:
\end{definition} \end{definition}
\begin{example} Examples include: \begin{example} Examples include:
\begin{enumerate} \begin{enumerate}
\item \emph{Set}: The coproduct of two sets $A$ and $B$ is the disjoint union $A + B = \{ (a,0)\;\vert\;a \in A \} \cup \{ (b, 1) \vert b \in B \}$. \item $\Set$: The coproduct of two sets $A$ and $B$ is the disjoint union $A + B = \{ (a,0)\;\vert\;a \in A \} \cup \{ (b, 1) \vert b \in B \}$.
\item \emph{Pos}: The coproduct of ordered sets $(A, \leq)$ and $(B, \leq)$ is the disjoint union $A+B$ where $z \leq z'$ iff $z, z' \in A$ and $z \leq z'$ or $z, z' \in B$ and $z \leq z'$. \item $\Pos$: The coproduct of ordered sets $(A, \leq)$ and $(B, \leq)$ is the disjoint union $A+B$ where $z \leq z'$ iff $z, z' \in A$ and $z \leq z'$ or $z, z' \in B$ and $z \leq z'$.
\item \emph{Gra}: Analogous to \emph{Pos}. \item $\Gra$: Analogous to $\Pos$.
\item Let $(X, \leq)$ be a poset, the coproduct of $a,b \in X$ is the least upper bound of $a$ and $b$. \item Let $(X, \leq)$ be a poset, the coproduct of $a,b \in X$ is the least upper bound of $a$ and $b$.
\item \emph{Rel}: Analogous to \emph{Set} the coproduct is the disjoint union. Since $\emph{Rel} \cong \dual{\emph{Rel}}$ we know that the product is also the disjoint union. \item $\Rel$: Analogous to $\Set$ the coproduct is the disjoint union. Since $\Rel \iso \dual{\Rel}$ we know that the product is also the disjoint union.
\end{enumerate} \end{enumerate}
\end{example} \end{example}
@ -219,7 +220,7 @@ Dual to products are:
\begin{proof} \begin{proof}
The usual proof is somewhat analogous to the proof of \autoref{prop:term_up_to}. Instead, we will prove it like this: The usual proof is somewhat analogous to the proof of \autoref{prop:term_up_to}. Instead, we will prove it like this:
Consider the category $\emph{span}_{\C}(A,B)$ where objects are triples $A \overset{f}{\leftarrow} C \overset{g}{\rightarrow} B$ and morphisms $(C, f, g) \rightarrow (C', f', g')$ are morphisms $k : C \rightarrow C'$ in $\C$ such that Consider the category $\emph{span}_{\CC}(A,B)$ where objects are triples $A \overset{f}{\leftarrow} C \overset{g}{\to} B$ and morphisms $(C, f, g) \to (C', f', g')$ are morphisms $k : C \to C'$ in $\CC$ such that
% https://q.uiver.app/#q=WzAsNCxbMiwwLCJDIl0sWzAsMiwiQSJdLFs0LDIsIkIiXSxbMiw0LCJDJyJdLFswLDEsImYiLDJdLFswLDIsImciXSxbMywxLCJmJyJdLFszLDIsImcnIiwyXSxbMCwzLCJrIl1d % https://q.uiver.app/#q=WzAsNCxbMiwwLCJDIl0sWzAsMiwiQSJdLFs0LDIsIkIiXSxbMiw0LCJDJyJdLFswLDEsImYiLDJdLFswLDIsImciXSxbMywxLCJmJyJdLFszLDIsImcnIiwyXSxbMCwzLCJrIl1d
\[ \[
\begin{tikzcd} \begin{tikzcd}
@ -235,7 +236,7 @@ Dual to products are:
\arrow["k", from=1-3, to=5-3] \arrow["k", from=1-3, to=5-3]
\end{tikzcd} \end{tikzcd}
\] \]
commutes. Products of $A$ and $B$ are the final objects in $\emph{span}_{\C}(A,B)$ and are thus unique up to isomorphism. commutes. Products of $A$ and $B$ are the final objects in $\emph{span}_{\CC}(A,B)$ and are thus unique up to isomorphism.
\end{proof} \end{proof}
By duality, we get: By duality, we get:
\begin{proposition} \begin{proposition}
@ -245,41 +246,41 @@ By duality, we get:
We can now characterize products (and later dually coproducts) as a commutative monoid: We can now characterize products (and later dually coproducts) as a commutative monoid:
\begin{proposition} \begin{proposition}
$1$ is a unit of $\times$, i.e.\ $A \cong A \times 1$ for any $A \in \obj{\C}$. $1$ is a unit of $\times$, i.e.\ $A \iso A \times 1$ for any $A \in \obj{\CC}$.
\end{proposition} \end{proposition}
\begin{proof} \begin{proof}
Take $\langle id_A , !_A \rangle : A \rightarrow A \times 1$ and $\pi_1 : A \times 1 \rightarrow A$, this indeed constitutes an isomorphism, since Take $\langle id_A , !_A \rangle : A \to A \times 1$ and $\pi_1 : A \times 1 \to A$, this indeed constitutes an isomorphism, since
\[\pi_1 \circ \langle id_A , !_A \rangle = id_A \] \[\pi_1 \comp \langle id_A , !_A \rangle = id_A \]
by definition and by definition and
\[\langle id_A , !_A \rangle \circ \pi_1 = \langle \pi_1 , !_A \rangle = \langle \pi_1 , \pi_2 \rangle = id_{A \times 1},\] \[\langle id_A , !_A \rangle \comp \pi_1 = \langle \pi_1 , !_A \rangle = \langle \pi_1 , \pi_2 \rangle = id_{A \times 1},\]
because $\pi_2 = !_A : A \times 1 \rightarrow 1$ by uniqueness of $!_A$. because $\pi_2 = !_A : A \times 1 \to 1$ by uniqueness of $!_A$.
\end{proof} \end{proof}
\begin{proposition} \begin{proposition}
$\times$ is associative, i.e.\ $A \times (B \times C) \cong (A \times B) \times C$ for any $A,B,C \in \obj{\C}$. $\times$ is associative, i.e.\ $A \times (B \times C) \iso (A \times B) \times C$ for any $A,B,C \in \obj{\CC}$.
\end{proposition} \end{proposition}
\begin{proof} \begin{proof}
Take \[\alpha = \langle \langle \pi_1 , \pi_1 \circ \pi_2 \rangle , \pi_2 \circ \pi_2 \rangle : A \times (B \times C) \rightarrow (A \times B) \times C\] Take \[\alpha = \langle \langle \pi_1 , \pi_1 \comp \pi_2 \rangle , \pi_2 \comp \pi_2 \rangle : A \times (B \times C) \to (A \times B) \times C\]
and \[\alpha^{-1} = \langle \pi_1 \circ \pi_1 , \langle \pi_2 \circ \pi_1 , \pi_2 \rangle \rangle : (A \times B) \times C \rightarrow A \times (B \times C).\] and \[\alpha^{-1} = \langle \pi_1 \comp \pi_1 , \langle \pi_2 \comp \pi_1 , \pi_2 \rangle \rangle : (A \times B) \times C \to A \times (B \times C).\]
The rest of the proof then amounts to simply rewriting. The rest of the proof then amounts to simply rewriting.
\end{proof} \end{proof}
\begin{proposition} \begin{proposition}
$\times$ is commutative, i.e.\ $A \times B \cong B \times A$ for any $A, B \in \obj{\C}$. $\times$ is commutative, i.e.\ $A \times B \iso B \times A$ for any $A, B \in \obj{\CC}$.
\end{proposition} \end{proposition}
\begin{proof} \begin{proof}
Take \[\langle \pi_2 , \pi_1 \rangle : A \times B \rightarrow B \times A\] Take \[\langle \pi_2 , \pi_1 \rangle : A \times B \to B \times A\]
and \[\langle \pi_2 , \pi_1 \rangle : B \times A \rightarrow A \times B.\] and \[\langle \pi_2 , \pi_1 \rangle : B \times A \to A \times B.\]
Indeed, $\langle \pi_2 , \pi_1 \rangle \circ \langle \pi_2 , \pi_1 \rangle = \langle \pi_2 \circ \langle \pi_2 , \pi_1 \rangle , \pi_1 \circ \langle \pi_2 , \pi_1 \rangle \rangle = \langle \pi_1 , \pi_2 \rangle = id$. Indeed, $\langle \pi_2 , \pi_1 \rangle \comp \langle \pi_2 , \pi_1 \rangle = \langle \pi_2 \comp \langle \pi_2 , \pi_1 \rangle , \pi_1 \comp \langle \pi_2 , \pi_1 \rangle \rangle = \langle \pi_1 , \pi_2 \rangle = id$.
\end{proof} \end{proof}
Duality instantly yields the commutative monoid structure of coproducts: Duality instantly yields the commutative monoid structure of coproducts:
\begin{proposition} \begin{proposition}
$0$ is the unit of $+$, i.e.\ $A \cong A + 0$ for any $A \in \obj{\C}$. $0$ is the unit of $+$, i.e.\ $A \iso A + 0$ for any $A \in \obj{\CC}$.
\end{proposition} \end{proposition}
\begin{proposition} \begin{proposition}
$+$ is associative, i.e.\ $A + (B + C) \cong (A + B) + C$ for any $A,B,C \in \obj{\C}$. $+$ is associative, i.e.\ $A + (B + C) \iso (A + B) + C$ for any $A,B,C \in \obj{\CC}$.
\end{proposition} \end{proposition}
\begin{proposition} \begin{proposition}
$+$ is commutative, i.e.\ $A + B \cong B + A$ for any $A, B \in \obj{\C}$. $+$ is commutative, i.e.\ $A + B \iso B + A$ for any $A, B \in \obj{\CC}$.
\end{proposition} \end{proposition}
\begin{remark} \begin{remark}
@ -289,36 +290,36 @@ Duality instantly yields the commutative monoid structure of coproducts:
\section{Functors} \section{Functors}
Functors are morphisms between categories, concretely: Functors are morphisms between categories, concretely:
\begin{definition}[Functor] \begin{definition}[Functor]
A functor $F : \C \rightarrow \D$ consists of A functor $F : \CC \to \CD$ consists of
\begin{itemize} \begin{itemize}
\item a mapping $F : \obj{\C} \rightarrow \obj{\D}$ on objects and \item a mapping $F : \obj{\CC} \to \obj{\CD}$ on objects and
\item a mapping $F : \C(A,B) \rightarrow \C(FA,FB)$ on morphisms, \item a mapping $F : \CC(A,B) \to \CC(FA,FB)$ on morphisms,
\end{itemize} \end{itemize}
such that $F(id_A) = id_{FA}$ and $F(g \circ f) = Fg \circ Ff$. such that $F(id_A) = id_{FA}$ and $F(g \comp f) = Fg \comp Ff$.
\end{definition} \end{definition}
\begin{example} Usual examples of functors include \begin{example} Usual examples of functors include
\begin{enumerate} \begin{enumerate}
\item Constant functors mapping to a single object: $D! : \C \rightarrow \D, D \in \obj{\D}$ with \item Constant functors mapping to a single object: $D! : \CC \to \CD, D \in \obj{\CD}$ with
\[D!(C) = D, \qquad D!(f) = id_D.\] \[D!(C) = D, \qquad D!(f) = id_D.\]
\item Identity functor: $Id_\C : \C \rightarrow \C$ with \item Identity functor: $\Id{}_{\CC} : \CC \to \CC$ with
\[Id_\C(C) = C, \qquad Id_\C(f) = f.\] \[\Id{}_{\CC} (C) = C, \qquad \Id{}_{\CC} (f) = f.\]
\item Composition of functors: $(FG)(X) = F(GX), (FG)(f) = F(Gf)$ \item Composition of functors: $(FG)(X) = F(GX), (FG)(f) = F(Gf)$
\item Square functor on $\emph{Set}$: $Q : \emph{Set} \rightarrow \emph{Set}$ with \item Square functor on $\Set$: $Q : \Set \to \Set$ with
\[QX = X \times X, \qquad Qf = f \times f.\] \[QX = X \times X, \qquad Qf = f \times f.\]
\item $list : Set \rightarrow Set$, see \autoref{subsec:lists}. \item $list : Set \to Set$, see \autoref{subsec:lists}.
\item For $A \in \obj{\C}$ there is the hom-functor $\C(A,-) : \C \rightarrow Set$ given by \item For $A \in \obj{\CC}$ there is the hom-functor $\CC(A,\argument) : \CC \to Set$ given by
\[ \C(A,B), \qquad \C(A,f : B \rightarrow B')(h : A \rightarrow B) = f \circ h : \C(A,B').\] \[ \CC(A,B), \qquad \CC(A,f : B \to B')(h : A \to B) = f \comp h : \CC(A,B').\]
\item Functors between posets are monotonous maps, which in turn are the morphisms in $\emph{Pos}$. \item Functors between posets are monotonous maps, which in turn are the morphisms in $\Pos$.
\item Functors between monoids are monoid homomorphisms, which in turn are the morphisms in $\emph{Mon}$. \item Functors between monoids are monoid homomorphisms, which in turn are the morphisms in $\Mon$.
\item The power set functor $\mathscr{P} : \emph{Set} \rightarrow \emph{Set}$ defined by \item The power set functor $\PSet{} : \Set \to \Set$ defined by
\begin{alignat*}{2} \begin{alignat*}{2}
& \mathscr{P}X & & = \{ Y\;\vert\; Y \subseteq X \} \\ & \PSet{}X & & = \{ Y\;\vert\; Y \subseteq X \} \\
& (\mathscr{P}f)Y & & = f[Y] \subseteq X', \text{ for } Y \subseteq X. & (\PSet{}f)Y & & = f[Y] \subseteq X', \text{ for } Y \subseteq X.
\end{alignat*} \end{alignat*}
\item If $\C$ is a category that adds some structure to sets (like \emph{Mon} or \emph{Pos}) one usually can construct a \emph{forgetful functor} $U : \C \rightarrow \emph{Set}$, e.g. \item If $\CC$ is a category that adds some structure to sets (like $\Mon$ or $\Pos$) one usually can construct a \emph{forgetful functor} $U : \CC \to \Set$, e.g.
\begin{alignat*}{4} \begin{alignat*}{4}
& U_{\emph{Pos}} : \emph{Pos} & & \rightarrow \emph{Set}; \qquad & & (X, \leq) & & \mapsto X \\ & U_{\Pos} : \Pos & & \to \Set; \qquad & & (X, \leq) & & \mapsto X \\
& U_{\emph{Mon}} : \emph{Mon} & & \rightarrow \emph{Set}; \qquad & & (M, \cdot, e) & & \mapsto M & U_{\Mon} : \Mon & & \to \Set; \qquad & & (M, \cdot, e) & & \mapsto M
\end{alignat*} \end{alignat*}
\setcounter{resumeenum}{\value{enumi}} \setcounter{resumeenum}{\value{enumi}}
\end{enumerate} \end{enumerate}
@ -328,34 +329,34 @@ Using functors as morphisms, one can \emph{almost} build a category $\emph{CAT}$
We can however consider structures like products and isomorphisms in the quasi-category $\emph{CAT}$: We can however consider structures like products and isomorphisms in the quasi-category $\emph{CAT}$:
\begin{definition}[Products of Categories] \begin{definition}[Products of Categories]
The product of two categories $\C, \D$ consists of The product of two categories $\CC, \CD$ consists of
\begin{itemize} \begin{itemize}
\item $\obj{\C \times \D} = \obj{\C} \times \obj{\D}$, \item $\obj{\CC \times \CD} = \obj{\CC} \times \obj{\CD}$,
\item $(\C \times \D)((A_1, A_2),(B_1,B_2)) = \C(A_1, B_1) \times D(A_2, B_2)$, \item $(\CC \times \CD)((A_1, A_2),(B_1,B_2)) = \CC(A_1, B_1) \times D(A_2, B_2)$,
\end{itemize} \end{itemize}
with projection functors $\pi_1 : \C \times \D \rightarrow \C, \pi_2 : \C \times \D \rightarrow \D$. with projection functors $\pi_1 : \CC \times \CD \to \CC, \pi_2 : \CC \times \CD \to \CD$.
\end{definition} \end{definition}
\begin{example} More examples of functors include: \begin{example} More examples of functors include:
\begin{enumerate} \setcounter{enumi}{\value{resumeenum}} \begin{enumerate} \setcounter{enumi}{\value{resumeenum}}
\item The Cartesian product functor: $-\times- : \emph{Set} \times \emph{Set} \rightarrow \emph{Set}$. \item The Cartesian product functor: $-\times- : \Set \times \Set \to \Set$.
\item The binary hom-functor $\C(-,-) : \dual{\C} \times \C \rightarrow \emph{Set}$ with \item The binary hom-functor $\CC(\argument,\argument) : \dual{\CC} \times \CC \to \Set$ with
\[\C(A,B), \qquad \C(g : X' \rightarrow X, f : Y \rightarrow Y')(h : X \rightarrow Y) = f \circ h \circ g : \C(X',Y').\] \[\CC(A,B), \qquad \CC(g : X' \to X, f : Y \to Y')(h : X \to Y) = f \comp h \comp g : \CC(X',Y').\]
\setcounter{resumeenum}{\value{enumi}} \setcounter{resumeenum}{\value{enumi}}
\end{enumerate} \end{enumerate}
\end{example} \end{example}
\begin{definition}[Covariant and Contravariant Functors] \begin{definition}[Covariant and Contravariant Functors]
A functor $F : \dual{\C} \rightarrow \D$ is called a \emph{contravariant} functor $\C \rightarrow \D$. For differentiation, we call `normal' functors $\C \rightarrow \D$ \emph{covariant}. A functor $F : \dual{\CC} \to \CD$ is called a \emph{contravariant} functor $\CC \to \CD$. For differentiation, we call `normal' functors $\CC \to \CD$ \emph{covariant}.
\end{definition} \end{definition}
\begin{example} Examples of contravariant functors include: \begin{example} Examples of contravariant functors include:
\begin{enumerate} \setcounter{enumi}{\value{resumeenum}} \begin{enumerate} \setcounter{enumi}{\value{resumeenum}}
\item For every $Y \in \obj{\C}$ there is a contravariant hom-functor $\C(-,Y) : \dual{\C} \rightarrow \emph{Set}$ given by \item For every $Y \in \obj{\CC}$ there is a contravariant hom-functor $\CC(\argument,Y) : \dual{\CC} \to \Set$ given by
\[\C(X,Y), \qquad \C(f : X' \rightarrow X, Y)(h : X \rightarrow Y) = h \circ f : \C(X', Y).\] \[\CC(X,Y), \qquad \CC(f : X' \to X, Y)(h : X \to Y) = h \comp f : \CC(X', Y).\]
\item $2^{(-)} : \dual{\emph{Set}} \rightarrow \emph{Set}$ where \item $2^{(\argument)} : \dual{\Set} \to \Set$ where
\[2^X = \{ f : X \rightarrow 2 \} \cong \mathscr{P}X\] \[2^X = \{ f : X \to 2 \} \iso \PSet{}X\]
and and
\[2^{(f : X \rightarrow Y)} : 2^Y \rightarrow 2^X \cong \mathscr{P}Y \rightarrow \mathscr{P}X, \qquad Z \mapsto \{ x \;\vert\; fx \in Z \} = f^{-1}[Z] \subseteq X.\] \[2^{(f : X \to Y)} : 2^Y \to 2^X \iso \PSet{}Y \to \PSet{}X, \qquad Z \mapsto \{ x \;\vert\; fx \in Z \} = f^{-1}[Z] \subseteq X.\]
\item For every functor $F : \C \rightarrow \D$ the identical functor $\dual{F} : \dual{\C} \rightarrow \dual{\D}$, given by \item For every functor $F : \CC \to \CD$ the identical functor $\dual{F} : \dual{\CC} \to \dual{\CD}$, given by
\[\dual{F}C = FC, \qquad \dual{F}f = Ff.\] \[\dual{F}C = FC, \qquad \dual{F}f = Ff.\]
\end{enumerate} \end{enumerate}
\end{example} \end{example}
@ -363,26 +364,26 @@ We can however consider structures like products and isomorphisms in the quasi-c
Isomorphisms of categories are the isomorphisms in the quasi-category $\emph{CAT}$, thus a functor is an isomorphism iff he is bijective on both objects and morphisms. However, oftentimes categories are not isomorphic but instead \emph{equivalent} in the following sense: Isomorphisms of categories are the isomorphisms in the quasi-category $\emph{CAT}$, thus a functor is an isomorphism iff he is bijective on both objects and morphisms. However, oftentimes categories are not isomorphic but instead \emph{equivalent} in the following sense:
\begin{definition}[Equivalence Functors] \begin{definition}[Equivalence Functors]
A functor $F : \C \rightarrow \D$ is called A functor $F : \CC \to \CD$ is called
\begin{itemize} \begin{itemize}
\item \emph{full} if every $F : \C(A,B) \rightarrow \D(FA,FB)$ is surjective, \item \emph{full} if every $F : \CC(A,B) \to \CD(FA,FB)$ is surjective,
\item \emph{faithful} if every $F : \C(A,B) \rightarrow \D(FA,FB)$ is injective, \item \emph{faithful} if every $F : \CC(A,B) \to \CD(FA,FB)$ is injective,
\item \emph{essentially surjective (dense)} if for every $D \in \D$ there exists a $C \in \C$ such that $D \cong FC$, \item \emph{essentially surjective (dense)} if for every $D \in \CD$ there exists a $C \in \CC$ such that $D \iso FC$,
\item an \emph{equivalence} if $F$ is full, faithful and dense. \item an \emph{equivalence} if $F$ is full, faithful and dense.
\end{itemize} \end{itemize}
\end{definition} \end{definition}
\begin{example} Let us consider two examples of equivalent categories: \begin{example} Let us consider two examples of equivalent categories:
\begin{enumerate} \begin{enumerate}
\item The category $\emph{Par}$ is equivalent to $\emph{Set}_p$, which is the category of pointed sets, where objects are tuples $(X,p), p \in X$ and morphisms are point-preserving. \item The category $\Par$ is equivalent to $\Set_p$, which is the category of pointed sets, where objects are tuples $(X,p), p \in X$ and morphisms are point-preserving.
\item The product category $\emph{Set} \times \emph{Set}$ is equivalent to the \emph{slice category} $\emph{Set}/2$, where objects are maps $X \rightarrow 2$ and morphisms $h : (X \overset{f}{\rightarrow} 2) \rightarrow (Y \overset{g}{\rightarrow} 2)$ are maps $h : X \rightarrow Y$ such that $g \circ h = f$. \item The product category $\Set \times \Set$ is equivalent to the \emph{slice category} $\Set/2$, where objects are maps $X \to 2$ and morphisms $h : (X \overset{f}{\to} 2) \to (Y \overset{g}{\to} 2)$ are maps $h : X \to Y$ such that $g \comp h = f$.
\end{enumerate} \end{enumerate}
\end{example} \end{example}
\section{Natural Transformations} \section{Natural Transformations}
Natural transformation are morphisms between functors. The definition of ``naturality'' was one of the original goals of category theory. Natural transformation are morphisms between functors. The definition of ``naturality'' was one of the original goals of category theory.
\begin{definition}[Natural Transformation] \begin{definition}[Natural Transformation]
Given two functors $F, G : \C \rightarrow \D$. Given two functors $F, G : \CC \to \CD$.
A natural transformation $\alpha : F \rightarrow G$ between these functors is a family of morphisms $(\alpha_C : FC \rightarrow GC)_{C\in\obj{\C}}$, such that for any $f : A \rightarrow B$ the diagram A natural transformation $\alpha : F \to G$ between these functors is a family of morphisms \[{(\alpha_C : FC \to GC)}_{C\in\obj{\CC}},\] such that for any $f : A \to B$ the diagram
% https://q.uiver.app/#q=WzAsNCxbMCwwLCJGQSJdLFsyLDAsIkZCIl0sWzAsMiwiR0EiXSxbMiwyLCJHQiJdLFswLDEsIkZmIl0sWzIsMywiR2YiXSxbMCwyLCJcXGFscGhhX0EiLDJdLFsxLDMsIlxcYWxwaGFfQiJdXQ== % https://q.uiver.app/#q=WzAsNCxbMCwwLCJGQSJdLFsyLDAsIkZCIl0sWzAsMiwiR0EiXSxbMiwyLCJHQiJdLFswLDEsIkZmIl0sWzIsMywiR2YiXSxbMCwyLCJcXGFscGhhX0EiLDJdLFsxLDMsIlxcYWxwaGFfQiJdXQ==
\[ \[
\begin{tikzcd} \begin{tikzcd}
@ -400,7 +401,7 @@ Natural transformation are morphisms between functors. The definition of ``natur
\begin{example} Examples of natural transformations include: \begin{example} Examples of natural transformations include:
\begin{enumerate} \begin{enumerate}
\item The obvious function $flatten : Tree\;A \rightarrow List\;A$: \item The obvious function $flatten : Tree\;A \to List\;A$:
% https://q.uiver.app/#q=WzAsNCxbMCwwLCJUcmVlXFw7QSJdLFsyLDAsIlRyZWVcXDtCIl0sWzAsMiwiTGlzdFxcO0EiXSxbMiwyLCJMaXN0XFw7QiJdLFswLDIsImZsYXR0ZW5fQSIsMl0sWzEsMywiZmxhdHRlbl9CIl0sWzIsMywibGlzdFxcO2YiXSxbMCwxLCJ0cmVlXFw7ZiJdXQ== % https://q.uiver.app/#q=WzAsNCxbMCwwLCJUcmVlXFw7QSJdLFsyLDAsIlRyZWVcXDtCIl0sWzAsMiwiTGlzdFxcO0EiXSxbMiwyLCJMaXN0XFw7QiJdLFswLDIsImZsYXR0ZW5fQSIsMl0sWzEsMywiZmxhdHRlbl9CIl0sWzIsMywibGlzdFxcO2YiXSxbMCwxLCJ0cmVlXFw7ZiJdXQ==
\[ \[
\begin{tikzcd} \begin{tikzcd}
@ -413,18 +414,18 @@ Natural transformation are morphisms between functors. The definition of ``natur
\arrow["{tree\;f}", from=1-1, to=1-3] \arrow["{tree\;f}", from=1-1, to=1-3]
\end{tikzcd} \end{tikzcd}
\] \]
\item For $Id, Q : Set \rightarrow Set$ we have $\delta : Id \rightarrow Q$ given by $\delta_X (x) = (x,x)$. \item For $\Id{}, Q : Set \to Set$ we have $\delta : \Id{} \to Q$ given by $\delta_X (x) = (x,x)$.
\item On $\mathcal{P}$ we can define natural transformations $\eta : Id \rightarrow \mathcal{P}$ and $\mu : \mathcal{P}\mathcal{P} \rightarrow \mathcal{P}$ by: \item On $\mathcal{P}$ we can define natural transformations $\eta : \Id{} \to \mathcal{P}$ and $\mu : \mathcal{P}\mathcal{P} \to \mathcal{P}$ by:
\begin{alignat*}{1} \begin{alignat*}{1}
\eta_X : X & \rightarrow \mathcal{P}X \\ \eta_X : X & \to \mathcal{P}X \\
x & \mapsto \{x\} x & \mapsto \{x\}
\end{alignat*} \end{alignat*}
and and
\begin{alignat*}{1} \begin{alignat*}{1}
\mu_X : \mathcal{P}\mathcal{P}X & \rightarrow \mathcal{P}X \\ \mu_X : \mathcal{P}\mathcal{P}X & \to \mathcal{P}X \\
Z & \mapsto \bigcup Z. Z & \mapsto \bigcup Z.
\end{alignat*} \end{alignat*}
\item Between $Q$ and $\mathcal{P}$ we can consider $\alpha,\beta : Q \rightarrow \mathcal{P}$ given by \item Between $Q$ and $\mathcal{P}$ we can consider $\alpha,\beta : Q \to \mathcal{P}$ given by
\begin{alignat*}{2} \begin{alignat*}{2}
& \alpha_X(x,y) & & = \{x,y\} \\ & \alpha_X(x,y) & & = \{x,y\} \\
& \beta_X(x,y) & & = \{x\}. & \beta_X(x,y) & & = \{x\}.
@ -432,12 +433,12 @@ Natural transformation are morphisms between functors. The definition of ``natur
\end{enumerate} \end{enumerate}
\end{example} \end{example}
Functors $\C \rightarrow \D$ together with natural transformations as morphisms form a quasi-category $[\C,\D]$, that is called the functor category. If $\C$ is small, then $[\C,\D]$ is a category, where identity and composition are defined component wise. Functors $\CC \to \CD$ together with natural transformations as morphisms form a quasi-category $[\CC,\CD]$, that is called the functor category. If $\CC$ is small, then $[\CC,\CD]$ is a category, where identity and composition are defined component wise.
\begin{example} Let us examine concrete examples of functor categories: \begin{example} Let us examine concrete examples of functor categories:
\begin{enumerate} \begin{enumerate}
\item $[2, \C] \cong \C \times \C$, where $2$ is the \emph{discrete} category with two objects, i.e.\ $2$ has no morphisms besides the identities. \item $[2, \CC] \iso \CC \times \CC$, where $2$ is the \emph{discrete} category with two objects, i.e.\ $2$ has no morphisms besides the identities.
\item Let $\rightarrow$ be the category with 2 objects and a single non-trivial morphism $m$. $[\rightarrow, \C$ is the \emph{category of morphisms} of $\C$, where morphisms $Fm \rightarrow Gm$ are pairs of morphisms $(f,g)$ where \item Let $\to$ be the category with 2 objects and a single non-trivial morphism $m$. $[\to, \CC]$ is the \emph{category of morphisms} of $\CC$, where morphisms $Fm \to Gm$ are pairs of morphisms $(f,g)$ where
% https://q.uiver.app/#q=WzAsNCxbMCwwLCJGMCJdLFsyLDAsIkYxIl0sWzAsMiwiRzAiXSxbMiwyLCJHMSJdLFswLDEsIkZtIl0sWzIsMywiR20iXSxbMCwyLCJmIiwyXSxbMSwzLCJnIl1d % https://q.uiver.app/#q=WzAsNCxbMCwwLCJGMCJdLFsyLDAsIkYxIl0sWzAsMiwiRzAiXSxbMiwyLCJHMSJdLFswLDEsIkZtIl0sWzIsMywiR20iXSxbMCwyLCJmIiwyXSxbMSwzLCJnIl1d
\[ \[
\begin{tikzcd} \begin{tikzcd}
@ -455,46 +456,305 @@ Functors $\C \rightarrow \D$ together with natural transformations as morphisms
\end{example} \end{example}
\begin{definition}[Natural Isomorphism] \begin{definition}[Natural Isomorphism]
Isomorphisms in $[\C,\D]$ are called \emph{natural isomorphisms}. Isomorphisms in $[\CC,\CD]$ are called \emph{natural isomorphisms}.
\end{definition} \end{definition}
\begin{proposition} \begin{proposition}
$\alpha : F \rightarrow G$ is a natural isomorphism \emph{iff} every $\alpha_C$ is an isomorphism. $\alpha : F \to G$ is a natural isomorphism \emph{iff} every $\alpha_C$ is an isomorphism.
\end{proposition} \end{proposition}
\begin{example} Let us consider some examples of natural isomorphisms: \begin{example} Let us consider some examples of natural isomorphisms:
\begin{enumerate} \begin{enumerate}
\item In $[\emph{Set},\emph{Set}]$ is $Id \cong \emph{Set}(1,-)$, since of course $Id\;X = X \cong X^1 = \emph{Set}(1,X)$. \item In $[\Set,\Set]$ is $\Id{} \iso \Set(1,\argument)$, since of course $\Id{}\;X = X \iso X^1 = \Set(1,X)$.
\item Also in $[\emph{Set},\emph{Set}]$ is $Q \cong \emph{Set}(2,-)$, similarly is $\lambda X.2\times X \cong \lambda X. X + X$. \item Also in $[\Set,\Set]$ is $Q \iso \Set(2,\argument)$, similarly is $\lambda X.2\times X \iso \lambda X. X + X$.
\item The forgetful functor $U : \emph{Pos} \rightarrow \emph{Set}$ is naturally isomorphic to $\emph{Pos}(1,-)$, because the constant mapping $x : 1 \rightarrow X$ is monotonous for every element $x$ of a poset. \item The forgetful functor $U : \Pos \to \Set$ is naturally isomorphic to $\Pos(1,\argument)$, because the constant mapping $x : 1 \to X$ is monotonous for every element $x$ of a poset.
\end{enumerate} \end{enumerate}
\end{example} \end{example}
\begin{proposition}[Yoneda Lemma] \begin{proposition}[Yoneda Lemma]
Let $A \in \obj{\C}$ and $G : \C \rightarrow \emph{Set}$. Then the natural transformations Let $A \in \obj{\CC}$ and $G : \CC \to \Set$. Then the natural transformations
\[\C(A,-) \rightarrow G\] \[\CC(A,\argument) \to G\]
are in bijection with the elements of the set $GA$. are in bijection with the elements of the set $GA$.
In other words
\[[\CC , \Set ](\CC(A,\argument), G) \iso GA \]
\end{proposition} \end{proposition}
\begin{proof} \begin{proof}
The mappings are The mappings are
\begin{alignat*}{1} \begin{alignat*}{1}
& Z : GA \rightarrow [\C , \emph{Set}](\C(A,-), G) \\ & Z : GA \to [\CC , \Set](\CC(A,\argument), G) \\
& Z\;x\;h = G\;h\;x & Z\;x\;h = G\;h\;x
\end{alignat*} \end{alignat*}
and and
\begin{alignat*}{1} \begin{alignat*}{1}
& Y : [\C , \emph{Set}](\C(A,-), G) \rightarrow GA \\ & Y : [\CC , \Set](\CC(A,\argument), G) \to GA \\
& Y\;\alpha = \alpha_A\;id_A & Y\;\alpha = \alpha_A\;id_A.
\end{alignat*} \end{alignat*}
We are left to check naturality of $Z\;x$ and that indeed $Z$ and $Y$ are inverse to each other, all of which follows by routine rewriting.
\end{proof} \end{proof}
\begin{example} \begin{example}
Let us consider an application of the Yoneda Lemma: how many natural transformations $Id \rightarrow Q$ are there? Let us consider an application of the Yoneda Lemma: how many natural transformations $\Id{} \to Q$ are there?
Recall that $Id \cong \emph{Set}(1,-)$, and by Yoneda there is exactly $\vert Q1 \vert = 1$ natural transformation $\emph{Set}(1,-) \rightarrow Q$, thus the number of natural transformations $Id \rightarrow Q$ is $1$. Recall that $\Id{} \iso \Set(1,\argument)$, and by Yoneda there is exactly $\vert Q1 \vert = 1$ natural transformation $\Set(1,\argument) \to Q$, thus the number of natural transformations $\Id{} \to Q$ is $1$.
Furthermore, consider the number of natural transformations $Q \rightarrow Q$. Recall that $Q \cong \emph{Set}(2, -)$, and by Yoneda there are $\vert Q2 \vert = 4$ natural transformations $\emph{Set}(2, -) \rightarrow Q$, thus the number of natural transformations $Q \rightarrow Q$ is $4$. Furthermore, consider the number of natural transformations $Q \to Q$. Recall that $Q \iso \Set(2, \argument)$, and by Yoneda there are $\vert Q2 \vert = 4$ natural transformations $\Set(2, \argument) \to Q$, thus the number of natural transformations $Q \to Q$ is $4$.
\end{example} \end{example}
\section{Functor Algebras} \section{Functor Algebras}
Recall the fold functions that we introduced in \autoref{chp:introduction} in the category $\Set$:
\begin{alignat*}{5}
& & foldn : \; & (1 \to C) \to (C \to C) & & \to Nat & & \to C \\
& & foldr : \; & (1 \to C) \to (A \times C \to C) & & \to List\;A & & \to C
\end{alignat*}
These are examples of special \emph{F-algebras} in $\Set$. In this section we will introduce this notion and examine what makes the fold functions special.
\begin{definition}[F-Algebras]
Let $F : \CC \to \CC$ be an endofunctor on $\CC$. An \emph{F-algebra} is a pair $(A \in \obj{\CC}, a : FA \to a)$. Homomorphisms between F-algebras $(A,a)$ and $(B,b)$ are morphisms $f : A \to B$ such that
% https://q.uiver.app/#q=WzAsNCxbMCwwLCJGQSJdLFswLDIsIkZCIl0sWzIsMCwiQSJdLFsyLDIsIkIiXSxbMSwzLCJiIl0sWzAsMiwiYSJdLFswLDEsIkZmIiwyXSxbMiwzLCJmIl1d
\[
\begin{tikzcd}
FA && A \\
\\
FB && B
\arrow["b", from=3-1, to=3-3]
\arrow["a", from=1-1, to=1-3]
\arrow["Ff"', from=1-1, to=3-1]
\arrow["f", from=1-3, to=3-3]
\end{tikzcd}
\]
commutes.
\end{definition}
\begin{proposition}
F-algebras together with their homomorphisms form a category that we call $\Alg{F}$.
\end{proposition}
\begin{proof}
Identities and composition are inherited by the underlying category $\CC$. We are left to show that the identities are homomorphisms:
% https://q.uiver.app/#q=WzAsNSxbMCwwLCJGQSJdLFswLDIsIkZBIl0sWzIsMCwiQSJdLFsyLDIsIkEiXSxbMSwxLCJcXGNvbW0iXSxbMSwzLCJhIl0sWzAsMiwiYSJdLFswLDEsIkZpZCIsMl0sWzIsMywiaWQiXV0=
\[
\begin{tikzcd}
FA && A \\
& \comm \\
FA && A
\arrow["a", from=3-1, to=3-3]
\arrow["a", from=1-1, to=1-3]
\arrow["Fid"', from=1-1, to=3-1]
\arrow["id", from=1-3, to=3-3]
\end{tikzcd}
\]
and that homomorphisms are closed under composition:
% https://q.uiver.app/#q=WzAsOCxbMCwwLCJGQSJdLFswLDIsIkZCIl0sWzIsMCwiQSJdLFsyLDIsIkIiXSxbMSwxLCJcXGNvbW0iXSxbMCw0LCJGQyJdLFsyLDQsIkMiXSxbMSwzLCJcXGNvbW0iXSxbMSwzLCJiIl0sWzAsMiwiYSJdLFswLDEsIkZmIiwyXSxbMiwzLCJmIl0sWzUsNiwiYyJdLFsxLDUsIkZnIiwyXSxbMyw2LCJnIl0sWzAsNSwiRihnIFxcY2lyYyBmKSIsMix7ImN1cnZlIjo0fV0sWzIsNiwiZyBcXGNpcmMgZiIsMCx7ImN1cnZlIjotNH1dXQ==
\[
\begin{tikzcd}
FA && A \\
& \comm \\
FB && B \\
& \comm \\
FC && C
\arrow["b", from=3-1, to=3-3]
\arrow["a", from=1-1, to=1-3]
\arrow["Ff"', from=1-1, to=3-1]
\arrow["f", from=1-3, to=3-3]
\arrow["c", from=5-1, to=5-3]
\arrow["Fg"', from=3-1, to=5-1]
\arrow["g", from=3-3, to=5-3]
\arrow["{F(g \circ f)}"', curve={height=24pt}, from=1-1, to=5-1]
\arrow["{g \circ f}", curve={height=-24pt}, from=1-3, to=5-3]
\end{tikzcd}
\]
\end{proof}
\begin{example}\label{ex:Falg} Let us now consider the structure of the data types Nat and List as F-algebras:
\begin{enumerate}
\item \textbf{Nat}: Take $\CC = \Set$ and $FX = 1 + X$, the F-algebras and their morphisms have the following form:
% https://q.uiver.app/#q=WzAsNCxbMCwwLCIxICsgQSJdLFsyLDAsIkEiXSxbMCwyLCIxICsgQiJdLFsyLDIsIkIiXSxbMCwxLCJbYyxoXSJdLFsyLDMsIltjJywgaCddIl0sWzAsMiwiXFxiYW5nICsgZiIsMl0sWzEsMywiZiJdXQ==
\[
\begin{tikzcd}
{1 + A} && A \\
\\
{1 + B} && B
\arrow["{[c,h]}", from=1-1, to=1-3]
\arrow["{[c', h']}", from=3-1, to=3-3]
\arrow["{\bang + f}"', from=1-1, to=3-1]
\arrow["f", from=1-3, to=3-3]
\end{tikzcd}
\]
Which is equivalent to:
% https://q.uiver.app/#q=WzAsNSxbMiwwLCJBIl0sWzQsMCwiQSJdLFsyLDIsIkIiXSxbNCwyLCJCIl0sWzAsMCwiMSJdLFs0LDAsImMiXSxbMCwxLCJoIl0sWzIsMywiaCciXSxbNCwyLCJjJyIsMl0sWzAsMiwiZiIsMl0sWzEsMywiZiJdXQ==
\[
\begin{tikzcd}
1 && A && A \\
\\
&& B && B
\arrow["c", from=1-1, to=1-3]
\arrow["h", from=1-3, to=1-5]
\arrow["{h'}", from=3-3, to=3-5]
\arrow["{c'}"', from=1-1, to=3-3]
\arrow["f"', from=1-3, to=3-3]
\arrow["f", from=1-5, to=3-5]
\end{tikzcd}
\]
\item \textbf{List\;A}: Take $\CC = \Set$ and $FX = 1 + A \times X$, where $A \in \obj{\Set}$. The F-algebras and their morphisms take the following form:
% https://q.uiver.app/#q=WzAsNCxbMCwwLCIxICsgQSBcXHRpbWVzIFgiXSxbMCwyLCIxICsgQSBcXHRpbWVzIFkiXSxbMiwwLCJYIl0sWzIsMiwiWSJdLFswLDIsIltjLGhdIl0sWzEsMywiW2MsaF0iXSxbMiwzLCJmIl0sWzAsMSwiXFxiYW5nICsgaWQgXFx0aW1lcyBmIiwyXV0=
\[
\begin{tikzcd}
{1 + A \times X} && X \\
\\
{1 + A \times Y} && Y
\arrow["{[c,h]}", from=1-1, to=1-3]
\arrow["{[c,h]}", from=3-1, to=3-3]
\arrow["f", from=1-3, to=3-3]
\arrow["{\bang + id \times f}"', from=1-1, to=3-1]
\end{tikzcd}
\]
Which again is equivalent to
% https://q.uiver.app/#q=WzAsNSxbMiwwLCJBIFxcdGltZXMgWCJdLFsyLDIsIkEgXFx0aW1lcyBZIl0sWzQsMCwiWCJdLFs0LDIsIlkiXSxbMCwwLCIxIl0sWzAsMiwiaCJdLFsxLDMsImgiXSxbMiwzLCJmIl0sWzAsMSwiaWQgXFx0aW1lcyBmIiwyXSxbNCwwLCJjIl0sWzQsMSwiYyciLDJdXQ==
\[
\begin{tikzcd}
1 && {A \times X} && X \\
\\
&& {A \times Y} && Y
\arrow["h", from=1-3, to=1-5]
\arrow["h", from=3-3, to=3-5]
\arrow["f", from=1-5, to=3-5]
\arrow["{id \times f}"', from=1-3, to=3-3]
\arrow["c", from=1-1, to=1-3]
\arrow["{c'}"', from=1-1, to=3-3]
\end{tikzcd}
\]
\end{enumerate}
\end{example}
\subsection{Initial F-algebras}
\emph{Initial F-algebras} (i.e.\ the initial object in $\Alg{F}$) are of special interest to us. More concretely an F-algebra $(I, i)$ is initial if for every $(A, a)$ there exists a unique $\cata{a} : I \to A$ such that
% https://q.uiver.app/#q=WzAsNCxbMCwwLCJGSSJdLFsyLDAsIkkiXSxbMCwyLCJGQSJdLFsyLDIsIkEiXSxbMCwxLCJpIl0sWzAsMiwiRlxcY2F0YXthfSIsMl0sWzEsMywiXFxleGlzdHMhXFxjYXRhe2F9IiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzIsMywiYSIsMl1d
\[
\begin{tikzcd}
FI && I \\
\\
FA && A
\arrow["i", from=1-1, to=1-3]
\arrow["{F\cata{a}}"', from=1-1, to=3-1]
\arrow["{\exists!\cata{a}}", dashed, from=1-3, to=3-3]
\arrow["a"', from=3-1, to=3-3]
\end{tikzcd}
\]
commutes.
The dual notion of \emph{terminal F-algebra} is usually not of interested, since it is just inherited from $\CC$:
% https://q.uiver.app/#q=WzAsNCxbMCwwLCJGQSJdLFsyLDAsIkEiXSxbMCwyLCJGMSJdLFsyLDIsIjEiXSxbMiwzLCIhIl0sWzAsMSwiYSJdLFswLDIsIkYhIiwyXSxbMSwzLCIhIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d
\[
\begin{tikzcd}
FA && A \\
\\
F1 && 1
\arrow["{!}", from=3-1, to=3-3]
\arrow["a", from=1-1, to=1-3]
\arrow["{F!}"', from=1-1, to=3-1]
\arrow["{!}", dashed, from=1-3, to=3-3]
\end{tikzcd}
\]
\begin{example}
Important examples of initial F-algebras include:
\begin{enumerate}
\item In \autoref{ex:Falg} (1) the data type $Nat$ is the initial algebra together with the function $foldn$ that we defined in the introduction. Where the following diagram expresses the defining equations for $foldn$:
% https://q.uiver.app/#q=WzAsNCxbMCwwLCIxICsgTmF0Il0sWzAsMiwiMSArIEMiXSxbMiwwLCJOYXQiXSxbMiwyLCJDIl0sWzAsMSwiISArZm9sZG4oYyxoKSIsMl0sWzAsMiwiW3plcm8sIHN1Y2NdIl0sWzIsMywiZm9sZG4oYyxoKSJdLFsxLDMsIltjLGhdIiwyXV0=
\[
\begin{tikzcd}
{1 + Nat} && Nat \\
\\
{1 + C} && C
\arrow["{id +foldn(c,h)}"', from=1-1, to=3-1]
\arrow["{[zero, succ]}", from=1-1, to=1-3]
\arrow["{foldn(c,h)}", from=1-3, to=3-3]
\arrow["{[c,h]}"', from=3-1, to=3-3]
\end{tikzcd}
\]
\item Similarly, in \autoref{ex:Falg} (2) the data type $List\;A$ is the initial algebra:
% https://q.uiver.app/#q=WzAsNCxbMCwwLCIxICsgQSBcXHRpbWVzIExpc3RcXDtBIl0sWzIsMCwiTGlzdFxcO0EiXSxbMCwyLCIxICsgQSBcXHRpbWVzIEMiXSxbMiwyLCJDIl0sWzAsMSwiW25pbCxjb25zXSJdLFsyLDMsIltjLGhdIl0sWzEsMywiZm9sZHIoYyxoKSJdLFswLDIsImlkICsgZm9sZHIoYyxoKSIsMl1d
\[
\begin{tikzcd}
{1 + A \times List\;A} && {List\;A} \\
\\
{1 + A \times C} && C
\arrow["{[nil,cons]}", from=1-1, to=1-3]
\arrow["{[c,h]}", from=3-1, to=3-3]
\arrow["{foldr(c,h)}", from=1-3, to=3-3]
\arrow["{id + foldr(c,h)}"', from=1-1, to=3-1]
\end{tikzcd}
\]
% TODO add other examples
\end{enumerate}
\end{example}
We can now abstract the fusion and identity laws that we defined for each data type in \autoref{sec:datatypes}:
\begin{proposition}
Let $(I,i)$ be an initial F-algebra. The following holds:
\begin{enumerate}
\item \customlabel{law:ident}{\textbf{Identity}}: $\cata{i} = id_I : I \to I$,
\item \customlabel{law:fusion}{\textbf{Fusion}}: Let $f : (A,a) \to (B,b)$ be a homomorphism between F-algebras, then
% https://q.uiver.app/#q=WzAsMyxbMCwwLCJJIl0sWzIsMCwiQSJdLFsyLDIsIkIiXSxbMCwxLCJcXGNhdGF7YX0iXSxbMSwyLCJmIl0sWzAsMiwiXFxjYXRhe2J9IiwyXV0=
\[
\begin{tikzcd}
I && A \\
\\
&& B
\arrow["{\cata{a}}", from=1-1, to=1-3]
\arrow["f", from=1-3, to=3-3]
\arrow["{\cata{b}}"', from=1-1, to=3-3]
\end{tikzcd}
\]
commutes.
\end{enumerate}
\end{proposition}
\begin{proof} Both follow by uniqueness of homomorphisms out of the initial object:
\begin{enumerate}
\item By uniqueness of homomorphisms $(I,i) \to (I,i)$
\item By uniqueness of homomorphisms $(I,i) \to (B,b)$
\end{enumerate}
\end{proof}
\begin{proposition}[Lambeks Lemma]
Let $(I,i)$ be an initial F-algebra. The F-algebra structure $i$ is an isomorphism.
\end{proposition}
\begin{proof}
Applying $F$ on $i$ yields another F-algebra $(FI, Fi)$, which induces a homomorphism $\cata{Fi} : I \rightarrow FI$. $\cata{FI}$ is the inverse to $i$. Consider
% https://q.uiver.app/#q=WzAsOCxbMCwwLCJGSSJdLFsyLDAsIkkiXSxbMiwyLCJGSSJdLFswLDIsIkZGSSJdLFsxLDEsIlxcY29tbSJdLFswLDQsIkZJIl0sWzIsNCwiSSJdLFsxLDMsIlxcY29tbSJdLFswLDEsImkiXSxbMywyLCJGaSJdLFswLDMsIkZcXGNhdGF7Rml9Il0sWzEsMiwiXFxjYXRhe0ZpfSIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFs1LDYsImkiXSxbMyw1LCJGaSJdLFsyLDYsImkiLDJdLFswLDUsIkYoaSBcXGNvbXAgXFxjYXRhe0ZpfSkiLDIseyJjdXJ2ZSI6M31dLFsxLDYsImkgXFxjb21wIFxcY2F0YXtGaX0iLDAseyJjdXJ2ZSI6LTN9XV0=
\[
\begin{tikzcd}
FI && I \\
& \comm \\
FFI && FI \\
& \comm \\
FI && I
\arrow["i", from=1-1, to=1-3]
\arrow["Fi", from=3-1, to=3-3]
\arrow["{F\cata{Fi}}", from=1-1, to=3-1]
\arrow["{\cata{Fi}}"', dashed, from=1-3, to=3-3]
\arrow["i", from=5-1, to=5-3]
\arrow["Fi", from=3-1, to=5-1]
\arrow["i"', from=3-3, to=5-3]
\arrow["{F(i \comp \cata{Fi})}"', curve={height=18pt}, from=1-1, to=5-1]
\arrow["{i \comp \cata{Fi}}", curve={height=-18pt}, from=1-3, to=5-3]
\end{tikzcd}
\]
from which we can follow that $i \comp \cata{Fi} = id_I : (I,i) \to (I,i)$ by uniqueness of the homomorphisms and thus also
\[\cata{Fi} \comp i = Fi \comp F\cata{Fi} = F(i \comp \cata{Fi}) = F {id}_I = {id}_{FI}.\]
\end{proof}
\subsection{Term Algebras}
% TODO
\subsection{Parametric Data Types}
% TODO
\section{Functor Coalgebras} \section{Functor Coalgebras}
\section{(co)Limits} % chktex 36 % TODO coalgebras
\subsection{Terminal F-Coalgebras}
\subsection{Corecursion and Coinduction}
\section{Limits}
\section{Colimits}