From f537b830a731733d9493a7f31256eda8db59eba0 Mon Sep 17 00:00:00 2001 From: Leon Vatthauer Date: Sun, 24 Mar 2024 13:50:49 +0100 Subject: [PATCH] Work on categorical prelims, added quiver.sty --- tex/.chktexrc | 4 + tex/.vscode/ltex.dictionary.en-US.txt | 5 + .../ltex.hiddenFalsePositives.en-US.txt | 3 + tex/.vscode/settings.json | 7 +- tex/main.pdf | Bin 85706 -> 106047 bytes tex/main.tex | 37 +-- tex/quiver.sty | 40 ++++ tex/sections/01_introduction.tex | 11 +- tex/sections/02_categories.tex | 225 +++++++++++++++--- tex/sections/03_constructions.tex | 8 +- 10 files changed, 283 insertions(+), 57 deletions(-) create mode 100644 tex/.chktexrc create mode 100644 tex/quiver.sty diff --git a/tex/.chktexrc b/tex/.chktexrc new file mode 100644 index 0000000..692aa0f --- /dev/null +++ b/tex/.chktexrc @@ -0,0 +1,4 @@ +CmdLine = { + -n18 + -n46 +} diff --git a/tex/.vscode/ltex.dictionary.en-US.txt b/tex/.vscode/ltex.dictionary.en-US.txt index ba9cbdd..3c1488d 100644 --- a/tex/.vscode/ltex.dictionary.en-US.txt +++ b/tex/.vscode/ltex.dictionary.en-US.txt @@ -27,3 +27,8 @@ Epimorphisms Isomorphisms cancellative Coproducts +poset +iff +posets +coproduct +monoid diff --git a/tex/.vscode/ltex.hiddenFalsePositives.en-US.txt b/tex/.vscode/ltex.hiddenFalsePositives.en-US.txt index d1b8528..cb4087c 100644 --- a/tex/.vscode/ltex.hiddenFalsePositives.en-US.txt +++ b/tex/.vscode/ltex.hiddenFalsePositives.en-US.txt @@ -5,3 +5,6 @@ {"rule":"MORFOLOGIK_RULE_EN_US","sentence":"^\\QFriedrich-Alexander-Universität Erlangen-Nürnberg\\E$"} {"rule":"COMMA_PARENTHESIS_WHITESPACE","sentence":"^\\QThis is a summary of the course “Algebra des Programmierens” taught by Prof. Dr. Stefan Milius in the winter term 2023/2024 at the FAU .\\E$"} {"rule":"MORFOLOGIK_RULE_EN_US","sentence":"^\\Q\\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q satisfies the following two rules law:natident Identity law:natfusion Fusion Lists.\\E$"} +{"rule":"COMMA_PARENTHESIS_WHITESPACE","sentence":"^\\Q[cases] mycase Case .\\E$"} +{"rule":"MORFOLOGIK_RULE_EN_US","sentence":"^\\QIt seems that these proofs should somehow be constructable from each other\\E$"} +{"rule":"ENGLISH_WORD_REPEAT_RULE","sentence":"^\\QEvery categorical notion can thus be dualized by viewing it in the dual category, some examples include: Notion Dual Notion Initial Object Terminal Object Monomorphism Epimorphism Isomorphism Isomorphism\\E$"} diff --git a/tex/.vscode/settings.json b/tex/.vscode/settings.json index fdce008..3659725 100644 --- a/tex/.vscode/settings.json +++ b/tex/.vscode/settings.json @@ -23,5 +23,10 @@ "latexmk-main" ] } - ] + ], + "ltex.latex.commands": { + "\\customlabel{}": "ignore", + "\\setminted[]{}": "ignore", + "\\setmintedinline[]{}": "ignore" + } } \ No newline at end of file diff --git a/tex/main.pdf b/tex/main.pdf index dcf70af5292416d95abab63341722260bbfac2ba..94387b59b2b9dc9f724ceca9ebf520cd9b017026 100644 GIT binary patch delta 79872 zcmb5VLy#t1&@|e1_q3;N+fSR*n6_=(*3-6a+qP}nw$17L#*Mr9zxe*m-JMO(fjHe_!XR2%@v!Osup|sW->0lV z)z{Z-h#qX=%Y%9!OodL;Tp>)xO|#A$Z?NhxmsAyLZTudpk+)`p1<1~sH1|c3&bW+D0u<~ap zHZ!IE&LLUCw$z?S>h_c%Cxa?Qna)wL{F8XrC-E!I$Y^P}z0m1%*V6l;(lk*UaKt~M zsz>Mx!VVJ^4rgL({QnK@zn7h%B^-0&zBPp?=l?lOB`RemWhM||5lAOeNGDb)K6t6% zBkept9^_KABJg&QdLY8%|8B&_oa&DUN)BWKF#Vs+t}`ZWH<^${U%%nL{nq#G-@w2a z;hG>jX!msHgRjYST;_jvlSso$Bs6}NFrbnO>j$NfJCYIT>N+lcSr~)w%W+`hT6Nj? 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z-0aaZJ6m~LN@{4ozI3d*;GxM}UYS|C=HU2EwaE3iNA&DvmA_`WufDL6=kF7Yg#c#p z>neT0H_(H2Swyqr?yc4-lxEz=2c5(`cb|V^jzv?0c8uOB@C${*aVarqwzbn%uf-#B xR%RinT=YZ7ON;C${WS*c-%+Cf`@eB>H*t0McD1lX7U1C*UlTnpL{$DoFT!a7s diff --git a/tex/main.tex b/tex/main.tex index fe024de..0981baa 100644 --- a/tex/main.tex +++ b/tex/main.tex @@ -1,9 +1,8 @@ -\documentclass[a4paper,11pt,titlepage]{scrartcl} +\documentclass[a4paper,11pt,titlepage,numbers=noenddot]{scrbook} %%%% Document Setup \usepackage[top=2cm,lmargin=1in,rmargin=1in,bottom=3cm,hmarginratio=1:1]{geometry} -\usepackage{scrlayer-fancyhdr} -\usepackage{extramarks} +\usepackage{scrlayer-scrpage} \usepackage[ngerman, english]{babel} \babeltags{german=ngerman} \usepackage{anyfontsize} @@ -13,14 +12,12 @@ \usepackage{booktabs} \usepackage[scale=.85]{noto-mono} % TODO find better unicode mono font \usepackage[final]{hyperref} -\pagestyle{fancy} -\lhead{\DocTitle} -\chead{\UniCourse} -\rhead{\firstxmark} -\lfoot{\lastxmark} -\cfoot{\thepage} -\renewcommand\headrulewidth{0.4pt} -\renewcommand\footrulewidth{0.4pt} +\usepackage{scrhack} +\automark{chapter} +\renewcommand{\sectionmark}[1]{\markright{\thesection\ #1}} +\renewcommand{\subsectionmark}[1]{\markright{\thesubsection\ #1}} +\pagestyle{scrheadings} + %%%% %%%% Metadata @@ -51,7 +48,13 @@ \setmintedinline[haskell]{ style=bw } +%%%% +%%%% Spacing +\setlength{\parindent}{0pt} +\setlength{\parskip}{6pt} +\setlength{\marginparsep}{0cm} +\AtBeginEnvironment{minted}{\setlength{\parskip}{0pt}} %%%% %%%% Math packages @@ -59,7 +62,8 @@ \usepackage{thmtools} \usepackage{tikz} \usetikzlibrary{cd, babel, quotes} -\declaretheorem[name=Definition,style=definition,numberwithin=section]{definition} +\usepackage{quiver} +\declaretheorem[name=Definition,style=definition,numberwithin=chapter]{definition} \declaretheorem[name=Example,style=definition,sibling=definition]{example} \declaretheorem[style=definition,numbered=no]{exercise} \declaretheorem[name=Remark,style=definition,sibling=definition]{remark} @@ -76,7 +80,7 @@ \makeatletter \hypersetup{ pdfauthor={\@author}, - pdftitle={\@title}, + pdftitle={\UniCourse}, % kill those ugly red rectangles around links hidelinks, } @@ -127,20 +131,23 @@ %%% Notation %% Category C \newcommand{\C}{\ensuremath{\mathscr{C}}} +%% Objects of category \newcommand{\obj}[1]{\ensuremath{\vert #1 \vert}} - +%% Dual category +\newcommand{\dual}[1]{\ensuremath{#1^{op}}} %%% %%%% \begin{document} %% Titlepage \maketitle +\thispagestyle{empty} +\null\newpage \setcounter{page}{1} %% %% TOC \tableofcontents -\pagebreak %% %% Contents diff --git a/tex/quiver.sty b/tex/quiver.sty new file mode 100644 index 0000000..b0b4e06 --- /dev/null +++ b/tex/quiver.sty @@ -0,0 +1,40 @@ +% *** quiver *** +% A package for drawing commutative diagrams exported from https://q.uiver.app. +% +% This package is currently a wrapper around the `tikz-cd` package, importing necessary TikZ +% libraries, and defining a new TikZ style for curves of a fixed height. +% +% Version: 1.4.2 +% Authors: +% - varkor (https://github.com/varkor) +% - AndréC (https://tex.stackexchange.com/users/138900/andr%C3%A9c) + +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{quiver}[2021/01/11 quiver] + +% `tikz-cd` is necessary to draw commutative diagrams. +\RequirePackage{tikz-cd} +% `amssymb` is necessary for `\lrcorner` and `\ulcorner`. +% \RequirePackage{amssymb} +% `calc` is necessary to draw curved arrows. +\usetikzlibrary{calc} +% `pathmorphing` is necessary to draw squiggly arrows. +\usetikzlibrary{decorations.pathmorphing} + +% A TikZ style for curved arrows of a fixed height, due to AndréC. +\tikzset{curve/.style={settings={#1},to path={(\tikztostart) + .. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$) + and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$) + .. (\tikztotarget)\tikztonodes}}, + settings/.code={\tikzset{quiver/.cd,#1} + \def\pv##1{\pgfkeysvalueof{/tikz/quiver/##1}}}, + quiver/.cd,pos/.initial=0.35,height/.initial=0} + +% TikZ arrowhead/tail styles. +\tikzset{tail reversed/.code={\pgfsetarrowsstart{tikzcd to}}} +\tikzset{2tail/.code={\pgfsetarrowsstart{Implies[reversed]}}} +\tikzset{2tail reversed/.code={\pgfsetarrowsstart{Implies}}} +% TikZ arrow styles. +\tikzset{no body/.style={/tikz/dash pattern=on 0 off 1mm}} + +\endinput diff --git a/tex/sections/01_introduction.tex b/tex/sections/01_introduction.tex index 6c76159..f61dffc 100644 --- a/tex/sections/01_introduction.tex +++ b/tex/sections/01_introduction.tex @@ -1,11 +1,10 @@ -% chktex-file 46 -\section{Introduction} +\chapter{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}. 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. -\subsection{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 $f$). More concretely $f$ is a relation $f \subseteq X \times Y$ which is \begin{itemize} @@ -32,7 +31,7 @@ Often, one is also interested in the symmetrical properties, a function is calle \end{enumerate} \end{example} -\subsection{Data Types} +\section{Data Types} 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}. @@ -54,7 +53,7 @@ These data types are declared by means of constructors, yielding concrete descri Such data types (parametric or non-parametric) usually come with a principle for defining functions called recursion and in richer type systems (e.g.\ in a dependently typed setting) with a principle for proving facts about recursive functions called induction. 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. -\subsubsection{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 \begin{alignat*}{2} & foldn\;c\;h\;zero & & = c \\ @@ -182,7 +181,7 @@ The type of natural numbers comes with a fold function $foldn : C \rightarrow (N \end{tikzcd}\] trivially commutes. \end{example} -\subsubsection{Lists} +\subsection{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 \begin{alignat*}{2} & foldr\;c\;h\;nil & & = c \\ diff --git a/tex/sections/02_categories.tex b/tex/sections/02_categories.tex index 8dffa83..7794157 100644 --- a/tex/sections/02_categories.tex +++ b/tex/sections/02_categories.tex @@ -1,10 +1,9 @@ -% chktex-file 46 -\section{Category Theory} +\chapter{Category Theory} Categories consist of objects and morphisms between those objects, that can be composed in a coherent way. -This yields a framework for abstracting many mathematical concepts for reasoning 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} A category \C consists of +\begin{definition}[Category] A category \C consists of \begin{itemize} \item a class of objects denoted $\obj{\C}$, \item for every pair of objects $A,B \in \obj{\C}$ a set of morphisms $\C(A,B)$ called the hom-set, @@ -28,13 +27,13 @@ This yields a framework for abstracting many mathematical concepts for reasoning \end{tabular} \end{center} \end{example} -\subsection{Special Objects} +\section{Special Objects} Special objects play an important role in category theory. In this chapter we will characterize (finite) products and coproducts, as well as special morphisms such as isomorphisms, monomorphisms and epimorphisms. -\subsubsection{Initial and Terminal Objects} +\section{Initial and Terminal Objects} -\begin{definition} 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} \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 $1 \in \obj{\C}$ is called terminal if for every $A \in \obj{C}$ there is a unique morphism $! : A \rightarrow 1$. @@ -53,10 +52,10 @@ In this chapter we will characterize (finite) products and coproducts, as well a \end{example} -\subsubsection{Special Morphisms} +\section{Special Morphisms} Now let us characterize special morphisms. -\begin{definition} +\begin{definition}[Special Morphisms] Let $f : A \rightarrow B$ be a morphism. $f$ is called \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 $g \circ f = id_A$; @@ -76,37 +75,201 @@ Now let us characterize special morphisms. \end{center} \end{example} -\begin{proposition} - % TODO - iso is mono and epi +\begin{proposition} Every isomorphism is a monomorphism and an epimorphism. \end{proposition} +\begin{proof} + Let $f$ be an isomorphism. + \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 $g \circ f = h \circ f$ implies $g = g \circ f \circ f^{-1} = h \circ f \circ f^{-1} = h$, thus $f$ is an epimorphism. + \end{itemize} +\end{proof} +\begin{proposition}\label{prop:monosplitting} If $f \circ m$ is a monomorphism then $m$ is also a monomorphism. +\end{proposition} +\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. +\end{proof} +\begin{proposition}\label{prop:episplitting} If $e \circ f$ is an epimorphism then $e$ is also an epimorphism. +\end{proposition} +\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. +\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''. + +\begin{proposition}\label{prop:init_up_to} Initial objects are unique up to isomorphism. +\end{proposition} +\begin{proof} + Let $0, 0' \in \obj{\C}$ be two initial objects of $\C$ with the unique morphisms $¡_A : 0 \rightarrow A$ and ${¡'}_A : 0' \rightarrow A$. + The isomorphism is: + % https://q.uiver.app/#q=WzAsMixbMCwwLCIwIl0sWzIsMCwiMCciXSxbMSwwLCJ7wqEnfV8wIiwyLHsiY3VydmUiOi0yfV0sWzAsMSwiwqFfezAnfSIsMCx7ImN1cnZlIjotMn1dXQ== + \[ + \begin{tikzcd} + 0 && {0'} + \arrow["{{¡'}_0}"', curve={height=-12pt}, from=1-3, to=1-1] + \arrow["{¡_{0'}}", curve={height=-12pt}, from=1-1, to=1-3] + \end{tikzcd} + \] + Note that by uniqueness $¡_{0'} \circ {¡'}_0 = {¡'}_{0'} = id_{0'}$ and ${¡'}_0 \circ ¡_{0'} = ¡_0 = id_0$. +\end{proof} +\begin{proposition}\label{prop:term_up_to} Terminal objects are unique up to isomorphism. +\end{proposition} +\begin{proof} + Let $1, 1' \in \obj{\C}$ be two terminal objects of $\C$ with the unique morphisms $!_A : A \rightarrow 1$ and ${!'}_A : A \rightarrow 1'$. + The isomorphism is: + % https://q.uiver.app/#q=WzAsMixbMCwwLCIxIl0sWzIsMCwiMSciXSxbMCwxLCJ7ISd9XzEiLDAseyJjdXJ2ZSI6LTJ9XSxbMSwwLCIhX3sxJ30iLDAseyJjdXJ2ZSI6LTJ9XV0= + \[ + \begin{tikzcd} + 1 && {1'} + \arrow["{{!'}_1}", curve={height=-12pt}, from=1-1, to=1-3] + \arrow["{!_{1'}}", curve={height=-12pt}, from=1-3, to=1-1] + \end{tikzcd} + \] + Note that by uniqueness ${!'}_1 \circ !_{1'} = {!'}_{1'} = id_{1'}$ and $!_{1'} \circ {!'}_1 = !_1 = id_1$. +\end{proof} + +\section{Duality} +Notice how similar the proofs of \autoref{prop:monosplitting} and \autoref{prop:episplitting} as well as \autoref{prop:init_up_to} and \autoref{prop:term_up_to} are to each other. +It seems that we should somehow be able to construct one proof from the other, such that the work required would be halved. +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] + Every category $\C$ has a \emph{dual category} $\dual{\C}$ defined by + \begin{itemize} + \item $\obj{\dual{\C}} = \obj{\C}$ + \item $\dual{\C}(A,B) = \C(B,A)$ + \end{itemize} +\end{definition} +\begin{example} Examples are: + \begin{enumerate} + \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 $\dual{(\dual{\C})} = \C$ + \end{enumerate} +\end{example} + +Every categorical notion can thus be dualized by viewing it in the dual category, some examples include: +\begin{center} + \begin{tabular}{l l} + Notion & Dual Notion \\\midrule + Initial Object & Terminal Object \\ + Monomorphism & Epimorphism \\ + Isomorphism & Isomorphism + \end{tabular} +\end{center} + +This yields a proof principle ``by duality'', where every theorem yields another theorem by duality. + +\section{Products and Coproducts} +The categorical abstraction of Cartesian products is: +\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: + % https://q.uiver.app/#q=WzAsNCxbMiwyLCJBIFxcdGltZXMgQiJdLFs0LDIsIkIiXSxbMCwyLCJBIl0sWzIsMCwiQyJdLFswLDIsIlxccGlfMSJdLFswLDEsIlxccGlfMiIsMl0sWzMsMiwiZiIsMl0sWzMsMSwiZyJdLFszLDAsIlxcZXhpc3RzIVxcbGFuZ2xlIGYsIGcgXFxyYW5nbGUiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0= + \[ + \begin{tikzcd} + && C \\ + \\ + A && {A \times B} && B + \arrow["{\pi_1}", from=3-3, to=3-1] + \arrow["{\pi_2}"', from=3-3, to=3-5] + \arrow["f"', from=1-3, to=3-1] + \arrow["g", from=1-3, to=3-5] + \arrow["{\exists!\langle f, g \rangle}", dashed, from=1-3, to=3-3] + \end{tikzcd} + \] +\end{definition} +\begin{example} Some examples include: + \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 \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 \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 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{example} +Dual to products are: +\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: + % https://q.uiver.app/#q=WzAsNCxbMiwwLCJBK0IiXSxbMCwwLCJBIl0sWzQsMCwiQiJdLFsyLDIsIkMiXSxbMSwwLCJpXzEiXSxbMiwwLCJpXzIiLDJdLFsxLDMsImYiLDJdLFsyLDMsImciXSxbMCwzLCJcXGV4aXN0cyFbZixnXSIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ== + \[ + \begin{tikzcd} + A && {A+B} && B \\ + \\ + && C + \arrow["{i_1}", from=1-1, to=1-3] + \arrow["{i_2}"', from=1-5, to=1-3] + \arrow["f"', from=1-1, to=3-3] + \arrow["g", from=1-5, to=3-3] + \arrow["{\exists![f,g]}", dashed, from=1-3, to=3-3] + \end{tikzcd} + \] +\end{definition} +\begin{example} Examples include: + \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 \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 \emph{Gra}: Analogous to \emph{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 \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. + \end{enumerate} +\end{example} \begin{proposition} - % TODO - $f \circ m$ mono then $m$ mono. + Products are unique up to isomorphism. \end{proposition} +\begin{proof} + 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 + % https://q.uiver.app/#q=WzAsNCxbMiwwLCJDIl0sWzAsMiwiQSJdLFs0LDIsIkIiXSxbMiw0LCJDJyJdLFswLDEsImYiLDJdLFswLDIsImciXSxbMywxLCJmJyJdLFszLDIsImcnIiwyXSxbMCwzLCJrIl1d + \[ + \begin{tikzcd} + && C \\ + \\ + A &&&& B \\ + \\ + && {C'} + \arrow["f"', from=1-3, to=3-1] + \arrow["g", from=1-3, to=3-5] + \arrow["{f'}", from=5-3, to=3-1] + \arrow["{g'}"', from=5-3, to=3-5] + \arrow["k", from=1-3, to=5-3] + \end{tikzcd} + \] + commutes. Products of $A$ and $B$ are the final objects in $\emph{span}_{\C}(A,B)$ and are thus unique up to isomorphism. +\end{proof} +By duality, we get: \begin{proposition} - % TODO - $e \circ f$ epi then $e$ epi. + Coproducts are unique up to isomorphism. \end{proposition} -% TODO explain up to iso + +We can now characterize products (and later dually coproducts) as a commutative monoid: \begin{proposition} - % TODO - Initial object up to iso + $1$ is a unit of $\times$, i.e.\ $A \cong A \times 1$ for any $A \in \obj{\C}$. +\end{proposition} +% TODO proof +\begin{proposition} + $\times$ is associative, i.e.\ $A \times (B \times C) \cong (A \times B) \times C$ for any $A,B,C \in \obj{\C}$. +\end{proposition} +% TODO proof +\begin{proposition} + $\times$ is commutative, i.e.\ $A \times B \cong B \times A$ for any $A, B \in \obj{\C}$. +\end{proposition} +% TODO proof + +Duality instantly yields to commutative monoid structure of coproducts: +\begin{proposition} + $0$ is the unit of $+$, i.e.\ $A \cong A + 0$ for any $A \in \obj{\C}$. \end{proposition} \begin{proposition} - % TODO - Terminal object up to iso + $+$ is associative, i.e.\ $A + (B + C) \cong (A + B) + C$ for any $A,B,C \in \obj{\C}$. +\end{proposition} +\begin{proposition} + $+$ is commutative, i.e.\ $A + B \cong B + A$ for any $A, B \in \obj{\C}$. \end{proposition} - -\subsubsection{Products and Coproducts} -% TODO - -\subsection{Duality} -\subsection{Functors} -\subsection{Natural Transformations} -\subsection{Functor Algebras} -\subsection{Functor Coalgebras} -\subsection{(co)Limits} % chktex 36 \ No newline at end of file +\section{Functors} +\section{Natural Transformations} +\section{Functor Algebras} +\section{Functor Coalgebras} +\section{(co)Limits} % chktex 36 \ No newline at end of file diff --git a/tex/sections/03_constructions.tex b/tex/sections/03_constructions.tex index 4190d84..ddf04f5 100644 --- a/tex/sections/03_constructions.tex +++ b/tex/sections/03_constructions.tex @@ -1,4 +1,4 @@ -\section{Constructions} -\subsection{CPO} -\subsection{Initial Algebra Construction} -\subsection{Terminal Coalgebra Construction} \ No newline at end of file +\chapter{Constructions} +\section{CPO} +\section{Initial Algebra Construction} +\section{Terminal Coalgebra Construction} \ No newline at end of file