From 64cbc5060138009654228322758d209061424a57 Mon Sep 17 00:00:00 2001 From: Leon Vatthauer Date: Tue, 26 Mar 2024 08:09:53 +0100 Subject: [PATCH] Finish functor section --- tex/.vscode/ltex.dictionary.en-US.txt | 1 + tex/main.pdf | Bin 121614 -> 131300 bytes tex/sections/02_categories.tex | 116 ++++++++++++++++++++++++++ 3 files changed, 117 insertions(+) diff --git a/tex/.vscode/ltex.dictionary.en-US.txt b/tex/.vscode/ltex.dictionary.en-US.txt index 81fe0a3..20768bc 100644 --- a/tex/.vscode/ltex.dictionary.en-US.txt +++ b/tex/.vscode/ltex.dictionary.en-US.txt @@ -36,3 +36,4 @@ hom-functor monoids n-ary Cocartesian +Yoneda diff --git a/tex/main.pdf b/tex/main.pdf index 6ae860e9133b5e1ad9874b9f41f2aad22cfd41c5..ea79009f504c2fff13628b51d37bfc7b223c33df 100644 GIT binary patch delta 65648 zcmZ5{Q*7W5)NF0r?zY_;Tifo|w#}{EUv1mAZQC}sw(aiy-!J#!%Y8eUoTrmaa%LtI z(*k|y4-+Q_`a52n0uWPA@yU8%@&*TAAB}@_s?hkU(`$2UbMvtCWw*zX;<`#!KPJJ`G)4541E6GJ9Hq#+k|#McJ(lls 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The definition of ``naturality'' was one of the original goals of category theory. +\begin{definition}[Natural Transformation] + Given two functors $F, G : \C \rightarrow \D$. + 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 + % https://q.uiver.app/#q=WzAsNCxbMCwwLCJGQSJdLFsyLDAsIkZCIl0sWzAsMiwiR0EiXSxbMiwyLCJHQiJdLFswLDEsIkZmIl0sWzIsMywiR2YiXSxbMCwyLCJcXGFscGhhX0EiLDJdLFsxLDMsIlxcYWxwaGFfQiJdXQ== + \[ + \begin{tikzcd} + FA && FB \\ + \\ + GA && GB + \arrow["Ff", from=1-1, to=1-3] + \arrow["Gf", from=3-1, to=3-3] + \arrow["{\alpha_A}"', from=1-1, to=3-1] + \arrow["{\alpha_B}", from=1-3, to=3-3] + \end{tikzcd} + \] + commutes. +\end{definition} + +\begin{example} Examples of natural transformations include: + \begin{enumerate} + \item The obvious function $flatten : Tree\;A \rightarrow List\;A$: + % https://q.uiver.app/#q=WzAsNCxbMCwwLCJUcmVlXFw7QSJdLFsyLDAsIlRyZWVcXDtCIl0sWzAsMiwiTGlzdFxcO0EiXSxbMiwyLCJMaXN0XFw7QiJdLFswLDIsImZsYXR0ZW5fQSIsMl0sWzEsMywiZmxhdHRlbl9CIl0sWzIsMywibGlzdFxcO2YiXSxbMCwxLCJ0cmVlXFw7ZiJdXQ== + \[ + \begin{tikzcd} + {Tree\;A} && {Tree\;B} \\ + \\ + {List\;A} && {List\;B} + \arrow["{flatten_A}"', from=1-1, to=3-1] + \arrow["{flatten_B}", from=1-3, to=3-3] + \arrow["{list\;f}", from=3-1, to=3-3] + \arrow["{tree\;f}", from=1-1, to=1-3] + \end{tikzcd} + \] + \item For $Id, Q : Set \rightarrow Set$ we have $\delta : Id \rightarrow Q$ 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: + \begin{alignat*}{1} + \eta_X : X & \rightarrow \mathcal{P}X \\ + x & \mapsto \{x\} + \end{alignat*} + and + \begin{alignat*}{1} + \mu_X : \mathcal{P}\mathcal{P}X & \rightarrow \mathcal{P}X \\ + Z & \mapsto \bigcup Z. + \end{alignat*} + \item Between $Q$ and $\mathcal{P}$ we can consider $\alpha,\beta : Q \rightarrow \mathcal{P}$ given by + \begin{alignat*}{2} + & \alpha_X(x,y) & & = \{x,y\} \\ + & \beta_X(x,y) & & = \{x\}. + \end{alignat*} + \end{enumerate} +\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. + +\begin{example} Let us examine concrete examples of functor categories: + \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 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 + % https://q.uiver.app/#q=WzAsNCxbMCwwLCJGMCJdLFsyLDAsIkYxIl0sWzAsMiwiRzAiXSxbMiwyLCJHMSJdLFswLDEsIkZtIl0sWzIsMywiR20iXSxbMCwyLCJmIiwyXSxbMSwzLCJnIl1d + \[ + \begin{tikzcd} + F0 && F1 \\ + \\ + G0 && G1 + \arrow["Fm", from=1-1, to=1-3] + \arrow["Gm", from=3-1, to=3-3] + \arrow["f"', from=1-1, to=3-1] + \arrow["g", from=1-3, to=3-3] + \end{tikzcd} + \] + commutes. + \end{enumerate} +\end{example} + +\begin{definition}[Natural Isomorphism] + Isomorphisms in $[\C,\D]$ are called \emph{natural isomorphisms}. +\end{definition} + +\begin{proposition} + $\alpha : F \rightarrow G$ is a natural isomorphism \emph{iff} every $\alpha_C$ is an isomorphism. +\end{proposition} + +\begin{example} Let us consider some examples of natural isomorphisms: + \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 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 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. + \end{enumerate} +\end{example} + +\begin{proposition}[Yoneda Lemma] + Let $A \in \obj{\C}$ and $G : \C \rightarrow \emph{Set}$. Then the natural transformations + \[\C(A,-) \rightarrow G\] + are in bijection with the elements of the set $GA$. +\end{proposition} +\begin{proof} + The mappings are + \begin{alignat*}{1} + & Z : GA \rightarrow [\C , \emph{Set}](\C(A,-), G) \\ + & Z\;x\;h = G\;h\;x + \end{alignat*} + and + \begin{alignat*}{1} + & Y : [\C , \emph{Set}](\C(A,-), G) \rightarrow GA \\ + & Y\;\alpha = \alpha_A\;id_A + \end{alignat*} +\end{proof} + +\begin{example} + Let us consider an application of the Yoneda Lemma: how many natural transformations $Id \rightarrow Q$ 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 $1$. + + 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 $4$. +\end{example} + \section{Functor Algebras} \section{Functor Coalgebras} \section{(co)Limits} % chktex 36 \ No newline at end of file