From 967a6e4427cae088651899032f2136963dcde213 Mon Sep 17 00:00:00 2001 From: zeramorphic Date: Mon, 16 Oct 2023 20:47:04 +0100 Subject: [PATCH] Augment category theory notes Signed-off-by: zeramorphic --- iii/alggeom/01_introduction.tex | 2 +- iii/cat/02_yoneda_lemma.tex | 104 ++++++++++++++++++++++---------- 2 files changed, 73 insertions(+), 33 deletions(-) diff --git a/iii/alggeom/01_introduction.tex b/iii/alggeom/01_introduction.tex index 0cb74ff..d935efb 100644 --- a/iii/alggeom/01_introduction.tex +++ b/iii/alggeom/01_introduction.tex @@ -269,7 +269,7 @@ \subsection{Distinguished opens and localisations} Now suppose \( U_f \subseteq U_g \), so \( \mathbb V(f) \supseteq \mathbb V(g) \). Hence, all prime ideals that contain \( g \) also contain \( f \). But since - \[ \sqrt{I} = \bigcap_{\mathfrak p \text{ prime} \supseteq I} \] + \[ \sqrt{I} = \bigcap_{\mathfrak p \text{ prime} \supseteq I} \mathfrak p \] we must have \[ \sqrt{(f)} \supseteq \sqrt{(g)} \] giving the result. diff --git a/iii/cat/02_yoneda_lemma.tex b/iii/cat/02_yoneda_lemma.tex index de8da60..1ed0341 100644 --- a/iii/cat/02_yoneda_lemma.tex +++ b/iii/cat/02_yoneda_lemma.tex @@ -121,31 +121,7 @@ \subsection{Statement and proof} \end{proof} This says that any locally small category \( \mathcal C \) is equivalent to a full subcategory of a functor category \( [\mathcal C^\cop, \mathbf{Set}] \). The category \( [\mathcal C^\cop, \mathbf{Set}] \) is sometimes called the category of \emph{presheaves} on \( \mathcal C \), so any category embeds into its category of presheaves. -\begin{definition} - Let \( \mathcal C \) be a locally small category. - A functor \( F : \mathcal C \to \mathbf{Set} \) is called \emph{representable} if it is isomorphic to \( \mathcal C(A, -) \) for some \( A \). - A \emph{representation} of \( F \) is a pair \( (A, x) \) where \( A \in \ob \mathcal C \), and \( x \in FA \) is such that - \[ \Psi(x) : \mathcal C(A, -) \to F \] - is a natural isomorphism. - In this case, we say that \( x \) is a \emph{universal element} of \( F \). -\end{definition} -\begin{corollary} - Suppose \( (A, x) \) and \( (B, y) \) are representations of \( F : \mathcal C \to \mathbf{Set} \). - Then there is a unique isomorphism \( f : A \to B \) such that \( Ff(x) = y \). -\end{corollary} -\begin{proof} - The Yoneda lemma shows that the elements of \( F A \) correspond to natural transformations \( \mathcal C(A, -) \to F \), and similarly for the elements of \( F B \). - Thus, \( Ff(x) = y \) equivalently says that - \[\begin{tikzcd} - {\mathcal C(B, -)} && {\mathcal C(A, -)} \\ - & F - \arrow["{\mathcal C(f, -)}", from=1-1, to=1-3] - \arrow["{\Psi(x)}", from=1-3, to=2-2] - \arrow["{\Psi(y)}"', from=1-1, to=2-2] - \end{tikzcd}\] - commutes. - But \( \Psi(x) \) and \( \Psi(y) \) are isomorphisms, so this holds if and only if \( f \) is the unique isomorphism sent by the Yoneda embedding to \( \Psi(x)^{-1} \Psi(y) \). -\end{proof} + We now explain and prove part (ii) of the Yoneda lemma. Suppose that \( \mathcal C \) were small, so \( [\mathcal C, \mathbf{Set}] \) were locally small. Then we have two functors @@ -173,7 +149,32 @@ \subsection{Statement and proof} as required. \end{proof} -\subsection{Examples of representable functors} +\subsection{Representable functors} +\begin{definition} + Let \( \mathcal C \) be a locally small category. + A functor \( F : \mathcal C \to \mathbf{Set} \) is called \emph{representable} if it is isomorphic to \( \mathcal C(A, -) \) for some \( A \). + A \emph{representation} of \( F \) is a pair \( (A, x) \) where \( A \in \ob \mathcal C \), and \( x \in FA \) is such that + \[ \Psi(x) : \mathcal C(A, -) \to F \] + is a natural isomorphism. + In this case, we say that \( x \) is a \emph{universal element} of \( F \). +\end{definition} +\begin{corollary} + Suppose \( (A, x) \) and \( (B, y) \) are representations of \( F : \mathcal C \to \mathbf{Set} \). + Then there is a unique isomorphism \( f : A \to B \) such that \( Ff(x) = y \). +\end{corollary} +\begin{proof} + The Yoneda lemma shows that the elements of \( F A \) correspond to natural transformations \( \mathcal C(A, -) \to F \), and similarly for the elements of \( F B \). + Thus, \( Ff(x) = y \) equivalently says that + \[\begin{tikzcd} + {\mathcal C(B, -)} && {\mathcal C(A, -)} \\ + & F + \arrow["{\mathcal C(f, -)}", from=1-1, to=1-3] + \arrow["{\Psi(x)}", from=1-3, to=2-2] + \arrow["{\Psi(y)}"', from=1-1, to=2-2] + \end{tikzcd}\] + commutes. + But \( \Psi(x) \) and \( \Psi(y) \) are isomorphisms, so this holds if and only if \( f \) is the unique isomorphism sent by the Yoneda embedding to \( \Psi(x)^{-1} \Psi(y) \). +\end{proof} \begin{enumerate} \item Consider the forgetful functor \( \mathbf{Gp} \to \mathbf{Set} \). This is representable by the free group on one generator, \( \mathbb Z \). @@ -230,7 +231,20 @@ \subsection{Separating and detecting families} We say that \begin{enumerate} \item \( \mathcal G \) is a \emph{separating family} for \( \mathcal C \) if the functors \( \mathcal C(G, -) \) for \( G \in \mathcal G \) are collectively faithful; that is, if \( f, g : A \rightrightarrows B \), the equations \( fh = gh \) for all \( h : G \to A \) with \( G \in \mathcal G \) imply \( f = g \). - \item \( \mathcal G \) is a \emph{detecting family} for \( \mathcal C \) if the functors \( \mathcal C(G, -) \) for \( G \in \mathcal G \) collectively \emph{reflect isomorphisms}; that is, if \( f : A \to B \) such that every \( h : G \to B \) with \( G \in \mathcal G \) factors uniquely through \( A \), then \( f \) is an isomorphism. + \[\begin{tikzcd} + G & A & B + \arrow["h", from=1-1, to=1-2] + \arrow["f", shift left=2, from=1-2, to=1-3] + \arrow["g"', shift right=2, from=1-2, to=1-3] + \end{tikzcd}\] + \item \( \mathcal G \) is a \emph{detecting family} for \( \mathcal C \) if the functors \( \mathcal C(G, -) \) for \( G \in \mathcal G \) collectively \emph{reflect isomorphisms}; that is, if \( f : A \to B \) is such that every \( h : G \to B \) with \( G \in \mathcal G \) factors uniquely through \( A \), then \( f \) is an isomorphism. + \[\begin{tikzcd} + G & A \\ + & B + \arrow["g", dashed, from=1-1, to=1-2] + \arrow["f", from=1-2, to=2-2] + \arrow["h"', from=1-1, to=2-2] + \end{tikzcd}\] \end{enumerate} If \( \mathcal G = \qty{G} \), we call \( G \) a \emph{separator} or \emph{detector} respectively. \end{definition} @@ -238,24 +252,50 @@ \subsection{Separating and detecting families} \begin{lemma} \begin{enumerate} \item If \( \mathcal C \) has equalisers, then any detecting family is separating. - \item If \( \mathcal C \) is balanced, then any separating family is detecting. + \item If \( \mathcal C \) is balanced, then any separating family is detecting. \end{enumerate} \end{lemma} \begin{proof} \emph{Part (i).} Suppose \( \mathcal G \) is detecting, and \( f, g : A \rightrightarrows B \) such that every morphism \( h : G \to A \) with \( G \in \mathcal G \) has \( fh = gh \). - Then every such \( h : G \to A \) with \( G \in \mathcal G \) factors uniquely through the equaliser of \( f \) and \( g \), so this equaliser must be an isomorphism as \( \mathcal G \) is detecting. - Hence \( f = g \). + Then every such \( h : G \to A \) with \( G \in \mathcal G \) factors uniquely through the equaliser of \( f \) and \( g \). + \[\begin{tikzcd} + G \\ + E & A & B + \arrow["h", from=1-1, to=2-2] + \arrow["f", shift left=2, from=2-2, to=2-3] + \arrow["g"', shift right=2, from=2-2, to=2-3] + \arrow["e"', from=2-1, to=2-2] + \arrow[dashed, from=1-1, to=2-1] + \end{tikzcd}\] + Thus this equaliser \( e \) must be an isomorphism as \( \mathcal G \) is detecting. + Since \( ef = eg \), we must have \( f = g \), as required. \emph{Part (ii).} Suppose \( \mathcal G \) is separating, and \( f : A \to B \) is such that every \( h : G \to B \) with \( G \in \mathcal G \) factors uniquely through \( f \). As \( \mathcal C \) is balanced, it suffices to show that \( f \) is both monic and epic. - If \( fg = fh \) for some \( g, h : C \rightrightarrows A \), then any \( h : G \to C \) with \( G \in \mathcal G \) satisfies \( gk = hk \), since both are factorisations of \( fgk = fhk \) through \( f \). + If \( fg = fh \) for some \( g, h : C \rightrightarrows A \), then any \( k : G \to C \) with \( G \in \mathcal G \) satisfies \( gk = hk \), since both are factorisations of \( fgk = fhk \) through \( f \). + \[\begin{tikzcd} + G & C & A & B + \arrow["g", shift left=2, from=1-2, to=1-3] + \arrow["h"', shift right=2, from=1-2, to=1-3] + \arrow["k", from=1-1, to=1-2] + \arrow["f", from=1-3, to=1-4] + \end{tikzcd}\] Since \( \mathcal G \) is separating, \( g = h \). As this is true for all pairs \( g, h \), we must have that \( f \) is monic. - + Similarly, if \( \ell, m : B \rightrightarrows D \) satisfy \( \ell f = mf \), then any \( n : G \to B \) with \( G \in \mathcal G \) satisfies \( \ell n = m n \), since it factors through \( f \). + \[\begin{tikzcd} + & G \\ + A & B & D + \arrow["\ell", shift left=2, from=2-2, to=2-3] + \arrow["m"', shift right=2, from=2-2, to=2-3] + \arrow["f", from=2-1, to=2-2] + \arrow["n", from=1-2, to=2-2] + \arrow[dashed, from=1-2, to=2-1] + \end{tikzcd}\] So \( \ell = m \), giving that \( f \) is epic. \end{proof} \begin{example}