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97 lines
3.6 KiB
TeX
97 lines
3.6 KiB
TeX
\documentclass[11pt]{amsart}
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\usepackage{geometry}
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\geometry{a4paper}
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\usepackage{graphicx}
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\usepackage{amssymb}
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\usepackage{epstopdf}
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\usepackage{mdframed}
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\usepackage{hyperref}
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\usepackage{tikz-cd}
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\usepackage{lmodern}
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% Reproducible builds
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\pdfinfoomitdate=1
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\pdftrailerid{}
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\pdfsuppressptexinfo=-1
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\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}
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\newmdtheoremenv{defn}{Definition}
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\newmdtheoremenv{thm}{Theorem}
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\newmdtheoremenv{motiv}{Motivation}
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\title{Representable functors and adjoint functors}
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\author{Patrick Stevens}
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\begin{document}
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\maketitle
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\tiny \begin{center} \url{https://www.patrickstevens.co.uk/misc/RepresentableFunctors/RepresentableFunctors.pdf} \end{center}
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\normalsize
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\emph{You should draw diagrams yourself throughout this document. It will be unreadable as mere symbols.}
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\begin{defn}A \emph{representable functor} is a functor $F: \mathcal{C} \to \mathbf{Sets}$ which is naturally isomorphic to $\text{Hom}_{\mathcal{C}}(A, \cdot)$ for some $A \in \mathcal{C}$.\end{defn}
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\
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It is an easy fact that representable functors preserve limits.
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Indeed, it is enough to show that they preserve products and equalisers, and both of these facts simply fall out of the relevant diagrams.
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What other functors preserve limits?
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Recall that morally, a functor has a left adjoint iff it preserves all limits.
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(There are set-theoretic issues here, so the statement isn't actually true as stated; these are dealt with by the General Adjoint Functor Theorem.)
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So we might hope that representable functors into $\mathbf{Sets}$ are precisely those which are right adjoints.
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\section{Right adjoints are representable}
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Suppose $G: \mathcal{C} \to \mathbf{Sets}$ is a right adjoint to $F: \mathbf{Sets} \to \mathcal{C}$.
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We want to find a representing object, and it's going to have to use $F$, so let's see what would happen if $G$ were naturally isomorphic to $\text{Hom}_{\mathcal{C}}(FA, \cdot)$.
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By the definition of an adjoint, for every $A \in \mathcal{C}$, have $$\text{Hom}_{\mathcal{C}}(FA, X) \cong \text{Hom}_{\mathbf{Sets}}(A, GX)$$
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But wait! We already have that $\text{Hom}_{\mathbf{Sets}}(\{ 1 \}, GX) \cong GX$.
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So actually $F1$ represents $G$.
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\section{Representable functors have left adjoints}
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This one is less immediate, so we'll motivate it.
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The most trivial example of a representable functor is $1_{\mathbf{Sets}}$, so we'll go for the next most trivial: $$\text{Hom}_{\mathbf{Sets}}(\{0, 1\}, \cdot) : \mathbf{Sets} \to \mathbf{Sets}$$
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What would it look like if this had an adjoint?
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We'll use the most concrete definition of an adjoint: as an initial object in an appropriate comma category.
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\[
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\begin{tikzcd}
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FA
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\arrow[r, dashrightarrow, "h"]
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& B
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\\
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\mathrm{Hom}(\{0,1\}, FA)
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\arrow[r, "h \circ \cdot"]
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& \mathrm{Hom}(\{0,1\}, B)
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\\
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A
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\arrow[u, "\eta_A"]
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\arrow[ur, "f"']
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&
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\end{tikzcd}
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\]
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What is $f: A \to \text{Hom}(\{0, 1\}, B)$?
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It is nothing more than an $A$-indexed collection of maps $\{0, 1\} \to B$.
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We want to make this unique $h: FA \to B$, given a collection of maps $\{0,1\} \to B$.
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That suggests we want to glue the maps together somehow, and $FA$ will be a ``glued object''.
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So we just take $FA$ to be the coproduct of $\{0,1\}$-many copies of $A$.
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That is, let $FA = \sqcup_{\{0, 1\}} A = \{a_0: a \in A\} \cup \{a_1: a \in A\}$, and define $\eta_A: A \to \text{Hom}(\{0,1\}, FA)$ by $a \mapsto (n \mapsto a_n)$.
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This construction generalises in exactly the obvious way to any representable functor $\mathbf{Sets} \to \mathbf{Sets}$, and then to $\mathcal{C} \to \mathbf{Sets}$ where $\mathbf{C}$ has coproducts.
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\end{document} |