\documentclass[% 12pt,% t,% xcolor=dvipsnames,% %handout% ]{beamer} \mode { \useoutertheme[right,height=25pt,hideothersubsections]{sidebar} \usecolortheme{seahorse} \usecolortheme{rose} \usefonttheme[stillsansserifsmall]{serif} \setbeamertemplate{navigation symbols}{} } \mode{ \usetheme{Pittsburgh} } \mode{ \setbeamercolor{alerted text}{fg=black} } \usepackage{pxfonts} \usepackage{amsmath} \DeclareMathOperator{\Topcat}{hTop} \DeclareMathOperator{\galg}{GAlg} \DeclareMathOperator{\alg}{Alg} \DeclareMathOperator{\gmod}{GMod} \DeclareMathOperator{\mdl}{Mod} \DeclareMathOperator{\gcoalg}{GCoalg} \DeclareMathOperator{\coalg}{Coalg} \DeclareMathOperator{\hopf}{Hopf} \DeclareMathOperator{\set}{Set} \DeclareMathOperator{\biring}{Biring} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\spc}[2][k]{\underline{#2}_{#1}} \newcommand{\lb}{[} \newcommand{\rb}{]} \newcommand{\hyp}{-\hspace{0pt}} \newcommand{\Hom}[3]{#1(#2,#3)} \newcommand{\m}[1]{\mathcal{#1}} \renewcommand{\wedge}{\lor} \newcommand{\scoprod}{\Perp} \newcommand{\co}{\colon\thinspace} \newtheorem{proposition}[theorem]{Proposition} \mode{ \definecolor{math}{named}{violet} \definecolor{str}{named}{WildStrawberry} } \mode{ \definecolor{math}{named}{black} \definecolor{str}{named}{black} } \newcommand{\str}[1]{{\color{str}#1}} \newlength{\smathgap} \newenvironment{smath}[1][0pt]{% \setlength{\smathgap}{#1} \par\vspace{\smathgap}\par% \hspace*{\fill}% \math\displaystyle% }{% \endmath% \hspace*{\fill}% \par\vspace{\smathgap}\par% } \newcounter{last}[framenumber] \newcounter{lastoffset}[framenumber] \newcommand{\last}{% \setcounter{last}{1}% \addtocounter{last}{\insertframeendpage}% \addtocounter{last}{-\insertframestartpage}% \addtocounter{last}{\value{lastoffset}}% } \newcommand{\lastlabel}[1]{% \last% \label<\value{last}| trans:1>{#1}% } \definecolor{last}{named}{white} \newcommand{\lastframetitle}[1]{% \last% \frametitle{% \raisebox{-9pt}[0pt][0pt]{\makebox[0pt][r]{\usebeamercolor[bg]{frametitle}\color<\value{last}| handout:0>{last}\rule{4pt}{25pt}\rule{4pt}{0pt}}}% \only<0| handout:1->{\arabic{framenumber}. }#1} } \newcommand{\lastoff}{% \renewcommand{\last}{% \setcounter{last}{1}% }} %\lastoff \bibliographystyle{halpha} \title{Describing Unstable Operations} \subtitle{Joint Algebra and Topology Seminar\\ Sheffield % } \author[Stacey \and Whitehouse]% {Andrew Stacey\inst{1} \and Sarah Whitehouse\inst{2}} \institute[Sheffield]% { University of Sheffield \\ \inst{2}Partially and \inst{1}fully supported by the EPSRC, grant no.: GR/S76823/01% } \date{19th October 2006} \begin{document} \begin{frame} \label{title} \titlepage \end{frame} \section{The Problem} \begin{frame}<1-3| handout:1>[label=problem] \only<4-6>{\setcounter{lastoffset}{3}} \only<7-9>{\setcounter{lastoffset}{6}} \only<10-11>{\setcounter{lastoffset}{9}} \only<12->{\setcounter{lastoffset}{11}} \lastframetitle{The Problem \uncover<8-| handout:3->{and Answers pt I\uncover<11-| handout:4->{I}\uncover<12-| handout:4->{I}}} \begin{overprint} \large \onslide<1-13| handout:1-5> \begin{center} To give a straightforward description of the algebraic structure of \end{center} \onslide<14-15| handout:0> \begin{center} Mon\makebox[6ex][l]{\alt<14>{adic}{oidal}} Description \end{center} \end{overprint} \begin{center} \Large \uncover<2-| handout:1-5>{Unstable operations} \end{center} {\color{str} \begin{overprint} \onslide<3-4| handout:1> \begin{center} \Large \color{black} and \end{center} \onslide<5-7| handout:2> \(E^*(\spc{E})\) acting, somehow, on the algebra \(E^*(X)\) but {\color{green}not} by morphisms of algebras. \onslide<8-10,14| handout:3> \begin{center} \(E^*(\spc[*]{E})\) represents a co\hyp{}monad on \(\galg\) \\ \(E^*(X)\) is a co\hyp{}module.\\ \end{center} \onslide<11-13,15| handout:4-> \begin{center} \(E^*(\spc[*]{E})\) is a graded, complete plethory \\ \(E^*(X)\) is a plethoric module.\\ \end{center} \end{overprint} } \begin{center} \Large \uncover<3-| handout:1-5>{Unstable co-operations} \end{center} {\color{str} \begin{overprint} \onslide<6-8| handout:2> Also want to factor \(E_*(\spc{E})\) and \(E_*(X)\) in to the story. \onslide<9-12,14| handout:3-4> \begin{center} \(E_*(\spc[*]{E})\) represents a co\hyp{}module functor.\\ \(E^*(X) \to \Hom{\gcoalg}{E_*(X)}{E_*(\spc[*]{E})}\)\\ is a morphism of co\hyp{}modules. \end{center} \onslide<13,15| handout:5> \begin{center} \(E_*(\spc[*]{E})\) is a plethoric module.\\ \(E^*(X) \to \Hom{\gcoalg}{E_*(X)}{E_*(\spc[*]{E})}\)\\ is a morphism of plethoric modules. \end{center} \end{overprint} } \end{frame} \begin{frame} \frametitle{Outline} \tableofcontents[hideallsubsections] \end{frame} \section{Preliminaries} \subsection{Operations} \begin{frame} \lastframetitle{Preliminaries: Operations} \lastlabel{prelim} \begin{itemize} \item \alert<.(1)>{Graded multiplicative cohomology theory} \\ \pause contravariant functors % \[ E^*(-) : \Topcat \to \galg \] \pause \item \alert<.(1)>{Operations} are \pause \alert<.(1)>{natural transformations} \pause \vspace{20pt} \item Forget structure: \(E^k_U(-) : \Topcat \to \set\) \vspace{20pt} \pause \item \alert<.(1)>{Unstable Operations:} \(E^k_U(-) \to E^l_U(-)\) \vspace{20pt} \pause \item \alert<.(1)>{Appears} to disregard the structure of \(E^*(X)\) \end{itemize} \end{frame} \subsection{Representation} \begin{frame} \lastframetitle{Preliminaries: Representation} \lastlabel{rep} \begin{itemize} \item \(E^*(-)\) is \alert<.(1)>{representable} \pause Spaces \(\spc{E}\), \(k \in \Z\), classes \(\iota_k \in E^k(\spc{E})\). % \begin{align*} \Hom{\Topcat}{X}{\spc{E}} &\xrightarrow{\cong} E^k(X) \\ % \alpha &\mapsto \alpha^* \iota_k \end{align*} \pause \item Structure of \(E^*(-)\) \(\leftrightarrow\) structure of \((\spc{E})_{k \in \Z}\) \pause \item Yoneda's Lemma: Unstable operations \(E^k(-) \to E^l(-)\) are % \[ \Hom{\Topcat}{\spc{E}}{\spc[l]{E}} \cong E^l(\spc{E}) \] \pause \item \(E^*(\spc{E})\) certainly has \alert{some} structure \end{itemize} \end{frame} \subsection{Homology} \begin{frame} \lastframetitle{Preliminaries: Homology} \begin{itemize} \item Associated homology:\\ covariant functor % \[ E_*(-) : \Topcat \to \gmod \quad (\gcoalg) \] \pause \vfill \item In ``good'' cases \(E^*(X)\) is \(E^*\)\hyp{}dual to \(E_*(X)\). \pause \vfill \item \alert<.(1)>{Unstable Co-operations:} \(E_*(\spc{E})\) \vfill \pause \item Considerable structure, but does \alert{not} act on \(E_*(X)\). \end{itemize} \end{frame} \subsection{Summary} \againframe<4-6| handout:2>{problem} \section{Algebra Actions} \begin{frame}<-11| handout:1>[label=monoiad] \lastframetitle{Algebras and Modules} \begin{overprint} \Large \onslide<1-4| handout:0> \[ A \only<1>{\times}\only<2-4>{\otimes} M \to M \] \onslide<5-7| handout:0> \[ A \to \Hom{\mdl}{M}{M} \] \onslide<8-10| handout:0> \[ M \to \Hom{\mdl}{A}{M} \] \end{overprint} \begin{columns}[t] \column{.33\textwidth}<3-> \uncover<11-| handout:1->{% \str{Monoid}} \vspace{12pt} \(A \otimes M \to M\) \vspace{12pt} \str{Pros:}\\ Intuitive \vspace{12pt} \uncover<4-| handout:1->{% \str{Cons:}\\ Monoidal} \column{.33\textwidth}<6-10| handout:1> \uncover<0| handout:0>{% \str{Monoid}} \vspace{12pt} \(A \to \Hom{\mdl}{M}{M}\) \vspace{12pt} \str{Pros:}\\ Simple \vspace{12pt} \uncover<7-| handout:1->{% \str{Cons:}\\ Algebras} \column{.33\textwidth}<9-| handout:1-> \uncover<11-| handout:1->{% \str{Co\hyp{}monad}} \vspace{12pt} \(M \to \Hom{\mdl}{A}{M}\) \vspace{12pt} \str{Pros:}\\ Least ``specials'' \vspace{12pt} \uncover<10-| handout:1->{% \str{Cons:}\\ Least intuitive} \end{columns} %\vspace{12pt} \end{frame} \section[Monad Story]{The Monad Story} \subsection{Monads and Co\hyp{}monads} \begin{frame} \lastframetitle{Monads and Co\hyp{}monads} \begin{definition} A \alert{co\hyp{}monad} on a category \(\m{C}\) consists of a functor \begin{smath}[6pt] \color{math} T : \m{C} \to \m{C} \end{smath} and natural transformations % \begin{smath}[6pt] \color{math} \mu : T \to T T \qquad \epsilon : T \to I \end{smath} % satisfying the obvious co\hyp{}associativity and co\hyp{}unit diagrams. \pause \vfill A \alert{co\hyp{}module} for a co\hyp{}monad \(T\) is an object \(X\) of \(\m{C}\) with a morphism % \begin{smath}[6pt] \color{math} \rho : X \to T(X) \end{smath} % satisfying the obvious co\hyp{}module diagrams. \end{definition} \end{frame} \subsection{Examples} \begin{frame} \lastframetitle{Examples} \begin{itemize} \item \(A\) an algebra. \(A_+ : \mdl \to \mdl\) by % \[ A_+(M) \coloneqq \Hom{\mdl}{A}{M} \] Natural transformations: % \begin{align*} \Hom{\mdl}{A}{M} &\to \Hom{\mdl}{A}{\Hom{\mdl}{A}{M}} \\ % \Big(f : A \to M\Big) &\mapsto \Big(a_1 \mapsto \big(a_2 \mapsto f(a_1 a_2) \big) \Big) \\ % \Big(f : A \to M\Big) &\mapsto f(1) \in M \end{align*} \vfill \pause \item \(M\) an \(A\)\hyp{}module. \(\hat{\rho} : M \to A_+(M)\) by % \[ m \mapsto \big(a \mapsto \rho(a \otimes m)\big). \] \end{itemize} \end{frame} \begin{frame} \lastframetitle{Examples (contd.)} \begin{itemize} \item \(C\) a co\hyp{}algebra. \(C_! : \mdl \to \mdl\) by % \[ C_!(M) \coloneqq C \otimes M \] Natural tranformations: % \begin{align*} C \otimes M &\xrightarrow{\Delta \otimes 1} C \otimes C \otimes M \\ % C \otimes M &\xrightarrow{\epsilon \otimes 1} k \otimes M \cong M \end{align*} \vfill \pause \item \(M\) a \(C\)\hyp{}co\hyp{}module. % \[ M \to C \otimes M \] \end{itemize} \end{frame} \subsection{Unstable Operations} \begin{frame} \lastframetitle{Operations as Co\hyp{}monads} \begin{theorem}[Boardman, Johnson, Wilson] \centering{% {\color{math}\(E^*(\spc[*]{E})\)} represents a \alert{co\hyp{}monad} in \(\galg\). \\[12pt] % {\color{math}\(E^*(X)\)} is a \alert{co\hyp{}module} for this co\hyp{}monad.} \end{theorem} \pause \vfill \str{Co\hyp{}module structure:} need a map % \begin{align*} E^{\color<3->{blue}\bigstar}(X) &\to \Hom{\galg}{E^{\color<3->{green}\bigstar} (\spc[{\color<3->{blue}\bigstar}]{E})}% {E^{\color<3->{green}\bigstar}(X)} \\ \uncover<4->{ \Big(\alpha \in E^k(X) = \Hom{\Topcat}{X}{\spc{E}}\Big) &\to \Big( \alpha^* : E^*(\spc{E}) \to E^*(X) \Big)} \end{align*} \end{frame} \subsection{Co\hyp{}operations} \begin{frame} \lastframetitle{Co\hyp{}operations} \vspace{-24pt} \begin{align*} E^k(X) = \Hom{\Topcat}{X}{\spc{E}} &\to \Hom{\galg}{E^*(\spc{E})}{E^*(X)} \\ % \alpha &\mapsto \alpha^* \\ % \uncover<2->{E^k(X) = \Hom{\Topcat}{X}{\spc{E}} &\to \Hom{\gcoalg}{E_*(X)}{E_*(\spc{E})} \\ % \alpha &\mapsto \alpha_*} \end{align*} \vfill \begin{theorem}[Ravenel, Wilson]<3-> \centering{{\color{math} \(E_*(\spc[*]{E})\)} is a \alert<.(1)>{Hopf ring}} \end{theorem} \vfill \uncover<4->{ \str{Question:} What is a Hopf ring? } \uncover<5->{ \str{Answer:} A co\hyp{}algebra \(H\) such that the contravariant functor % \[ H^+ : C \to \Hom{\coalg}{C}{H} \] % actually ends up in \(\alg\). } \end{frame} \begin{frame} \lastframetitle{Module Functors} \begin{definition} Let \(T\) be a co\hyp{}monad on a category \(\m{C}\). A \alert<.(1)>{(left) co\hyp{}module functor} of \(T\) is a functor % \begin{smath}[6pt] F : \m{D} \to \m{C} \end{smath} % with a natural transformation % \begin{smath}[6pt] \rho : F \to T F \end{smath} % satisfying the obvious diagrams. \end{definition} \vfill \pause \begin{block}{Silly Example} \(A_+(-) = \Hom{\mdl}{A}{-}\), \(M_+(-) = \Hom{\mdl}{M}{-}\) % \begin{smath}[6pt] \Big(f : M \to N\Big) \mapsto \Big(a \mapsto \big(m \mapsto f(a m)\big) \Big) \end{smath} \end{block} \end{frame} \begin{frame} \lastframetitle{Module Functors (contd.)} \begin{proposition} A co\hyp{}module functor factors through the subcategory of co\hyp{}modules. \end{proposition} \end{frame} \begin{frame} \lastframetitle{Enriched Hopf Rings} \begin{theorem}[Boardman, Johnson, Wilson] \centering{\(E_*(\spc[*]{E})\) is an \alert<.(1)>{enriched} Hopf ring} \end{theorem} \pause \vfill \begin{proposition} The \alert<.(1)>{enriched} bit means that the functor \(\gcoalg \to \galg\) % \[ C_* \mapsto \Hom{\gcoalg}{C_*}{E_*(\spc[*]{E})} \] % is a \alert<.(1)>{co\hyp{}module functor} for the co\hyp{}monad represented by \(E^*(\spc[*]{E})\). % \pause % The map % \[ E^*(X) \to \Hom{\gcoalg}{E_*(X)}{E_*(\spc[*]{E})} \] % is a \alert<.(1)>{morphism of co\hyp{}modules.} \end{proposition} \end{frame} \againframe<7-9| handout:3>{problem} \section[Monoidal Story]{The Monoidal Story} \subsection{Tensor Products} \begin{frame} \lastframetitle{The Monoidal Story} \vfill \begin{center} \begin{minipage}{.9\textwidth} \emph{% The problem \dots is \dots the tensor product \dots that is simply unavailable for operations that are not additive \alert{(not that this has stopped us from trying)}.% } \end{minipage} \end{center} \begin{flushright} Boardman, Johnson, Wilson \end{flushright} \end{frame} \againframe<12| handout:2>{monoiad} \subsection{Freyd's Theorem} \begin{frame}<1-5| handout:1>[label=freyd] \lastframetitle{Freyd's Theorem\only<6-| handout:2>{ pt II}} \begin{theorem}[Freyd] Let \(\m{C}\) be a category with small \alert<1,6-7>{\alt<1-6| handout:1>{colimits}{limits}}; \(\m{V}\) a variety of algebras; \(F : \m{C} \to \m{V}\) a \alert<1,6-7>{\alt<1-6| handout:1>{covariant}{contravariant}} functor. The following are equivalent. \begin{enumerate} \item<2-6,8-| handout:1-> \(F\) \alert<2,8>{\alt<1-6| handout:1>{has a left adjoint}{is one of a mutually right adjoint pair}} \item<3-6,9-| handout:1-> \(F\) is representable by a \alert<3,9>{\only<3-6| handout:1>{co\hyp{}}\(\m{V}\)\hyp{}object} in \(\m{C}\) \item<4-6,10-| handout:1-> \(\alert<4,10>{F_U : \m{C} \to \set}\) is representable. \end{enumerate} \end{theorem} \vfill \begin{overprint} \onslide<5,6| handout:1> \str{Corollary:} Compositions of covariant representable functors are representable \onslide<11| handout:2> \str{Corollary:} Composition of a contravariant representable functor followed by a covariant one is representable \end{overprint} \only<5| handout:0>{ \hfill\hyperlink{biring}{\beamergotobutton{Skip example}} \hfill\hyperlink{apps}{\beamergotobutton{Skip all examples}} } \end{frame} \subsection{Examples} \begin{frame} \lastframetitle{Examples} \str{Variety:} groups \str{Source:} \(\Topcat'\) (\emph{based}) The \alert<.(1)>{circle} is a \alert<.(1)>{co\hyp{}group} object in \(\Topcat'\) with maps % \begin{align*} S^1 &\xrightarrow{\mu} S^1 \wedge S^1, &&\text{pinch} \\ % S^1 &\xrightarrow{\nu} S^1, &&\text{reverse} \\ % S^1 &\xrightarrow{\varepsilon} \text{pt}. \end{align*} These make \(\pi_1(X) := \Hom{\Topcat'}{S^1}{X}\) into a group: \pause % \begin{align*} f + g &\text{ is } S^1 \xrightarrow{\mu} S^1 \wedge S^1 \xrightarrow{f \wedge g} X \wedge X \to X \\ % - f &\text{ is } S^1 \xrightarrow{\nu} S^1 \xrightarrow{f} X \\ % 1 &\text{ is } S^1 \xrightarrow{\varepsilon} \text{pt} \to X \end{align*} \end{frame} \begin{frame}[label=biring] \lastframetitle{Examples\alt<1-2>{ (contd.)}{: Birings}} \str{Variety:} \(k\)\hyp{}algebras \str{Source:} \alt<1-2>{arbitrary, \(\m{C}\); initial object \(I\).}{\(k\)\hyp{}algebras} \str{Operations:} \(\lambda \in k\) % \begin{align*} X &\xrightarrow{\alpha} X \makebox[1em]{\(\alt<1-2>{\scoprod}{\otimes}\)} X &&\text{co\hyp{}addition} \\ % X &\xrightarrow{\mu} X \makebox[1em][c]{\(\alt<1-2>{\scoprod}{\otimes}\)} X &&\text{co\hyp{}multiplication} \\ % X &\xrightarrow{\lambda} \alt<1-2>{I}{k} &&\text{co\hyp{}\(\lambda\)\hyp{}action}. \end{align*} \pause \begin{overprint} \onslide<2-3| handout:0> \str{Structure:} \(f, g \in \Hom{\alt<1-2>{\m{C}}{\alg}}{X}{Y}\) % \begin{align*} f + g&\co X \xrightarrow{\alpha} X \makebox[1em][c]{\(\alt<1-2>{\scoprod}{\otimes}\)} X \xrightarrow{f \makebox[1em][c]{\(\alt<1-2>{\scoprod}{\otimes}\)} g} Y \makebox[1em][c]{\(\alt<1-2>{\scoprod}{\otimes}\)} Y \xrightarrow{\uncover<3->{m}} Y \\ % f g &\co X \xrightarrow{\mu} X \makebox[1em][c]{\(\alt<1-2>{\scoprod}{\otimes}\)} X \xrightarrow{f \makebox[1em][c]{\(\alt<1-2>{\scoprod}{\otimes}\)} g} Y \makebox[1em][c]{\(\alt<1-2>{\scoprod}{\otimes}\)} Y \xrightarrow{\uncover<3->{m}} Y \\ % \lambda &\co X \xrightarrow{\lambda} \makebox[1em][c]{\(\alt<1-2>{I}{k}\)} \xrightarrow{\iota_Y} Y \end{align*} \onslide<4| handout:0> \str{Structure:} \(f, g \in \Hom{\alt<1-2>{\m{C}}{\alg}}{X}{Y}\) % \begin{align*} f + g \co x &\xrightarrow{\alpha} \sum_i x_i^{[1]} \otimes x_i^{[2]} \xrightarrow{f \otimes g} \sum_i f(x_i^{[1]}) \otimes g(x_i^{[2]}) \\ &\xrightarrow{m} \sum_i f(x_i^{[1]}) g(x_i^{[2]}) \end{align*} \onslide<5-| handout:1> \str{Operations:} The biring structure of \(E^*(\spc[*]{E})\) is: % \begin{itemize} \item Co\hyp{}addition: \(H\)\hyp{}map \(\spc{E} \times \spc{E} \to \spc{E}\) \item Co\hyp{}multiplication: ring maps \(\spc{E} \times \spc[l]{E} \to \spc[k+l]{E}\) \item Co\hyp{}\(E^*\)\hyp{}action: \(v \in E^k = E^k(\text{pt}) = \Hom{\Topcat}{\text{pt}}{\spc{E}}\) induces \(v^* : E^*(\spc{E}) \to E^*(\text{pt}) = E^*\). \end{itemize} \end{overprint} \end{frame} \subsection{Unstable Operations} \begin{frame}[label=apps] \lastframetitle{Applications pt I: Products} Compositions of representable functors are representable. \hspace*{\fill} \(A_+ B_+ : \alg \to \set\) \hspace*{\fill} \({B_1}_+ {B_2}_+ : \alg \to \alg\) \hspace*{\fill} \pause \begin{proposition}[Tall, Wraith (1970)] There is a product % \begin{smath}[5pt] \biring \times \alg \to \alg, \qquad (B,A) \to B \odot A \end{smath} % and a natural isomorphism % \begin{smath}[5pt] \Hom{\alg}{B \odot A}{A'} \cong \Hom{\alg}{A}{\Hom{\alg}{B}{A'}} \end{smath} % and if \(B_1, B_2\) are birings then \(B_1 \odot B_2\) is a biring. \end{proposition} \pause \begin{itemize}[<+->] \item Linear in \(B\) but \alert<.>{not} in \(A\) \item \alert<.>{Not} symmetric \end{itemize} \end{frame} \begin{frame} \lastframetitle{Applications pt II: Plethories} Representable co\hyp{}monad: \(P_+ \to P_+P_+\), \(P_+ \to I\). \begin{definition}[Tall and Wraith, Borger and Wieland] A \alert<.(1)>{plethory} consists of a biring \(P\) and maps of birings % \begin{smath}[6pt] P \odot P \to P, \qquad I \to P \end{smath} % satisfying the obvious diagrams. \end{definition} \pause \begin{itemize}[<+->] \item \(I\) is the initial \alert<.>{biring}, \(k \lb e \rb\) \item For a ring \(R\), \(\Hom{\set}{R}{R}\) is a plethory \item For a group \(G\) there is a notion of a \alert<.>{free plethory} \(P(G)\) \end{itemize} \end{frame} \begin{frame} \lastframetitle{Applications pt III: Modules} Co\hyp{}module over a representable co\hyp{}monad: % \[ \rho \in \Hom{\alg}{A}{P_+(A)} = \Hom{\alg}{A}{\Hom{\alg}{P}{A}} \cong \Hom{\alg}{P \odot A}{A} \] \begin{definition} Let \(P\) be a plethory. A \alert<.(1)>{\(P\)\hyp{}module} is an algebra \(A\) with a map % \begin{smath}[6pt] P \odot A \to A \end{smath} % satisfying the obvious diagrams. \end{definition} \pause \str{Example:} If \(G\) acts on an algebra \(A\) then \(A\) is a \(P(G)\)\hyp{}module. \end{frame} \begin{frame} \lastframetitle{Unstable Operations} \vfill \begin{theorem} For ``good'' cohomology theories, the set of unstable operations of a cohomology theory is a \\ \centering{\alert<.(1)>{graded, completed plethory}.} \vspace{20pt} The cohomology of a space is a \alert<.(1)>{module} for this plethory. \end{theorem} \end{frame} \againframe<10-11| handout:4>{problem} \subsection{Unstable Co\hyp{}operations} \againframe<6-| handout:2>{freyd} \begin{frame} \lastframetitle{Pairings} \( C \mapsto \Hom{\alg}{A}{\Hom{\coalg}{C}{H}} \) % \alert<.(1)>{is} representable % \pause \begin{lemma} There is a pairing % \[ \alg \times \hopf \to \coalg, \qquad (A,H) \to A \boxtimes H \] % and a natural bijection % \[ \Hom{\alg}{A}{\Hom{\coalg}{C}{H}} \cong \Hom{\coalg}{C}{A \boxtimes H} \] \end{lemma} \pause \str{Caveat:} The pairing is \alert{contravariant} in \(A\) and \alert{covariant} in \(H\). \end{frame} \begin{frame} \lastframetitle{Pairing Properties} \color<4| handout:0>{gray} From: % \begin{align*} % \Hom{\alg}{B}{\Hom{\coalg}{C}{H}} &= \Hom{\coalg}{C}{B \boxtimes H}\\[12pt] % \Hom{\alg}{A}{\Hom{\alg}{B}{\Hom{\coalg}{C}{H}}} &= \Hom{\coalg}{C}{A \boxtimes (B \boxtimes H)}\\ % \Hom{\alg}{% {\color<4>{black}B} \odot% {\color<4>{black}A}% }{% \Hom{\coalg}{% {\color<4>{black}C}% }{ {\color<4>{black}H}% }} % &= \Hom{\coalg}{C}{(B \odot A) \boxtimes H} \end{align*} We deduce: \pause \begin{lemma} \begin{enumerate}[<+->] \item If \(B\) is a biring, \(B \boxtimes H\) is a Hopf ring \item There is a natural isomorphism \((B \odot A) \boxtimes H \cong A \boxtimes (B \boxtimes H)\). \end{enumerate} \end{lemma} \uncover<3| handout:0>{ \hfill\hyperlink{hopf}{\beamergotobutton{Skip appalling pun}} } \end{frame} \begin{frame}[label=hopf] \lastframetitle{Plethories and Hopf Rings} \str{Question:} When does \(H_* : C \mapsto \Hom{\coalg}{C}{H}\) land in the subcategory of \(P\)\hyp{}modules? \vspace{12pt} \pause \str{Answer:} When \(H\) is a \(P\)\hyp{}co\hyp{}module. % \[ H \to P \boxtimes H \] \pause \vfill \begin{lemma} An \alert{enriched Hopf ring} is a Hopf ring with an action of a plethory. \end{lemma} \vfill \pause \str{Problem:} \(P \boxtimes H\) is \alert{contravariant} in \(P\) so \alert{tricky} to work with. \end{frame} \begin{frame} \lastframetitle{Enriched Hopf Rings} \vfill \begin{theorem} There is a pairing \(\biring \times \hopf \to \hopf\), \((B,H) \mapsto B \circledcirc H\), \alert{covariant} in both, and a natural isomorphism % \[ \Hom{\hopf}{H_1}{B \boxtimes H_2} \cong \Hom{\hopf}{B \circledcirc H_1}{H_2}. \] \end{theorem} \end{frame} \begin{frame} \lastframetitle{Remarks} \vfill \begin{itemize} \item Existence does \alert<.(1)>{not} come from Freyd's theorem. \vfill \pause \item But related to pairing % \begin{smath}[6pt] \alg \times \coalg \to \hopf \end{smath} % from the representable functor \begin{smath}[6pt] H \mapsto \Hom{\alg}{A}{\Hom{\coalg}{C}{H}} \end{smath} % which does come from Freyd's theorem. \vfill \pause \item A plethoric action on a Hopf ring is now in the more usual form: % \begin{smath}[6pt] P \circledcirc H \to H \end{smath} \end{itemize} \end{frame} \section{The Answers} \againframe<12-| handout:5>{problem} \end{document}