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\usepackage{amsmath}
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\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}
\DeclareMathOperator{\abg}{Ab}
\DeclareMathOperator*{\bstar}{\bigstar}
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\newcommand{\co}{\colon}%\thinspace}
\newcommand{\wotimes}{\widetilde{\otimes}}

\mode<article>{
\renewcommand{\inst}[1]{}
}

\title{Describing Unstable Operations}

\subtitle{Geometry and Topology Seminar, Glasgow}

\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{30th October 2006}

\begin{document}

\begin{frame}<beamer>
\titlepage
\end{frame}

\maketitle

\begin{frame}<beamer>[shrink]
\tableofcontents[hideallsubsections]
\end{frame}

\mode<article>{
\setlength{\parindent}{0pt}
\setlength{\parskip}{1ex plus 0.5ex minus 0.2ex}
}

\section{The Problem}

\begin{frame}<handout>
\frametitle{The Problem}


Describe the structure on the set of unstable operations.

\vfill

\begin{itemize}
\item Simple
\item Practical
\item Intuitive
\item Complete
\item Elegant
\end{itemize}
\end{frame}


To give a good description of the structure on the set of unstable
operations of a nice cohomology theory.

``Good'' description:
%
\begin{itemize}
\item Simple
\item Practical
\item Intuitive
\item Complete
\item Elegant
\end{itemize}


\section{Preliminaries}

\begin{frame}<handout>
\frametitle{Preliminaries}

Graded, multiplicative, commutative cohomology theory:
%
\[
  E^*(-) \co \Topcat \to \galg
\]
%
coefficients: \(E^* \coloneqq E^*(\text{pt})\)

\vfill

Unstable operations: 
%
\[
  r \co E^k_U(-) \to E^l_U(-)
\]
\end{frame}

Our fundamental object of study is \(E^*(-)\); a graded,
multiplicative, commutative, generalised cohomology theory.
It is a
\emph{contravariant} functor
%
\[
E^*(-) \co \Topcat \to \galg
\]
%
The coefficient ring is \(E^* \coloneqq E^*(\text{pt})\).

Unstable operations on \(E^*(-)\) are natural transformations
\(E^k_U(-) \to E^l_U(-)\) where \(E^k_U(-) \co \Topcat \to \set\) is
the underlying set of \(k\)th component.

Algebraic topologists use cohomology theories to convert topological
problems into algebraic ones.  One common problem is to distinguish
between two spaces; which turns into detecting a difference between
\(E^*(X)\) and \(E^*(Y)\).  Even if they are isomorphic as algebras,
they may have different actions of operations.  Unstable operations
are the largest set of operations that can act, and so are the most
powerful tools.  Thus studying unstable operations tells us about the
power of the theory \(E^*(-)\).

\begin{frame}<handout| article>[shrink]
\begin{examples}
Some examples of unstable operations:
%
\begin{itemize}
\item \(v \in E^*\), \(x \mapsto v x\);
\item \(x \mapsto 1_X\);
\item \(x \mapsto x^2\);
\item More generally, for \(p(t) \in E^* \lb t \rb\), get \(x \mapsto
p(x)\);
\item On \(K^0(X) = K(\text{iso classes of vector bundles over } X)\)
\[
(E \to X) \mapsto \begin{cases}
\hspace*{\fill}
(E^* &\to X) \\
\hspace*{\fill}
(E \otimes E &\to X) \\
\hspace*{\fill}
(\hom(E,E) &\to X) \\
\hspace*{\fill}
(\Lambda^k E &\to X) \\
\hspace*{\fill}
(S^k E &\to X)
  \end{cases}
\]

\item On \(K^0(X)\), define the \(k\)th Adams operation \(\Psi^k\) by:
%
\[
  \Psi^k(L) = L^{\otimes k}, \qquad \Psi^k(E \oplus F) = \Psi^k(E)
  \oplus \Psi^k(F)
\]
\end{itemize}
\end{examples}
\end{frame}


\begin{frame}<handout>
\frametitle{Representability}

\(E^*(-)\) is representable.

Spaces \(\spc{E}\), classes \(\iota_k \in E^k(\spc{E})\).

\[
\Hom{\Topcat}{X}{\spc{E}} \xrightarrow{\cong} E^k(X), \qquad  \alpha
  \mapsto \alpha^*\iota_k \qquad (\alpha^* = E^*(\alpha))
\]

\vfill

Yoneda:
%
\[
\{r \co E^k_U(-) \to E^l_U(-)\} \cong E^l(\spc{E}) \qquad \text{(as sets)}
\]

\vfill
\begin{enumerate}
\item Underlying sets
\item How to determine structure
\end{enumerate}
\end{frame}

What makes this problem tractable is the fact that \(E^*(-)\) is
\emph{representable}.
There are spaces \(\spc{E}\) and classes
\(\iota_k \in E^k(\spc{E})\) such that the map \(\alpha \mapsto
\alpha^* \iota_k\) is a natural bijection
%
\[
  \Hom{\Topcat}{X}{\spc{E}} \to E^k(X)
\]
%
Yoneda's Lemma tells us that:
%
\[
  \{\text{Unstable operations: } E^k_U(-) \to E^l_U(-)\} \cong
  E^l(\spc{E})
\]
%
and that the structure on \(E^*(-)\) is determined by maps on the
\((\spc{E})\).

This tells us
%
\begin{enumerate}
\item The underlying sets
\item How to determine the structure
\end{enumerate}


\section{Answer I: The Structure}

The first answer to our problem is simply to list all the available
structure.  This answer is certainly complete; one could even make a
case for it being practical.  However it is certainly not simple,
elegant, or intuitive.

\begin{frame}<handout| article>
\begin{enumerate}
\item \(E^*(\spc{E})\) is a graded algebra over \(E^*\)

\item \(E^*(X)\) is an abelian group, so \(\spc{E}\) is an
\(H\)\hyp{}space:
%
\[
  \spc{E} \times \spc{E} \to \spc{E}
\]


\item \(E^*(X)\) is a graded algebra, so get maps
%
\[
  \spc{E} \times \spc[l]{E} \to \spc[k+l]{E}
\]

\item \(E^*(X)\) is an \(E^*\)\hyp{}module, so for each \(v \in E^k\) have
a map
%
\[
  \xi v \co \text{pt} \to \spc{E}
\]

\item \(E^l(\spc{E})\) acts on \(E^k(X)\), so we have composition
%
\[
  \Hom{\Topcat}{\spc[l]{E}}{\spc[k]{E}} \times
  \Hom{\Topcat}{\spc[m]{E}}{\spc[l]{E}} \to
  \Hom{\Topcat}{\spc[m]{E}}{\spc{E}}
\]
\end{enumerate}
\end{frame}

In summary we get the following maps, which have to satisfy certain
compatibility relationships that we won't specify:

\begin{frame}
\frametitle{The Structure}
\begin{enumerate}

\only<presentation>{\vfill}

\item \(E^*(\spc{E})\) is a graded algebra

\only<presentation>{\vfill}

\item Co\hyp{}addition: \(\Delta^+ \co E^*(\spc{E}) \to E^*(\spc{E}) \wotimes
E^*(\spc{E})\)

\only<presentation>{\vfill}

\item Co\hyp{}multiplication: \(\Delta^\times \co E^*(\spc[k+l]{E}) \to
E^*(\spc{E}) \wotimes E^*(\spc[l]{E})\)

\only<presentation>{\vfill}

\item Co\hyp{}linear: \(\epsilon^v \co E^*(\spc{E}) \to E^*(\text{pt}) =
E^*\)

\only<presentation>{\vfill}

\item Composition: \(E^k(\spc[l]{E}) \times E^l(\spc[m]{E}) \to
E^k(\spc[m]{E})\)

\only<presentation>{\vfill}
\end{enumerate}
\end{frame}

\section{Answer II: Co\hyp{}monads}

\begin{frame}<handout>

\begin{theorem}[Boardman, Johnson, Wilson]
\(E^*(\spc[*]{E})\) represents a \emph{co\hyp{}monad} in \(\galg\).
\(E^*(X)\) is a \emph{co\hyp{}module} for this co\hyp{}monad.
\end{theorem}

\vfill

\begin{enumerate}
\item \(A^* \to \Hom{\galg}{E^*(\spc[*]{E})}{A^*}\) lifts to \(U \co
\galg \to \galg\)

\item \(\mu \co U \to U^2\), \(\epsilon \co U \to I\) (obvious diagrams)

\item co\hyp{}action \(\rho \co E^*(X) \to U(E^*(X)\)
\end{enumerate}

\vfill
%
\[
  E^k(X) \cong \Hom{\Topcat}{X}{\spc{E}} \ni \alpha \mapsto \alpha^* \co
  E^*(\spc{E}) \to E^*(X)
\]

\vfill

simple, complete, elegant, \ldots impractical

\end{frame}

This structure is certainly rich, but simply listing it does not offer
any great insight as to how it fits together or what it is for.  Thus
we search for some simple statement that encapsulates all of this
structure.  The first such answer is due to Boardman, Johnson, and
Wilson in their paper in the Handbook of Algebraic Topology in 1995.

\begin{theorem}[Boardman, Johnson, Wilson]
\(E^*(\spc[*]{E})\) represents a \emph{co\hyp{}monad} in \(\galg\).
\(E^*(X)\) is a \emph{co\hyp{}module} for this co\hyp{}monad.
\end{theorem}

What this means is
%
\begin{enumerate}
\item The hom\hyp{}functor \(\galg \to \set\)
%
\[
  A^* \mapsto \Hom{\galg}{E^*(\spc[*]{E})}{A^*}
\]
%
has a natural lift to a functor \(U \co \galg \to \galg\)

\item There are natural transformations \(\mu \co U \to U^2\),
\(\epsilon \co U \to I\) satisfying obvious co\hyp{}associativity and
co\hyp{}unit diagrams.

\item There is a co\hyp{}action map \(\rho \co E^*(X) \to U(E^*(X))\)
satisfying the obvious diagrams.
\end{enumerate}

We'll look at the how the lift works a bit later.
The action map is
straightfoward; it is
%
\[
  E^k(X) \cong \Hom{\Topcat}{X}{\spc{E}} \ni \alpha \mapsto \alpha^* \co
  E^*(\spc{E}) \to E^*(X)
\]

This description is \ldots
complete, simple, elegant, \ldots impractical.


This is authors' opinion of this answer:

\begin{frame}<handout| article>

\begin{quote}
This is our elegant but extremely terse answer, \ldots [later] we
translate this answer into practical language, in the context of Hopf
rings, that we can use for computation.
\end{quote}

\begin{flushright}
\emph{Boardman, Johnson, Wilson}
\end{flushright}

\end{frame}

\section{Answer III: Hopf Rings}

\begin{frame}<handout>

Homology: \(E_*(-) \co \Topcat \to \gmod\)

\vfill

\(X\) ``good'', \(E^*(X) = \Hom{\gmod}{E_*(X)}{E^*}\)

\vfill

\(E_*(\spc[*]{E})\) ``pre\hyp{}dual'' to \(E^*(\spc[*]{E})\)

\vfill

%
\[
  E^k(X) \cong \Hom{\Topcat}{X}{\spc{E}} \ni \alpha \mapsto \alpha_* \co
   E_*(X) \to E_*(\spc{E})
\]

\vfill

\(E_*(\spc[*]{E})\) is an ``enriched Hopf ring''

\end{frame}

The ``practical language'' that Boardman, Johnson, and
  Wilson refer to is one that uses the associated homology theory,
\(E_*(-)\).  This is a \emph{covariant} functor
%
\[
  E_*(-) \co \Topcat \to \gmod
\]


For ``good'' spaces, \(E^*(X)\) is the
\(E^*\)\hyp{}linear dual of \(E_*(X)\) (and \(E_*(X)\)
  is a graded coalgebra).  We shall call a cohomology theory ``good''
  if each \(\spc{E}\) is a ``good'' space in this sense.

So \(E_*(\spc{E})\) is ``pre\hyp{}dual'' to
  \(E^*(\spc{E})\) and is simpler in some respects.  Analogous to the
co\hyp{}action map
%
\[
  E^k(X) \cong \Hom{\Topcat}{X}{\spc{E}} \ni \alpha \mapsto \alpha^* \co
  E^*(\spc{E}) \to E^*(X)
\]
%
we have
%
\[
  E^k(X) \cong \Hom{\Topcat}{X}{\spc{E}} \ni \alpha \mapsto \alpha_* \co
   E_*(X) \to E_*(\spc{E})
\]
%
and this shows how \(E_*(\spc[*]{E})\) might fit into
  the picture.



Just as \(E^*(\spc[*]{E})\) has considerable structure, so also does
\(E_*(\spc[*]{E})\).  Boardman, Johnson, and Wilson call it an
\emph{enriched Hopf ring}.  Its structure is as follows.

\begin{frame}
\frametitle{Enriched Hopf Ring Structure}

\only<presentation>{\vfill}

\begin{enumerate}
\item \(E_*(\spc{E})\) is a graded \(E^*\)\hyp{}coalgebra

\only<presentation>{\vfill}

\item \(*\)\hyp{}multiplication
%
\(
  E_*(\spc{E}) \otimes E_*(\spc{E}) \to E_*(\spc{E})
\)

\only<presentation>{\vfill}

\item \(\circ\)\hyp{}multiplication
%
\(
  E_*(\spc{E}) \otimes E_*(\spc[l]{E}) \to E_*(\spc[k+l]{E})
\)

\only<presentation>{\vfill}

\item extra linear structure, \(v \in E^k\)
%
\(
  (\xi v)_* \co E^* = E_*(\text{pt}) \to E_*(\spc{E})
\)

\only<presentation>{\vfill}

\item co\hyp{}composition (``mposition''), \(r \in E^l(\spc{E})\)
%
\(
  r_* \co E_*(\spc{E}) \to E_*(\spc[l]{E})
\)

\only<presentation>{\vfill}
\end{enumerate}
\end{frame}


\begin{frame}<handout>

(1 - 4): Hopf ring \emph{aka} algebra object in coalgebras

(5): enriched structure

\vfill

Hopf ring: sums of \(*\)\hyp{}products of \(\circ\)\hyp{}products of
generators subject to relations

\[
  \sum_i \bstar_j \bcirc_k g_{i j k}
\]

\vfill

Operations \(r,s\)
%
\[
  \langle s r, c \rangle = \langle s, r_* c \rangle
\]

Need to know \(r_* c\) all \(c\).
\end{frame}

(1 - 4): Hopf ring \emph{aka} algebra object in coalgebras

(5): enriched structure

The ``enriched'' part is slightly messy and complicated to work with
-- but it is necessary if one is interested in operations.

A Hopf ring is usually specified by giving a list of generators and
relations.  Elements of the Hopf ring are
sums of \(*\)\hyp{}products
of \(\circ\)\hyp{}products of these generators subject to these
relations.  A typical element thus has the form
%
\[
  \sum_i \bstar_j \bcirc_k g_{i j k}
\]

The structure of an enriched Hopf ring is dual to that on the set of
unstable operations.  For most of structure there is not overmuch to
choose between the two pictures, though one might prefer the Hopf ring
as it has one co\hyp{}multiplication and two multiplications against
the operations having two co\hyp{}multiplications and one
multiplication.  However, what really distinguishes them is
composition.

Given two operations, \(r, s\), expressed as linear functionals on the
Hopf ring, the composition \(s r\) is determined by the formula
%
\[
  \langle s r, c \rangle = \langle s, r_* c \rangle
\]
%
so we need to know \(r_* c\) for all \(c\).

Generally there is some
explicit -- but complicated -- formula for \(r_* c\) for each of the
generators and then one uses certain formulae for
\(\circ\)\hyp{}products and \(*\)\hyp{}products to get \(r_* c\) for
all \(c\).


\begin{frame}
\frametitle{Push Forward Formulae}

Example: \(K(1)^0(\spc[0]{K(1)})\)

\(*\)\hyp{}generators: \(b^J\), \(J = (j_0, j_1, \dotsc)\) with \(0
\le j_i \le p - 1\), almost all zero, and \(\sum j_i = 0 \mod p - 1\)

relations: \((b^J)^{* p} = 0\)

\(r_* b^J\): \(b^J\) built by \(\circ\)\hyp{}multiplication of
elements \(b_k\); \(r_* b_k\) is the coefficient of \(x^k\) in the
formal identity:
%
\[
  r_* b(x) = \lb \langle r, 1_2 \rangle \rb * \bstar_{j=1}^\infty
  b(x)^{\circ j} \circ \lb \langle r, b_j \rangle \rb
\]

After which, we use the formulae:
%
\begin{align*}
r_*(a \circ c) &= \sum_i \sum_j \pm \bstar_\alpha {r_{\alpha}'}_*
a_{i, \alpha} \circ {r_{\alpha}''}_* c_{j, \alpha} \\
%
r_*(a * c) &= \sum_i \sum_j \pm \bstar_\alpha {r_{\alpha}'}_*
a_{i, \alpha} \circ {r_{\alpha}'''}_* c_{j, \alpha}
\end{align*}
\end{frame}

\section{Interlude: Algebras and Modules}

\begin{frame}<handout>

\(\abg\) abelian groups.

\(T \co \abg \to \abg\) representable co\hyp{}monad, object \(R\).

\vfill

\(T^2\) also representable:
%
\[
  T^2(M) = \Hom{\abg}{R}{\Hom{\abg}{R}{M}} = \Hom{\abg}{R \otimes
    R}{M}
\]
%
\begin{align*}
T &\to T^2 & R \otimes R &\to R \\
%
T &\to I & \Z &\to R
\end{align*}

\(R\) is a \emph{ring}.

\vfill

\(M\) \(T\)\hyp{}co\hyp{}module
%
%
\begin{align*}
  \rho &\in \Hom{\abg}{M}{T(M)} \\
%
&= \Hom{\abg}{M}{\Hom{\abg}{R}{M}} \\
%
&= \Hom{\abg}{R \otimes M}{M}
\end{align*}
%
\(M\) is an \(R\)\hyp{}module 

\vfill

Key: \(T\) has a left adjoint, and so \(T^2\) is representable
\end{frame}


Boardman, Johnson, and Wilson take some care to justify their choice
of co\hyp{}monads to describe the structure of unstable operations.
At one point, they make the following remark 

\begin{frame}<handout| article>

\begin{center}
\begin{minipage}{0.68\textwidth}
The problem \ldots is \ldots the tensor product \ldots that is simply
unavailable for operations that are not additive \textbf{(not that
  this has stopped us from trying)}.
\end{minipage}
\end{center}

\begin{flushright}
Boardman, Johnson, Wilson (emphasis added)
\end{flushright}

\end{frame}

This, together with the earlier quote, suggests that they are aware of
the disadvantages of the co\hyp{}monad description and looked for one
more akin to the usual action of an algebra on a module but without
success.

Co\hyp{}monads are certainly less familiar than algebras so let us
look at a simpler example to see what is going on.

Let \(\abg\) be the category of abelian groups.  Let \(T \co \abg \to
\abg\) be a representable co\hyp{}monad on \(\abg\) with representing
object \(R\).

The functor \(T^2\) is also representable:
%
\[
  T^2(M) = \Hom{\abg}{R}{\Hom{\abg}{R}{M}} = \Hom{\abg}{R \otimes
    R}{M}
\]
%
so the co\hyp{}monad structure of \(T\) translates into morphisms:
%
\begin{align*}
T &\to T^2 & R \otimes R &\to R \\
%
T &\to I & \Z &\to R
\end{align*}
%
The diagrams translate into the axioms for \(R\) to be a ring.

If \(M\) is a \(T\)\hyp{}co\hyp{}module, we have a morphism of abelian
groups
%
\begin{align*}
  \rho &\in \Hom{\abg}{M}{T(M)} \\
%
&= \Hom{\abg}{M}{\Hom{\abg}{R}{M}} \\
%
&= \Hom{\abg}{R \otimes M}{M}
\end{align*}
%
so \(M\) is an \(R\)\hyp{}module in the usual sense.


Thus we can either talk of representable co\hyp{}monads and their
co\hyp{}modules or of rings and their modules.  The latter is more
intuitive.

The key feature of the correspondence was the fact that \(T^2\) was
representable, equivalently the existence of a left adjoint to the
hom\hyp{}functor \(M \to \Hom{\abg}{R}{M}\).


\section{Birings and Plethories}

\begin{frame}<handout>

\(T \co \alg \to \alg\) representable co\hyp{}monad, object \(B\)

\(\Hom{\alg}{B}{A}\) not ``naturally'' in \(\alg\)

\(f, g \co B \to A\) define
%
\begin{align*}
f + g &\co B \to A \\
%
f g &\co B \to A \\
%
\lambda &\co B \to A \\
%
\text{co\hyp{}addition: } \Delta^+ &\co B \to B \otimes B \\
%
\text{co\hyp{}multiplication: } \Delta^\times &\co B \to B \otimes B \\
%
\text{co\hyp{}scalar: } \epsilon^\lambda &\co B \to R \\
%
\end{align*}
\end{frame}

\begin{frame}<handout>
\begin{align*}
f + g &\co B \xrightarrow{\Delta^+} B \otimes B \xrightarrow{f \otimes
  g} A \otimes A \xrightarrow{m} A \\
%
f + g(b) &= \sum_i f(b^{(1)}_i)g(b^{(2)}_i) \\
%
f  g &\co B \xrightarrow{\Delta^\times} B \otimes B \xrightarrow{f \otimes
  g} A \otimes A \xrightarrow{m} A \\
%
f g(b) &= \sum_i f(b^{[1]}_i)g(b^{[2]}_i) \\
%
\lambda &\co B \xrightarrow{\epsilon^\lambda} R \xrightarrow{\iota_A}
A \\
%
\lambda(b) &= \epsilon^\lambda(b) 1_A \\
%
\Delta^+ b &= \sum_i b_i^{(1)} \otimes b_i^{(2)} \\
%
\Delta^\times b &= \sum_i b_i^{[1]} \otimes b_i^{[2]}
\end{align*}

\end{frame}

Now let us return to algebras over a coefficient ring \(R\).  We want
to consider a representable functor \(T \co \alg \to \alg\),
with representing object \(B\).

The first thing to observe is that with abelian groups, the
hom\hyp{}sets were again naturally abelian groups so we could consider
things like
%
\[
  \Hom{\abg}{M_1}{\Hom{\abg}{M_2}{M_3}}
\]
%
for arbitrary abelian groups \(M_1,M_2,M_3\).
This \emph{doesn't} hold for algebras.  So if \(T \co \alg \to \alg\) is
representable, the representing object must be somewhat special for
the hom\hyp{}functor to lift to algebras.

Given \(f, g \co B \to A\) and \(\lambda \in R\) we need to define
%
\begin{align*}
f + g &\co B \to A \\
%
f g &\co B \to A \\
%
\lambda &\co B \to A
\end{align*}
%
The techniques of universal algebra tell us that we need the following
maps to exist on \(B\)
%
\begin{align*}
\text{co\hyp{}addition: } \Delta^+ &\co B \to B \otimes B \\
%
\text{co\hyp{}multiplication: } \Delta^\times &\co B \to B \otimes B \\
%
\text{co\hyp{}scalar: } \epsilon^\lambda &\co B \to R
\end{align*}
%
writing \(\Delta^+ b = \sum_i b_i^{(1)} \otimes b_i^{(2)}\) and
\(\Delta^\times b = \sum_i b_i^{[1]} \otimes b_i^{[2]}\), these give
us the required operations as follows
%
\begin{align*}
f + g &\co B \xrightarrow{\Delta^+} B \otimes B \xrightarrow{f \otimes
  g} A \otimes A \xrightarrow{m} A \\
%
f + g(b) &= \sum_i f(b^{(1)}_i)g(b^{(2)}_i) \\
%
f  g &\co B \xrightarrow{\Delta^\times} B \otimes B \xrightarrow{f \otimes
  g} A \otimes A \xrightarrow{m} A \\
%
f g(b) &= \sum_i f(b^{[1]}_i)g(b^{[2]}_i) \\
%
\lambda &\co B \xrightarrow{\epsilon^\lambda} R \xrightarrow{\iota_A}
A \\
%
\lambda(b) &= \epsilon^\lambda(b) 1_A
\end{align*}
%
Note: the tensor product appears because it is the categorical
coproduct of algebras.

The structure maps have to satisfy some obvious compatibility
relations which we won't state here.  The object \(B\) is officially
known as a \emph{coalgebra object in the category of algebras} and
more commonly as a \emph{biring}.  It was first identified by Tall and
Wraith in 1970.

\begin{frame}<handout| article>
\begin{examples}
~

\begin{enumerate}
\item Let \(R\) be a finite ring; \(\Hom{\set}{R}{R}\) is a biring
with co\hyp{}structure coming from the first factor of \(R\).

\item For a set \(X\), the free algebra on \(X\) is a biring with
co\hyp{}structure defined by:
%
\begin{align*}
\Delta^+ x &= 1 \otimes x + x \otimes 1 \\
%
\Delta^\times x &= x \otimes x \\
%
\epsilon^\lambda(x) &= \lambda
\end{align*}
%
(\(x\) is called \emph{ring\hyp{}like})

\item in particular the polynomial algebra, \(R\lb x \rb\), is a
biring.
\end{enumerate}
\end{examples}
\end{frame}

\begin{frame}<handout>

Question: represent \(T^2\)? \\
Answer: Yes, \(B \odot B\)

Plethory:
%
\[
B \odot B \to B, \qquad I = R \lb e \rb \to B
\]

\(T\)\hyp{}co\hyp{}module:
%
\[
  B \odot A \to A
\]

\(B \odot A\): free algebra on \((b,a)\) subject to:
%
\begin{align*}
(b_1 + \lambda b_2, a) &= (b_1, a) + \lambda (b_2, a) \\
%
(b, a_1 + a_2) &= \sum_i (b_i^{(1)}, a_1)(b_2^{(2)}, a_2) \\
%
(b, a_1 a_2) &= \sum_i (b_i^{[1]}, a_1)(b_2^{[2]}, a_2) \\
%
(b, \lambda 1_A) &= \epsilon^\lambda(b) 1
\end{align*}
\end{frame}

The next question is: can we represent \(T^2\)?  The techniques of
universal algebra say that certain combinations of representable
functors are representable and \(T^2\) is one of these so, yes,
\(T^2\) is representable.  Let us write \(B \odot B\) for the
representing object, it must again be a biring.

The structure of a co\hyp{}monad on \(T\) induces biring maps
%
\[
  B \odot B \to B, \qquad I \to B
\]
%
where \(I = R\lb x \rb\) (which represents the identity functor);
these satisfy the obvious diagrams.

A biring with this structure is called, variously, a \emph{biring
  triple}, a \emph{Tall\hyp{}Wraith biring triple}, or a
\emph{plethory}.

Similarly, there is a product \(B \odot A\) for an algebra \(A\) and
the structure of a \(T\)\hyp{}co\hyp{}module on  \(A\) induces an
action map
%
\[
  B \odot A \to A
\]

Thus we have a picture much like the intuitive picture of an algebra
acting on a module, only it is a plethory acting on an algebra.

What does \(B \odot A\) look like?  Tall and Wraith gave the first
construction as the free algebra on symbols \((b,a)\) subject to the
relations:
%
\begin{align*}
(b_1 + \lambda b_2, a) &= (b_1, a) + \lambda (b_2, a) \\
%
(b, a_1 + a_2) &= \sum_i (b_i^{(1)}, a_1)(b_2^{(2)}, a_2) \\
%
(b, a_1 a_2) &= \sum_i (b_i^{[1]}, a_1)(b_2^{[2]}, a_2) \\
%
(b, \lambda 1_A) &= \epsilon^\lambda(b) 1
\end{align*}
%
These should be compared with the formula for \((f + g)(b)\) and \((f
g)(b)\) which make \(\Hom{\alg}{B}{A}\) into an algebra.

Unlike the tensor product, this is neither symmetric nor bilinear
(though it is linear in \(B\)).

\begin{frame}<handout>
\begin{examples}
\begin{enumerate}
\item \(\Hom{\set}{R}{R}\) composition, \(\Hom{\set}{X}{R}\) a
module

\item \(R \lb x \rb\) composition,  \(R \lb x \rb \odot A \cong A\) so
any \(A\)

\item \(M\) monoid, \(P = R \lb \lvert M \rvert \rb\), \(m\) \emph{ring-like}
\begin{enumerate}
\item \(M = \N_0\), \(P = R \lb x_j | j \ge 0\rb\); \(A\) with algebra
endomorphism

\item \(M = \Z\), \(P = R \lb x_j | j \in \Z \rb\); \(A\) with algebra
automorphism

\item \(M = \Z/2\), \(P = R \lb x, x^{-1} \rb\); \(A\) with involution
\end{enumerate}
\end{enumerate}
\end{examples}
\end{frame}

\begin{examples}
~

\begin{enumerate}
\item Let \(R\) be a finite ring, \(\Hom{\set}{R}{R}\) is a plethory
with composition as the action map.  For any set \(X\),
\(\Hom{\set}{X}{R}\) is a module over this plethory with action map
composition.

\item \(R\lb x \rb\) is a plethory by composition.  It is, in fact,
the initial plethory and there is a canonical isomorphism \(R \lb x
\rb \odot A \cong A\) for an \(R\)\hyp{}algebra \(A\).  Thus all
\(R\)\hyp{}algebras are \(R \lb x \rb\)\hyp{}modules.

\item Let \(M\) be a monoid; the free algebra on the underlying set of
\(M\) is a plethory with \(m \in M\) ring\hyp{}like and composition
from the monoid product: \(m_1 \odot m_2 \to m_1 m_2\).  This is the
``free plethory on \(M\)''.  Let us write this as \(P(M)\).

Any algebra with an action of \(M\) by algebra morphisms is a module
for this plethory.

\begin{examples}
~

\begin{enumerate}
\item \(M = \N_0\); \(P(M) = R \lb x_0, x_1, x_2, \dotsc \rb\).  A
\(P\)\hyp{}module is an algebra with a specified algebra endomorphism,
\((A, \psi)\).  \(x_j\) acts as \(\psi^j\) thus, say,
%
\[
  x_1^{2} x_2(a) = \psi(a)^2 \psi^2(a) = \psi(a) \psi(a) \psi(\psi(a))
\]

\item \(M = \Z\); \(P(M) = R \lb \dotsc, x_{-1}, x_0, x_1, \dotsc
\rb\).  A \(P\)\hyp{}module is an algebra with a specified
algebra automorphism.

\item \(M = \Z/2\); \(P(M) = R \lb x_{-1}, x_1 \rb\).  A
\(P\)\hyp{}module is an algebra with a specified involution.
\end{enumerate}
\end{examples}
\end{enumerate}
\end{examples}


\section{Answer IV: Plethories and Modules}

\begin{frame}<handout| article>

\begin{theorem}[S-Whitehouse]
\(E^*(\spc[*]{E})\) is a \emph{graded, completed, plethory};
\(E^*(X)\) is an \(E^*(\spc[*]{E})\)\hyp{}module.
\end{theorem}
\end{frame}

\begin{frame}
\frametitle{Graded Plethories}

\only<presentation>{\vfill}

\begin{enumerate}
\item \(P_k^*\) is a graded algebra

\only<presentation>{\vfill}

\item Co\hyp{}addition: \(\Delta^+ \co P_k^* \to P_k^* \wotimes P_k^*\)

\only<presentation>{\vfill}

\item Co\hyp{}multiplication: \(\Delta^\times \co P_{k+l}^* \to P_k^*
\wotimes P_l^*\)

\only<presentation>{\vfill}

\item Co\hyp{}linear: \(\epsilon^\lambda \co P_k^* \to k^*\)

\only<presentation>{\vfill}

\item Composition: \(P_l^k \odot P^l_m \to P_m^k\)

\only<presentation>{\vfill}
\end{enumerate}
\end{frame}

\begin{frame}<handout>

Identity: \(I_*^* \coloneqq R \lb \iota_k \co k \in \Z\rb\), \(\iota_k
\in I^k_k\)

\vfill

\begin{examples}
\begin{enumerate}
\item ungraded \(P\), extends to \(+\)vely graded \(P_*^*\) by:
%
\[
P_k^l = \begin{cases}
P & k = l = 0 \\
%
I_k^l & \text{otherwise}
\end{cases}
\]
%
\(A^*\) with \(A^0\) \(P\)\hyp{}module is \(P^*_*\)\hyp{}module: \(P\)
action on \(A^0\), identity on rest.

\item \(H = H\Q\), \(H^*(\spc[*]{H})\) is this for \(P =
\Hom{\set}{\Q}{\Q}\); \(E^0(X) = \Hom{\set}{\pi_0(X)}{\Q}\)

\item \(K(1)^0(\spc[0]{K(1)})\) free completed plethory on monoid
%
\[
  \langle x, y | x y = y x = y^2 = y \rangle
\]
%
\(x = \Psi^{\tilde{q}}\), \(y = \Psi^0\)
\end{enumerate}
\end{examples}
\end{frame}

A graded plethory is actually bigraded, \(P_*^*\).  Following the
example of cohomology, we call the upper degree the \emph{homological}
degree and the lower one the \emph{spacial} degree.  The identity
functor is represented by
%
\[
  I_*^* \coloneqq R \lb \iota_k \co k \in \Z \rb
\]
%
with \(\iota_k \in I^k_k\); thus it is polynomial in even spacial
degree and exterior in odd.


The ``completed'' refers to the fact that the algebras are completed
with respect to the filtration topology, and in particular tensor
products must be completed appropriately.

\begin{examples}
~

\begin{enumerate}
\item There is a natural extension from ungraded plethories to
positively graded plethories.  Assume that the coefficient ring is
concentrated in degree \(0\).  Let \(P\) be an ungraded plethory.
Define \(P^*_*\) as the plethory
%
\[
  P^k_l \coloneqq \begin{cases}
  P &\text{if } (k,l) = (0,0) \\
%
  I^k_l &\text{otherwise}
  \end{cases}
\]
%
(note that \(I^k_0 = \{0\}\) unless \(k = 0\)).  Examining the
structure of a graded plethory, we can see that the only interaction
between the two parts of the structures is as follows
%
\begin{gather*}
\Delta^\times \co P_k^l \to P_k^l \wotimes P_0^0 \\
%
P_0^0 \odot P_m^0 \to P^0_m
\end{gather*}
%
In the former, if \(k \ne 0\) we use the canonical map \(I_0^0 = R \lb
\iota_0 \rb \to P\)
%
\[
  \Delta^\times \co I_k^* \to I_k^* \wotimes I_0^0 \to I_k^* \wotimes
  P
\]
%
In the latter, \(P^0_m\) is canonically isomorphic to \(R\) and there
is a canonical isomorphism \(P \odot R \to R\) which we use.


Now let \(A^*\) be a positively graded algebra such that \(A^0\) is a
\(P\)\hyp{}module.  Then \(A^*\) is a \(P^*_*\) module with \(P\)
acting on \(A^0\) and \(I_*^*\) acting via the identity on the rest.

\item \(H = H \Q\).
This is a specific case of the previous extension with \(P =
\Hom{\set}{\Q}{\Q}\).  Since \(H^0(X) = \Hom{\set}{\pi_0(X)}{\Q}\),
\(H^*(X)\) satisfies the required conditions.

\item \(K(1)^0(\spc[0]{K(1)})\).  This is the free completed plethory
on the submonoid of \(\{\N_0, \times\}\) generated by \(\{0,
  \tilde{q}\}\); equivalently,
%
\[
  \langle x, y | x y = y x = y^2 = y\rangle
\]
%
Then \(x\) acts as \(\Psi^{\tilde{q}}\), \(y\) as \(\Psi^0\).  The
completion is slightly non-standard as we take formal sums in \(x -
1\) rather than \(x\).
\end{enumerate}
\end{examples}

\section{Bonus: Enriched Hopf Rings}

\begin{frame}<handout>

Question: What is a Hopf ring? \\
Answer: Algebra object in coalgebras

\vfill

\(H^* \co C \to \Hom{\coalg}{C}{H}\) lifts to \(\alg\)

\vfill

\(P\) plethory, co\hyp{}monad \(P_*\), category of \(P\)\hyp{}modules

\vfill

Question: When does \(H^*\) factor through \(P\)\hyp{}modules? \\
Answer: natural transformation \(H^* \to P_* H^*\)

\vfill

\(P_* H^*\) representable, \(P \boxtimes H\); get \(P_* \co \hopf \to
\hopf\)

\vfill

Answer: when \(H\) is a \(P\)\hyp{}module

\vfill

Issues:
\begin{enumerate}
\item contravariant in \(P\)
\item limits in \(\coalg\) complicated
\end{enumerate}
\end{frame}


Finally, as an added bonus we get a simpler description of an enriched
Hopf ring.

\noindent Question: What is a Hopf ring? \\
Answer: An algebra object in coalgebras.

That is, to say that a coalgebra \(H\) is a Hopf ring is the same as
saying that the contravariant hom\hyp{}functor \(H^* \co C \to
\Hom{\coalg}{C}{H}\) lifts to a functor into algebras.

Suppose we have a plethory \(P\) representing a functor \(P_* \co \alg
\to \alg\); then we have a category \(P-\alg\) of \(P\)\hyp{}modules,
which is a subcategory of \(\alg\).

\noindent Question: When does \(H^* \co \coalg \to \alg\) actually land in
\(P-\alg\)? \\
Answer: When we have a natural transformation \(H^* \to P_* H^*\).

This answer is co\hyp{}monadic in form so we would like to convert it
to a module\hyp{}like setting.  We need to represent the functor \(P_*
H^* \co \coalg \to \alg\).  Again, universal algebra tells us that we
can.  Write the representing object as \(P \boxtimes H\).

This is covariant in \(H\) and contravariant in \(P\); these variances
mean that the action map is:
%
\[
  H \to P \boxtimes H.
\]
%
This is like saying that \(H\) is a \(P\)\hyp{}module, except that
\(P\) and \(H\) now lie in different categories.  There is no problem
with this, however.

We can reformulate our answer:

\noindent Question: When does \(H^* \co \coalg \to \alg\) actually land in
\(P-\alg\)? \\
Answer: When \(H\) is a \(P\)\hyp{}module.

There are two issues with this answer:
%
\begin{enumerate}
\item The assignment \((P,H) \to P \boxtimes H\) is covariant in \(H\)
but \emph{contravariant} in \(P\).  This makes it somewhat tricky to
work with.

\item It is constructed using limits in the category \(\coalg\), but
arbitrary small limits in \(\coalg\) are complicated.

As an example, let \(C_r \coloneqq \Z \langle \beta_0, \dots, \beta_{2^r}
\rangle\) with co\hyp{}multiplication
%
\[
  \Delta \beta_k = \sum_{i + j = k} \beta_i \otimes \beta_j
\]
%
Define maps \(f_r \co C_r \to C_{r-1}\) by \(f_r(\beta_{2k}) = \beta_k\)
and \(f_r(\beta_{2k+1}) = 0\).

The limit of this system as abelian groups is
%
\[
 \prod_{\N \times \N} \Z
\]
%
but the limit as coalgebras is just \(\Z\).
\end{enumerate}

\begin{frame}<handout| article>
\begin{theorem}[S-Whitehouse]
The functor \(H \to P \boxtimes H\) has a left adjoint, \(H \to P
\circledast H\).
\end{theorem}

\begin{onlyenv}<article>
This converts the action map \(H \to P \boxtimes H\) into the more
familiar form \(P \circledast H \to H\).  It is covariant in both
arguments and uses colimits in \(\coalg\) which are much nicer.
\end{onlyenv}

\begin{onlyenv}<handout>
\begin{enumerate}
\item covariant in \(P\)
\item uses colimits
\end{enumerate}
\end{onlyenv}

\begin{theorem}[S-Whitehouse]
The \emph{enriched} part of \(E_*(\spc[*]{E})\) says that it is an
\(E^*(\spc[*]{E})\)\hyp{}module.  The natural map
%
\[
  E^*(X) \to \Hom{\gcoalg}{E_*(\spc[*]{E})}{E_*(X)}
\]
%
is a morphism of \(E^*(\spc[*]{E})\)\hyp{}modules.
\end{theorem}

In fact, the most general statement is that the following is a map of
\(E^*(\spc[*]{E})\)\hyp{}modules:
%
\begin{align*}
E^*(X) &\to \Hom{\gcoalg}{F_*(\spc[*]{E})}{F_*(X)} \\
%
  E^*(X) \ni \alpha &\mapsto \alpha_* \co F_*(\spc[*]{E}) \to F_*(X).
\end{align*}
\end{frame}

\begin{frame}<beamer>
\end{frame}

\end{document}