\section{Meta-Hopf Algebras and the Drinfel'd Double Construction}

\begin{quote} \footnotesize
Sections~\ref{sec:RVK}--\ref{sec:SolvApp} of this paper can be regarded
as independet paperlets which can be read in any order.
\end{quote}

\begin{Aside}\fbox{\begin{minipage}{0.9\linewidth}\sl
If $U$ is a vector space over $\bbQ$ (or another field) we form
$U^{\otimes n}=U\otimes\cdots\otimes U$ ($n$ times), and in particular
$U^{\otimes 0}=\bbQ$, $U^{\otimes 1}=U$, $U^{\otimes 2}=U\otimes U$,
etc. We identify $U^{\otimes 1}\cong U^{\otimes 1}\otimes U^{\otimes
0}\cong U^{\otimes 0}\otimes U^{\otimes 1}$. With these conventions,
a {\em Hopf Algebra} is a vector space $U$ endowed with maps $m\colon
U^{\otimes 2}\to U^{\otimes 1}$, $\Delta\colon U^{\otimes 1}\to U^{\otimes
2}$, $\eta\colon U^{\otimes 0}\to U^{\otimes 1}$, $\epsilon\colon
U^{\otimes 1}\to U^{\otimes 0}$, and an invertible $S\colon U^{\otimes
1}\to U^{\otimes 1}$ such that:

\begin{multicols}{2}
\begin{enumerate}[leftmargin=*,labelindent=0pt,itemsep=0pt,topsep=0pt]
\item $(m\otimes\Id)\act m = (\Id\otimes m)\act m$.
\item % In $\Hom((U_0\otimes U_1=U_1\otimes U_0=U_1)\to U_1)$,
  $(\eta\otimes\Id)\act m = (\Id\otimes\eta)\act m = \Id$.
\item $\Delta\act(\Delta\otimes\Id) = \Delta\act(\Id\otimes\Delta)$.
\item $\Delta\act(\epsilon\otimes\Id) = \Delta\act(\Id\otimes\epsilon) = \Id$.
\item $m\act\Delta = (\Delta\otimes\delta)\act(\Id\otimes\sigma\otimes\Id)\act(m\otimes m)$, where 
$\sigma\colon U_2\to U_2$ is the transposition.
\item $\eta = (\eta\otimes\eta)\act m$.
\item $\epsilon = \Delta\act(\epsilon\otimes\epsilon)$.
\item $\Delta\act(S\otimes\Id)\act m = \Delta\act(\Id\otimes S)\act m = \epsilon\act\eta$.
\end{enumerate}
\end{multicols}
Note that we are not assuming $m=\sigma\act m$ (``commutativity'') or $\Delta=\Delta\act\sigma$
(``cocommutativity'').
\caption{Ordinary Hopf Algebras.} \label{aside:OrdinaryHopf}
\end{minipage}}\end{Aside}

A Hopf algebra is a vector space $U$ with some operations which
satisfy some axioms (see Aside~\ref{aside:OrdinaryHopf}). These axioms,
labeled (1)--(9) in the aside, never directly mention the vector space
structure of $U$; that structure is only used for the formation of
the spaces $U^{\otimes n}$ on which the operations are defined and in
which the axioms are stated. But this calls for a generalization ---
why not replace $U^{\otimes n}$ with sets $U_n$ that do not need to be
vector spaces, and replace operations such as $\Id\otimes\cdots\otimes
m\otimes\cdots\otimes\Id\colon U^{\otimes n}\to U^{\otimes(n-1)}$ with
arbitrary maps $m_n\colon U_n\to U_{n-1}$ in such a manner that the axioms
(1)--(9) would still make sense?

A bit of further reflection\footnoteT{A fuller but longer explanation
is at~\cite[Section~10.3]{KBH}.} leads one to realize that $m$ should
be replaced with a family of operations $m^{ij}_k$ which generalize
``multiply the content of the $i$th tensor factor with the content of
the $j$th tensor factor putting the result as a $k$th tensor factor'',
and that in fact, the restriction that the labels of the tensor factors
would be natural numbers is a (minor) handicap.  Hence we come to the
following convention and definition:

\begin{convention} If $S$, $A$, and $B$ are finite sets, $A\subset S$, and
$(S\setminus A)\cap B=\emptyset$ we write $S\setminus A\cup B$ for $(S\setminus
A)\sqcup B$. In fact, whenever we write $S\setminus A\cup B$ we automatically add the
assumptions that $A\subset S$ and $(S\setminus A)\cap B=\emptyset$,
even if this is not explicitly stated. In this context we often
suppress braces and commas when referring to sets with a small number of
elements, and automatically assume that these elements are distinct. Hence
for example $S\setminus ij\cup kl=(S\setminus\{i,j\})\sqcup\{k,l\}$,
and the assumptions $i\neq j$, $k\neq l$, $\{i,j\}\subset S$, and
$(S\setminus\{i,j\})\cap\{k,l\}=\emptyset$ are silently made. Finally, if $A$ or $B$ are omitted
from the notation, the omitted set is assumed to be the empty set and all further conventions still
apply.
\end{convention}

\begin{definition} A meta-Hopf algebra (in the category of sets)
is an assignment $U\colon S\mapsto U_S$ that assigns a (possibly big)
set $U_S$ to every finite set $S$ (we ignore the easily-resolved issues
that come with the likes of ``the set of all finite sets''), along with
the following families of operations and axioms:

{\em Most Interesting.} For any finite $S$,
operations $m^{ij}_k\colon U_S\to U_{S\setminus ij\cup k}$ called
``meta-multiplications'' or ``stitchings'', $\Delta^i_{jk}\colon U_S\to
U_{S\setminus i\cup jk}$ ``meta-comultiplications'' or ``doublings'',
$\eta_i\colon U_S\to U_{S\cup i}$ ``meta-units'', $\epsilon^i\colon U_S\to U_{S\setminus i}$
``meta-counits'', and $S_i\colon U_S\to U_S$ ``meta-antipodes'', satisfying the following axioms
(compare with Aside~\ref{aside:OrdinaryHopf}):

\begin{multicols}{2}\begin{enumerate}[leftmargin=*,labelindent=0pt,itemsep=0pt,topsep=0pt]
\item $m^{ij}_i\act m^{ik}_i = m^{jk}_j\act m^{ij}_i$.
\item $\eta_i\act m^{ij}_j = \eta_j\act m^{ij}_i = \Id$.
\item $\Delta^i_{ik}\act\Delta^i_{ij} = \Delta^i_{ij}\act\Delta^j_{jk}$.
\item $\Delta^j_{ij}\act\epsilon^i = \Delta^i_{ij}\act\epsilon^j = \Id$.
\item $m^{ij}_k\act\Delta^k_{ij} = \Delta^i_{kl}\act\Delta^j_{mn}\act m^{km}_i\act m^{ln}_j$.
\item $\eta_k = \eta_i\act\eta_j\act m^{ij}_k$.
\item $\epsilon^k = \Delta^k_{ij}\act\epsilon_i\act\epsilon_j$.
\item $\Delta^i_{jk}\act S_j\act m^{jk}_i = \Delta^i_{jk}\act S_k\act m^{jk}_i = \epsilon^i\act\eta_i$.
\end{enumerate}\end{multicols}
Note that we are not assuming $m^{ij}_k=m^{ji}_k$ (``commutativity'') or $\Delta^i_{jk}=\Delta^i_{kj}$
(``cocommutativity'').

\end{definition}

{\red MORE.}

\subsection{Meta-monoids vs.\ monoid objectss in a monoidal category}

{\red MORE.}