679 We will aim for a little bit more generality than we need and not assume that the objects

   680 of our auxiliary category are sets with extra structure.

   681 First we must specify requirements for the auxiliary category.

   682 It should have a {\it distributive monoidal structure} in the sense of 

   683 \nn{Stolz and Teichner, Traces in monoidal categories, 1010.4527}.

   684 This means that there is a monoidal structure $\otimes$ and also coproduct $\oplus$,

   685 and these two structures interact in the appropriate way.

   686 Examples include 

   687 \begin{itemize}

   688 \item vector spaces (or $R$-modules or chain complexes) with tensor product and direct sum; and

   689 \item topological spaces with product and disjoint union.

   690 \end{itemize}

   691 Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category,

   692 we need a preliminary definition.

   693 Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the 

   694 category $\bbc$ of {\it $n$-balls with boundary conditions}.

   695 Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition".

   696 Its morphisms are homeomorphisms $f:X\to X$ such that $f|_{\bd X}(c) = c$.

   697  

   698 \begin{axiom}[Enriched $n$-categories]

   699 \label{axiom:enriched}

   700 Let $\cS$ be a distributive symmetric monoidal category.

   701 An $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$,

   702 and modifies the axioms for $k=n$ as follows:

   703 \begin{itemize}

   704 \item Morphisms. We have a functor $\cC_n$ from $\bbc$ ($n$-balls with boundary conditions) to $\cS$.

   705 \item Composition. Let $B = B_1\cup_Y B_2$ as in Axiom \ref{axiom:composition}.

   706 Let $Y_i = \bd B_i \setmin Y$.  

   707 Note that $\bd B = Y_1\cup Y_2$.

   708 Let $c_i \in \cC(Y_i)$ with $\bd c_1 = \bd c_2 = d \in \cl\cC(E)$.

   709 Then we have a map

   710 \[

   711 	\gl_Y : \bigoplus_c \cC(B_1; c_1 \bullet c) \otimes \cC(B_2; c_2\bullet c) \to \cC(B; c_1\bullet c_2),

   712 \]

   713 where the sum is over $c\in\cC(Y)$ such that $\bd c = d$.

   714 This map is natural with respect to the action of homeomorphisms and with respect to restrictions.

   715 \item Product morphisms. \nn{Hmm... not sure what to say here. maybe we need sets with structure after all.}

   716 \end{itemize}

   717 \end{axiom}

   718 

   719 

   720 

   721 

   722 \nn{a-inf starts by enriching over inf-cat; maybe simple version first, then peter's version (attrib to peter)}

   723 

   724 \nn{blarg}

   725