674 and all the structure maps of the $n$-category are compatible with the auxiliary

   674 and all the structure maps of the $n$-category are compatible with the auxiliary

   675 category structure.

   675 category structure.

   676 Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then 

   676 Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then 

   677 $\cC(Y; c)$ is just a plain set.

   677 $\cC(Y; c)$ is just a plain set.

   678 

   678 

   681 First we must specify requirements for the auxiliary category.

   681 First we must specify requirements for the auxiliary category.

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

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

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

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

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

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

   685 and these two structures interact in the appropriate way.

   685 and these two structures interact in the appropriate way.

   686 Examples include 

   686 Examples include 

   687 \begin{itemize}

   687 \begin{itemize}

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

   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.

   689 \item topological spaces with product and disjoint union.

   690 \end{itemize}

   690 \end{itemize}

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

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

   692 we need a preliminary definition.

   695 we need a preliminary definition.

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

   696 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}.

   697 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".

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

   710 \[

   713 \[

   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),

   714 	\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 \]

   715 \]

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

   716 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.

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

   716 \end{itemize}

   719 \end{itemize}

   717 \end{axiom}

   720 \end{axiom}

   718 

   721 

   719 

   722 

   720 

   723