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 |
679 We will aim for a little bit more generality than we need and not assume that the objects |
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. |
680 %of our auxiliary category are sets with extra structure. |
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} |
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691 For convenience, we will also assume that the objects of our auxiliary category are sets with extra structure. |
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692 (Otherwise, stating the axioms for identity morphisms becomes more cumbersome.) |
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693 |
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. |
715 \item Product morphisms. \nn{Hmm... not sure what to say here. maybe we need sets with structure after all.} |
718 %\item Product morphisms. \nn{Hmm... not sure what to say here. maybe we need sets with structure after all.} |
716 \end{itemize} |
719 \end{itemize} |
717 \end{axiom} |
720 \end{axiom} |
718 |
721 |
719 |
722 |
720 |
723 |