125 Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
125 Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
126 |
126 |
127 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
127 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
128 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
128 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
129 all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
129 all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
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130 with sufficient limits and colimits |
130 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
131 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
131 \nn{actually, need both disj-union/sub and product/tensor-product; what's the name for this sort of cat?} |
132 %\nn{actually, need both disj-union/sum and product/tensor-product; what's the name for this sort of cat?} |
132 and all the structure maps of the $n$-category should be compatible with the auxiliary |
133 and all the structure maps of the $n$-category should be compatible with the auxiliary |
133 category structure. |
134 category structure. |
134 Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then |
135 Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then |
135 $\cC(Y; c)$ is just a plain set. |
136 $\cC(Y; c)$ is just a plain set. |
136 |
137 |