819 but (string diagrams)/(relations) is isomorphic to |
819 but (string diagrams)/(relations) is isomorphic to |
820 (pasting diagrams composed of smaller string diagrams)/(relations)} |
820 (pasting diagrams composed of smaller string diagrams)/(relations)} |
821 } |
821 } |
822 |
822 |
823 |
823 |
824 \begin{example}[The bordism $n$-category, ordinary version] |
824 \begin{example}[The bordism $n$-category of $d$-manifolds, ordinary version] |
825 \label{ex:bord-cat} |
825 \label{ex:bord-cat} |
826 \rm |
826 \rm |
827 \label{ex:bordism-category} |
827 \label{ex:bordism-category} |
828 For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional PL |
828 For a $k$-ball $X$, $k<n$, define $\Bord^{n,d}(X)$ to be the set of all $(d{-}n{+}k)$-dimensional PL |
829 submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W = W \cap \bd X \times \Real^\infty$. |
829 submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W = W \cap \bd X \times \Real^\infty$. |
830 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such $n$-dimensional submanifolds; |
830 For an $n$-ball $X$ define $\Bord^{n,d}(X)$ to be homeomorphism classes (rel boundary) of such $d$-dimensional submanifolds; |
831 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
831 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
832 $W \to W'$ which restricts to the identity on the boundary. |
832 $W \to W'$ which restricts to the identity on the boundary. |
|
833 For $n=1$ we have the familiar bordism 1-category of $d$-manifolds. |
|
834 The case $n=d$ captures the $n$-categorical nature of bordisms. |
|
835 The case $n > 2d$ captures the full symmetric monoidal $n$-category structure. |
833 \end{example} |
836 \end{example} |
834 |
837 |
835 %\nn{the next example might be an unnecessary distraction. consider deleting it.} |
838 %\nn{the next example might be an unnecessary distraction. consider deleting it.} |
836 |
839 |
837 %\begin{example}[Variation on the above examples] |
840 %\begin{example}[Variation on the above examples] |
888 is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
891 is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
889 It's easy to see that with $n=0$, the corresponding system of fields is just |
892 It's easy to see that with $n=0$, the corresponding system of fields is just |
890 linear combinations of connected components of $T$, and the local relations are trivial. |
893 linear combinations of connected components of $T$, and the local relations are trivial. |
891 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
894 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
892 |
895 |
893 \begin{example}[The bordism $n$-category, $A_\infty$ version] |
896 \begin{example}[The bordism $n$-category of $d$-manifolds, $A_\infty$ version] |
894 \rm |
897 \rm |
895 \label{ex:bordism-category-ainf} |
898 \label{ex:bordism-category-ainf} |
896 As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,\infty}(X)$ |
899 As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,d}_\infty(X)$ |
897 to be the set of all $k$-dimensional |
900 to be the set of all $(d{-}n{+}k)$-dimensional |
898 submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W$ is |
901 submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W = W \cap \bd X \times \Real^\infty$. |
899 contained in $\bd X \times \Real^\infty$. |
|
900 For an $n$-ball $X$ with boundary condition $c$ |
902 For an $n$-ball $X$ with boundary condition $c$ |
901 define $\Bord^{n,\infty}(X; c)$ to be the space of all $k$-dimensional |
903 define $\Bord^{n,d}_\infty(X; c)$ to be the space of all $d$-dimensional |
902 submanifolds $W$ of $X\times \Real^\infty$ such that |
904 submanifolds $W$ of $X\times \Real^\infty$ such that |
903 $W$ coincides with $c$ at $\bd X \times \Real^\infty$. |
905 $W$ coincides with $c$ at $\bd X \times \Real^\infty$. |
904 (The topology on this space is induced by ambient isotopy rel boundary. |
906 (The topology on this space is induced by ambient isotopy rel boundary. |
905 This is homotopy equivalent to a disjoint union of copies $\mathrm{B}\!\Homeo(W')$, where |
907 This is homotopy equivalent to a disjoint union of copies $\mathrm{B}\!\Homeo(W')$, where |
906 $W'$ runs though representatives of homeomorphism types of such manifolds.) |
908 $W'$ runs though representatives of homeomorphism types of such manifolds.) |