237 \begin{axiom}[Boundaries]\label{nca-boundary} |
237 \begin{axiom}[Boundaries]\label{nca-boundary} |
238 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
238 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
239 These maps, for various $X$, comprise a natural transformation of functors. |
239 These maps, for various $X$, comprise a natural transformation of functors. |
240 \end{axiom} |
240 \end{axiom} |
241 |
241 |
242 For $c\in \cl{\cC}_{k-1}(\bd X)$ we let $\cC_k(X; c)$ denote the preimage $\bd^{-1}(c)$. |
242 For $c\in \cl{\cC}_{k-1}(\bd X)$ we define $\cC_k(X; c) = \bd^{-1}(c)$. |
243 |
243 |
244 Many of the examples we are interested in are enriched in some auxiliary category $\cS$ |
244 Many of the examples we are interested in are enriched in some auxiliary category $\cS$ |
245 (e.g. $\cS$ is vector spaces or rings, or, in the $A_\infty$ case, chain complex or topological spaces). |
245 (e.g. $\cS$ is vector spaces or rings, or, in the $A_\infty$ case, chain complex or topological spaces). |
246 This means (by definition) that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure |
246 This means (by definition) that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure |
247 of an object of $\cS$, and all of the structure maps of the category (above and below) are |
247 of an object of $\cS$, and all of the structure maps of the category (above and below) are |
754 %% \caption{Almost Sharp Front}\label{afoto} |
754 %% \caption{Almost Sharp Front}\label{afoto} |
755 %% \end{figure} |
755 %% \end{figure} |
756 |
756 |
757 |
757 |
758 \begin{figure} |
758 \begin{figure} |
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759 \centering |
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760 \begin{tikzpicture}[%every label/.style={green} |
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761 ] |
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762 \node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {}; |
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763 \node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {}; |
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764 \draw (S) arc (-90:90:1); |
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765 \draw (N) arc (90:270:1); |
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766 \node[left] at (-1,1) {$B_1$}; |
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767 \node[right] at (1,1) {$B_2$}; |
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768 \end{tikzpicture} |
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769 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
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770 |
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771 \begin{figure} |
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772 \centering |
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773 \begin{tikzpicture}[%every label/.style={green}, |
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774 x=1.5cm,y=1.5cm] |
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775 \node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; |
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776 \node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; |
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777 \draw (S) arc (-90:90:1); |
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778 \draw (N) arc (90:270:1); |
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779 \draw (N) -- (S); |
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780 \node[left] at (-1/4,1) {$B_1$}; |
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781 \node[right] at (1/4,1) {$B_2$}; |
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782 \node at (1/6,3/2) {$Y$}; |
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783 \end{tikzpicture} |
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784 \caption{From two balls to one ball.}\label{blah5}\end{figure} |
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785 |
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786 \begin{figure} |
759 \begin{equation*} |
787 \begin{equation*} |
760 \mathfig{.23}{ncat/zz2} |
788 \mathfig{.23}{ncat/zz2} |
761 \end{equation*} |
789 \end{equation*} |
762 \caption{A small part of $\cell(W)$} |
790 \caption{A small part of $\cell(W)$.} |
763 \label{partofJfig} |
791 \label{partofJfig} |
764 \end{figure} |
792 \end{figure} |
765 |
793 |
766 \begin{figure} |
794 \begin{figure} |
767 $$\mathfig{.4}{deligne/manifolds}$$ |
795 $$\mathfig{.4}{deligne/manifolds}$$ |
768 \caption{An $n$-dimensional surgery cylinder}\label{delfig2} |
796 \caption{An $n$-dimensional surgery cylinder.}\label{delfig2} |
769 \end{figure} |
797 \end{figure} |
770 |
798 |
771 |
799 |
772 %% For Tables, put caption above table |
800 %% For Tables, put caption above table |
773 %% |
801 %% |