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490 \end{tikzpicture} |
490 \end{tikzpicture} |
491 $$ |
491 $$ |
492 \caption{Five examples of unions of pinched products}\label{pinched_prod_unions} |
492 \caption{Five examples of unions of pinched products}\label{pinched_prod_unions} |
493 \end{figure} |
493 \end{figure} |
494 |
494 |
495 The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product |
495 Note that $\bd X$ has a (possibly trivial) subdivision according to |
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496 the dimension of $\pi\inv(x)$, $x\in \bd X$. |
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497 Let $\cC(X)\trans{}$ denote the morphisms which are splittable along this subdivision. |
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498 |
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499 The product axiom will give a map $\pi^*:\cC(X)\trans{}\to \cC(E)$ for each pinched product |
496 $\pi:E\to X$. |
500 $\pi:E\to X$. |
497 Morphisms in the image of $\pi^*$ will be called product morphisms. |
501 Morphisms in the image of $\pi^*$ will be called product morphisms. |
498 Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories. |
502 Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories. |
499 In the case where $\cC(X) = \{f: X\to T\}$, we define $\pi^*(f) = f\circ\pi$. |
503 In the case where $\cC(X) = \{f: X\to T\}$, we define $\pi^*(f) = f\circ\pi$. |
500 In the case where $\cC(X)$ is the set of all labeled embedded cell complexes $K$ in $X$, |
504 In the case where $\cC(X)$ is the set of all labeled embedded cell complexes $K$ in $X$, |
504 |
508 |
505 %\addtocounter{axiom}{-1} |
509 %\addtocounter{axiom}{-1} |
506 \begin{axiom}[Product (identity) morphisms] |
510 \begin{axiom}[Product (identity) morphisms] |
507 \label{axiom:product} |
511 \label{axiom:product} |
508 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
512 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
509 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
513 there is a map $\pi^*:\cC(X)\trans{}\to \cC(E)$. |
510 These maps must satisfy the following conditions. |
514 These maps must satisfy the following conditions. |
511 \begin{enumerate} |
515 \begin{enumerate} |
512 \item |
516 \item |
513 If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and |
517 If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and |
514 if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
518 if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
527 (See Figure \ref{pinched_prod_unions}.) |
531 (See Figure \ref{pinched_prod_unions}.) |
528 Note that $X_1$ and $X_2$ can be identified with subsets of $X$, |
532 Note that $X_1$ and $X_2$ can be identified with subsets of $X$, |
529 but $X_1 \cap X_2$ might not be codimension 1, and indeed we might have $X_1 = X_2 = X$. |
533 but $X_1 \cap X_2$ might not be codimension 1, and indeed we might have $X_1 = X_2 = X$. |
530 We assume that there is a decomposition of $X$ into balls which is compatible with |
534 We assume that there is a decomposition of $X$ into balls which is compatible with |
531 $X_1$ and $X_2$. |
535 $X_1$ and $X_2$. |
532 Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$. |
536 Let $a\in \cC(X)\trans{}$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$. |
533 (We assume that $a$ is splittable with respect to the above decomposition of $X$ into balls.) |
537 (We assume that $a$ is splittable with respect to the above decomposition of $X$ into balls.) |
534 Then |
538 Then |
535 \[ |
539 \[ |
536 \pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
540 \pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
537 \] |
541 \] |