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395 \draw[->,red,line width=2pt] (-2.83,-1.5) -- (-2.83,-2.5); |
395 \draw[->,red,line width=2pt] (-2.83,-1.5) -- (-2.83,-2.5); |
396 \end{tikzpicture} |
396 \end{tikzpicture} |
397 $$ |
397 $$ |
398 \caption{Examples of pinched products}\label{pinched_prods} |
398 \caption{Examples of pinched products}\label{pinched_prods} |
399 \end{figure} |
399 \end{figure} |
400 (The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs} |
400 The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs} |
401 where we construct a traditional category from a disk-like category.) |
401 where we construct a traditional category from a disk-like category. |
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402 For example, ``half-pinched" products of 1-balls are used to construct weak identities for 1-morphisms |
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403 in 2-categories. |
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404 We also need fully-pinched products to define collar maps below (see Figure \ref{glue-collar}). |
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405 |
402 Define a {\it pinched product} to be a map |
406 Define a {\it pinched product} to be a map |
403 \[ |
407 \[ |
404 \pi: E\to X |
408 \pi: E\to X |
405 \] |
409 \] |
406 such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled |
410 such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled |