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27 Let $f: P\times X \to T$, as above. |
27 Let $f: P\times X \to T$, as above. |
28 Then there exists |
28 Then there exists |
29 \[ |
29 \[ |
30 F: I \times P\times X \to T |
30 F: I \times P\times X \to T |
31 \] |
31 \] |
32 such that |
32 such that the following conditions hold. |
33 \begin{enumerate} |
33 \begin{enumerate} |
34 \item $F(0, \cdot, \cdot) = f$ . |
34 \item $F(0, \cdot, \cdot) = f$. |
35 \item We can decompose $P = \cup_i D_i$ so that |
35 \item We can decompose $P = \cup_i D_i$ so that |
36 the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$. |
36 the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$. |
37 \item If $f$ has support $S\sub X$, then |
37 \item If $f$ has support $S\sub X$, then |
38 $F: (I\times P)\times X\to T$ (a $k{+}1$-parameter family of maps) also has support $S$. |
38 $F: (I\times P)\times X\to T$ (a $k{+}1$-parameter family of maps) also has support $S$. |
39 Furthermore, if $Q$ is a convex linear subpolyhedron of $\bd P$ and $f$ restricted to $Q$ |
39 Furthermore, if $Q$ is a convex linear subpolyhedron of $\bd P$ and $f$ restricted to $Q$ |