487 	\draw[green!50!brown] (\x,-2) -- (\x,2);

   487 	\draw[green!50!brown] (\x,-2) -- (\x,2);

   488 }

   488 }

   489 \end{scope}

   489 \end{scope}

   490 \end{tikzpicture}

   490 \end{tikzpicture}

   491 $$

   491 $$

   493 \end{figure}

   493 \end{figure}

   494 

   494 

   495 Note that $\bd X$ has a (possibly trivial) subdivision according to 

   495 Note that $\bd X$ has a (possibly trivial) subdivision according to 

   496 the dimension of $\pi\inv(x)$, $x\in \bd X$.

   496 the dimension of $\pi\inv(x)$, $x\in \bd X$.

   497 Let $\cC(X)\trans{}$ denote the morphisms which are splittable along this subdivision.

   497 Let $\cC(X)\trans{}$ denote the morphisms which are splittable along this subdivision.

  1406 Then there exist

  1406 Then there exist

  1407 \begin{itemize}

  1407 \begin{itemize}

  1408 \item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$;

  1408 \item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$;

  1409 \item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and

  1409 \item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and

  1410 \item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, 

  1410 \item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, 

  1412 \end{itemize}

  1412 \end{itemize}

  1413 In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$.

  1413 In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$.

  1414 The idea of the proof is to produce a similar zig-zag where everything antirefines to the same

  1414 The idea of the proof is to produce a similar zig-zag where everything antirefines to the same

  1415 disjoint union of balls, and then invoke Axiom \ref{nca-assoc} which ensures associativity.

  1415 disjoint union of balls, and then invoke Axiom \ref{nca-assoc} which ensures associativity.

  1416 

  1416