equal
deleted
inserted
replaced
1202 Then there exist |
1202 Then there exist |
1203 \begin{itemize} |
1203 \begin{itemize} |
1204 \item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$; |
1204 \item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$; |
1205 \item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and |
1205 \item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and |
1206 \item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, |
1206 \item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, |
1207 such that $b_i$ and $b_{i+1}$both map to (glue up to) $a_i$. |
1207 such that $b_i$ and $b_{i+1}$ both map to (glue up to) $a_i$. |
1208 \end{itemize} |
1208 \end{itemize} |
1209 In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$. |
1209 In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$. |
1210 The idea of the proof is to produce a similar zig-zag where everything antirefines to the same |
1210 The idea of the proof is to produce a similar zig-zag where everything antirefines to the same |
1211 disjoint union of balls, and then invoke Axiom \ref{nca-assoc} which ensures associativity. |
1211 disjoint union of balls, and then invoke Axiom \ref{nca-assoc} which ensures associativity. |
1212 |
1212 |