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458 |
458 |
459 \begin{thm} |
459 \begin{thm} |
460 \label{thm:CH-associativity} |
460 \label{thm:CH-associativity} |
461 The $\CH{X \to Y}$ actions defined above are associative. |
461 The $\CH{X \to Y}$ actions defined above are associative. |
462 That is, the following diagram commutes up to homotopy: |
462 That is, the following diagram commutes up to homotopy: |
463 \[ \xymatrix{ |
463 \[ \xymatrix@C=5pt{ |
464 & \CH{Y\to Z} \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ |
464 & \CH{Y\to Z} \ot \bc_*(Y) \ar[drr]^{e_{YZ}} & &\\ |
465 \CH{X \to Y} \ot \CH{Y \to Z} \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ |
465 \CH{X \to Y} \ot \CH{Y \to Z} \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & & \bc_*(Z) \\ |
466 & \CH{X \to Z} \ot \bc_*(X) \ar[ur]_{e_{XZ}} & |
466 & \CH{X \to Z} \ot \bc_*(X) \ar[urr]_{e_{XZ}} & & |
467 } \] |
467 } \] |
468 Here $\mu:\CH{X\to Y} \ot \CH{Y \to Z}\to \CH{X \to Z}$ is the map induced by composition |
468 Here $\mu:\CH{X\to Y} \ot \CH{Y \to Z}\to \CH{X \to Z}$ is the map induced by composition |
469 of homeomorphisms. |
469 of homeomorphisms. |
470 \end{thm} |
470 \end{thm} |
471 \begin{proof} |
471 \begin{proof} |