583 $A_\infty$ $1$-category $A$ as follows. |
583 $A_\infty$ $1$-category $A$ as follows. |
584 The objects of $A$ are $\cC(pt)$. |
584 The objects of $A$ are $\cC(pt)$. |
585 The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$ |
585 The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$ |
586 ($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$). |
586 ($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$). |
587 For simplicity we will now assume there is only one object and suppress it from the notation. |
587 For simplicity we will now assume there is only one object and suppress it from the notation. |
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588 Henceforth $A$ will also denote its unique morphism space. |
588 |
589 |
589 A choice of homeomorphism $I\cup I \to I$ induces a chain map $m_2: A\otimes A\to A$. |
590 A choice of homeomorphism $I\cup I \to I$ induces a chain map $m_2: A\otimes A\to A$. |
590 We now have two different homeomorphisms $I\cup I\cup I \to I$, but they are isotopic. |
591 We now have two different homeomorphisms $I\cup I\cup I \to I$, but they are isotopic. |
591 Choose a specific 1-parameter family of homeomorphisms connecting them; this induces |
592 Choose a specific 1-parameter family of homeomorphisms connecting them; this induces |
592 a degree 1 chain homotopy $m_3:A\ot A\ot A\to A$. |
593 a degree 1 chain homotopy $m_3:A\ot A\ot A\to A$. |
608 where $\beta \in \Homeo(I\to I)$ acts via $(f, a) \mapsto (f\circ \beta\inv, \beta_*(a))$. |
609 where $\beta \in \Homeo(I\to I)$ acts via $(f, a) \mapsto (f\circ \beta\inv, \beta_*(a))$. |
609 Note that $\cC(J) \cong A$ (non-canonically) for all intervals $J$. |
610 Note that $\cC(J) \cong A$ (non-canonically) for all intervals $J$. |
610 We define a $\Homeo(J)$ action on $\cC(J)$ via $g_*(f, a) = (g\circ f, a)$. |
611 We define a $\Homeo(J)$ action on $\cC(J)$ via $g_*(f, a) = (g\circ f, a)$. |
611 The $C_*(\Homeo(J))$ action is defined similarly. |
612 The $C_*(\Homeo(J))$ action is defined similarly. |
612 |
613 |
613 Let $J_1$ and $J_2$ be intervals. |
614 Let $J_1$ and $J_2$ be intervals, and let $J_1\cup J_2$ denote their union along a single boundary point. |
614 We must define a map $\cC(J_1)\ot\cC(J_2)\to\cC(J_1\cup J_2)$. |
615 We must define a map $\cC(J_1)\ot\cC(J_2)\to\cC(J_1\cup J_2)$. |
615 Choose a homeomorphism $g:I\to J_1\cup J_2$. |
616 Choose a homeomorphism $g:I\to J_1\cup J_2$. |
616 Let $(f_i, a_i)\in \cC(J_i)$. |
617 Let $(f_i, a_i)\in \cC(J_i)$. |
617 We have a parameterized decomposition of $I$ into two intervals given by |
618 We have a parameterized decomposition of $I$ into two intervals given by |
618 $g\inv \circ f_i$, $i=1,2$. |
619 $g\inv \circ f_i$, $i=1,2$. |