209 In this section, we make contact between the usual definition of an $A_\infty$ category |
209 In this section, we make contact between the usual definition of an $A_\infty$ category |
210 and our definition of a topological $A_\infty$ $1$-category, from \S \ref{ss:n-cat-def}. |
210 and our definition of a topological $A_\infty$ $1$-category, from \S \ref{ss:n-cat-def}. |
211 |
211 |
212 \medskip |
212 \medskip |
213 |
213 |
214 Given a topological $A_\infty$ $1$-category $\cC$, we define an ``$m_k$-style |
214 Given a topological $A_\infty$ $1$-category $\cC$, we define an ``$m_k$-style" |
215 $A_\infty$ $1$-category $A$ as follows. |
215 $A_\infty$ $1$-category $A$ as follows. |
216 The objects of $A$ are $\cC(pt)$. |
216 The objects of $A$ are $\cC(pt)$. |
217 The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$ |
217 The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$ |
218 ($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$). |
218 ($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$). |
219 For simplicity we will now assume there is only one object and suppress it from the notation. |
219 For simplicity we will now assume there is only one object and suppress it from the notation. |
231 an algebra $A$ for the $A_\infty$ operad. |
231 an algebra $A$ for the $A_\infty$ operad. |
232 (For simplicity, we are assuming our $A_\infty$ 1-category has only one object.) |
232 (For simplicity, we are assuming our $A_\infty$ 1-category has only one object.) |
233 We are free to choose any operad with contractible spaces, so we choose the operad |
233 We are free to choose any operad with contractible spaces, so we choose the operad |
234 whose $k$-th space is the space of decompositions of the standard interval $I$ into $k$ |
234 whose $k$-th space is the space of decompositions of the standard interval $I$ into $k$ |
235 parameterized copies of $I$. |
235 parameterized copies of $I$. |
236 Note in particular that when $k=1$ this implies a $\Homeo(I)$ action on $A$. |
236 Note in particular that when $k=1$ this implies a $C_*(\Homeo(I))$ action on $A$. |
237 (Compare with Example \ref{ex:e-n-alg} and preceding discussion.) |
237 (Compare with Example \ref{ex:e-n-alg} and the discussion which precedes it.) |
238 Given a non-standard interval $J$, we define $\cC(J)$ to be |
238 Given a non-standard interval $J$, we define $\cC(J)$ to be |
239 $(\Homeo(I\to J) \times A)/\Homeo(I\to I)$, |
239 $(\Homeo(I\to J) \times A)/\Homeo(I\to I)$, |
240 where $\beta \in \Homeo(I\to I)$ acts via $(f, a) \mapsto (f\circ \beta\inv, \beta_*(a))$. |
240 where $\beta \in \Homeo(I\to I)$ acts via $(f, a) \mapsto (f\circ \beta\inv, \beta_*(a))$. |
241 \nn{check this} |
241 \nn{check this} |
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242 Note that $\cC(J) \cong A$ (non-canonically) for all intervals $J$. |
242 We define a $\Homeo(J)$ action on $\cC(J)$ via $g_*(f, a) = (g\circ f, a)$. |
243 We define a $\Homeo(J)$ action on $\cC(J)$ via $g_*(f, a) = (g\circ f, a)$. |
243 The $C_*(\Homeo(J))$ action is defined similarly. |
244 The $C_*(\Homeo(J))$ action is defined similarly. |
244 |
245 |
245 Let $J_1$ and $J_2$ be intervals. |
246 Let $J_1$ and $J_2$ be intervals. |
246 We must define a map $\cC(J_1)\ot\cC(J_2)\to\cC(J_1\cup J_2)$. |
247 We must define a map $\cC(J_1)\ot\cC(J_2)\to\cC(J_1\cup J_2)$. |
248 Let $(f_i, a_i)\in \cC(J_i)$. |
249 Let $(f_i, a_i)\in \cC(J_i)$. |
249 We have a parameterized decomposition of $I$ into two intervals given by |
250 We have a parameterized decomposition of $I$ into two intervals given by |
250 $g\inv \circ f_i$, $i=1,2$. |
251 $g\inv \circ f_i$, $i=1,2$. |
251 Corresponding to this decomposition the operad action gives a map $\mu: A\ot A\to A$. |
252 Corresponding to this decomposition the operad action gives a map $\mu: A\ot A\to A$. |
252 Define the gluing map to send $(f_1, a_1)\ot (f_2, a_2)$ to $(g, \mu(a_1\ot a_2))$. |
253 Define the gluing map to send $(f_1, a_1)\ot (f_2, a_2)$ to $(g, \mu(a_1\ot a_2))$. |
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254 Operad associativity for $A$ implies that this gluing map is independent of the choice of |
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255 $g$ and the choice of representative $(f_i, a_i)$. |
253 |
256 |
254 It is straightforward to verify the remaining axioms for a topological $A_\infty$ 1-category. |
257 It is straightforward to verify the remaining axioms for a topological $A_\infty$ 1-category. |
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260 |