210 and our definition of a topological $A_\infty$ $1$-category, from \S \ref{???}.

   210 and our definition of a topological $A_\infty$ $1$-category, from \S \ref{ss:n-cat-def}.

   211 

   212 \medskip

   213 

   214 Given a topological $A_\infty$ $1$-category $\cC$, we define an ``$m_k$-style 

   215 $A_\infty$ $1$-category $A$ as follows.

   216 The objects of $A$ are $\cC(pt)$.

   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 $y$).

   219 For simplicity we will now assume there is only one object and suppress it from the notation.

   220 

   221 A choice of homeomorphism $I\cup I \to I$ induces a chain map $m_2: A\times A\to A$.

   222 We now have two different homeomorphisms $I\cup I\cup I \to I$, but they are isotopic.

   223 Choose a specific 1-parameter family of homeomorphisms connecting them; this induces

   224 a degree 1 chain homotopy $m_3:A\ot A\ot A\to A$.

   225 Proceeding in this way we define the rest of the $m_i$'s.

   226 It is straightforward to verify that they satisfy the necessary identities.

   227 

   228 \medskip

   229 

   230 In the other direction, we start with an alternative conventional definition of an $A_\infty$ algebra:

   231 an algebra $A$ for the $A_\infty$ operad.

   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

   234 whose $k$-th space is the space of decompositions of the standard interval $I$ into $k$

   235 parameterized copies of $I$.

   236 Note in particular that when $k=1$ this implies a $\Homeo(I)$ action on $A$.

   237 (Compare with Example \ref{ex:e-n-alg} and preceding discussion.)

   238 Given a non-standard interval $J$, we define $\cC(J)$ to be

   239 $(\Homeo(I\to J) \times A)/\Homeo(I\to I)$,

   240 where $\beta \in \Homeo(I\to I)$ acts via $(f, a) \mapsto (f\circ \beta\inv, \beta_*(a))$.

   241 \nn{check this}

   242 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 

   245 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 Choose a homeomorphism $g:I\to J_1\cup J_2$.

   248 Let $(f_i, a_i)\in \cC(J_i)$.

   249 We have a parameterized decomposition of $I$ into two intervals given by

   250 $g\inv \circ f_i$, $i=1,2$.

   251 Corresponding to this decomposition the operad action gives a map $\mu: A\ot A\to A$.

   252 Define the gluing map to send $(f_1, a_1)\ot (f_2, a_2)$ to $(g, \mu(a_1\ot a_2))$.

   253 

   254 It is straightforward to verify the remaining axioms for a topological $A_\infty$ 1-category.

   255 

   256 

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   261 

   262 \noop { %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%