198 \nn{to be continued...} |
198 \nn{to be continued...} |
199 \medskip |
199 \medskip |
200 |
200 |
201 \subsection{$A_\infty$ $1$-categories} |
201 \subsection{$A_\infty$ $1$-categories} |
202 \label{sec:comparing-A-infty} |
202 \label{sec:comparing-A-infty} |
203 In this section, we make contact between the usual definition of an $A_\infty$ algebra |
203 In this section, we make contact between the usual definition of an $A_\infty$ category |
204 and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}. |
204 and our definition of a topological $A_\infty$ $1$-category, from \S \ref{???}. |
205 |
205 |
206 We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$, |
206 That definition associates a chain complex to every interval, and we begin by giving an alternative definition that is entirely in terms of the chain complex associated to the standard interval $[0,1]$. |
207 which we can alternatively characterise as: |
|
208 \begin{defn} |
207 \begin{defn} |
209 A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, |
208 A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, |
210 and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with |
209 and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with |
211 \begin{itemize} |
210 \begin{itemize} |
212 \item an action of the operad of $\Obj(\cC)$-labeled cell decompositions |
211 \item an action of the operad of $\Obj(\cC)$-labeled cell decompositions |
220 them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points |
219 them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points |
221 of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. |
220 of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. |
222 In the $X$-labeled case, we insist that the appropriate labels match up. |
221 In the $X$-labeled case, we insist that the appropriate labels match up. |
223 Saying we have an action of this operad means that for each labeled cell decomposition |
222 Saying we have an action of this operad means that for each labeled cell decomposition |
224 $0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain |
223 $0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain |
225 map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC(a_0,a_{k+1})$$ and these |
224 map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC_{a_0,a_{k+1}}$$ and these |
226 chain maps compose exactly as the cell decompositions. |
225 chain maps compose exactly as the cell decompositions. |
227 An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad |
226 An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad |
228 if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which |
227 if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which |
229 is supported on the subintervals determined by $\pi$, then the two possible operations |
228 is supported on the subintervals determined by $\pi$, then the two possible operations |
230 (glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms |
229 (glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms |