10 (c) $A_\infty$ 1-cat; (b) $A_\infty$ 1-cat module?; (e) tensor products?} |
10 (c) $A_\infty$ 1-cat; (b) $A_\infty$ 1-cat module?; (e) tensor products?} |
11 |
11 |
12 \subsection{$1$-categories over $\Set$ or $\Vect$} |
12 \subsection{$1$-categories over $\Set$ or $\Vect$} |
13 \label{ssec:1-cats} |
13 \label{ssec:1-cats} |
14 Given a topological $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$. |
14 Given a topological $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$. |
15 This construction is quite straightforward, but we include the details for the sake of completeness, because it illustrates the role of structures (e.g. orientations, spin structures, etc) on the underlying manifolds, and |
15 This construction is quite straightforward, but we include the details for the sake of completeness, |
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16 because it illustrates the role of structures (e.g. orientations, spin structures, etc) |
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17 on the underlying manifolds, and |
16 to shed some light on the $n=2$ case, which we describe in \S \ref{ssec:2-cats}. |
18 to shed some light on the $n=2$ case, which we describe in \S \ref{ssec:2-cats}. |
17 |
19 |
18 Let $B^k$ denote the \emph{standard} $k$-ball. |
20 Let $B^k$ denote the \emph{standard} $k$-ball. |
19 Let the objects of $c(\cX)$ be $c(\cX)^0 = \cX(B^0)$ and the morphisms of $c(\cX)$ be $c(\cX)^1 = \cX(B^1)$. The boundary and restriction maps of $\cX$ give domain and range maps from $c(\cX)^1$ to $c(\cX)^0$. |
21 Let the objects of $c(\cX)$ be $c(\cX)^0 = \cX(B^0)$ and the morphisms of $c(\cX)$ be $c(\cX)^1 = \cX(B^1)$. |
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22 The boundary and restriction maps of $\cX$ give domain and range maps from $c(\cX)^1$ to $c(\cX)^0$. |
20 |
23 |
21 Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$. |
24 Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$. |
22 Define composition in $c(\cX)$ to be the induced map $c(\cX)^1\times c(\cX)^1 \to c(\cX)^1$ (defined only when range and domain agree). |
25 Define composition in $c(\cX)$ to be the induced map $c(\cX)^1\times c(\cX)^1 \to c(\cX)^1$ |
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26 (defined only when range and domain agree). |
23 By isotopy invariance in $\cX$, any other choice of homeomorphism gives the same composition rule. |
27 By isotopy invariance in $\cX$, any other choice of homeomorphism gives the same composition rule. |
24 Also by isotopy invariance, composition is strictly associative. |
28 Also by isotopy invariance, composition is strictly associative. |
25 |
29 |
26 Given $a\in c(\cX)^0$, define $\id_a \deq a\times B^1$. |
30 Given $a\in c(\cX)^0$, define $\id_a \deq a\times B^1$. |
27 By extended isotopy invariance in $\cX$, this has the expected properties of an identity morphism. |
31 By extended isotopy invariance in $\cX$, this has the expected properties of an identity morphism. |
28 |
32 |
29 |
33 |
30 If the underlying manifolds for $\cX$ have further geometric structure, then we obtain certain functors. The base case is for oriented manifolds, where we obtain no extra algebraic data. |
34 If the underlying manifolds for $\cX$ have further geometric structure, then we obtain certain functors. |
31 |
35 The base case is for oriented manifolds, where we obtain no extra algebraic data. |
32 For 1-categories based on unoriented manifolds (somewhat confusingly, we're thinking of being unoriented as requiring extra data beyond being oriented, namely the identification between the orientations), there is a map $*:c(\cX)^1\to c(\cX)^1$ |
36 |
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37 For 1-categories based on unoriented manifolds (somewhat confusingly, we're thinking of being |
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38 unoriented as requiring extra data beyond being oriented, namely the identification between the orientations), |
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39 there is a map $*:c(\cX)^1\to c(\cX)^1$ |
33 coming from $\cX$ applied to an orientation-reversing homeomorphism (unique up to isotopy) |
40 coming from $\cX$ applied to an orientation-reversing homeomorphism (unique up to isotopy) |
34 from $B^1$ to itself. |
41 from $B^1$ to itself. |
35 Topological properties of this homeomorphism imply that |
42 Topological properties of this homeomorphism imply that |
36 $a^{**} = a$ (* is order 2), * reverses domain and range, and $(ab)^* = b^*a^*$ |
43 $a^{**} = a$ (* is order 2), * reverses domain and range, and $(ab)^* = b^*a^*$ |
37 (* is an anti-automorphism). |
44 (* is an anti-automorphism). |
189 \nn{to be continued...} |
198 \nn{to be continued...} |
190 \medskip |
199 \medskip |
191 |
200 |
192 \subsection{$A_\infty$ $1$-categories} |
201 \subsection{$A_\infty$ $1$-categories} |
193 \label{sec:comparing-A-infty} |
202 \label{sec:comparing-A-infty} |
194 In this section, we make contact between the usual definition of an $A_\infty$ algebra and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}. |
203 In this section, we make contact between the usual definition of an $A_\infty$ algebra |
195 |
204 and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}. |
196 We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$, which we can alternatively characterise as: |
205 |
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206 We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$, |
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207 which we can alternatively characterise as: |
197 \begin{defn} |
208 \begin{defn} |
198 A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with |
209 A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, |
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210 and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with |
199 \begin{itemize} |
211 \begin{itemize} |
200 \item an action of the operad of $\Obj(\cC)$-labeled cell decompositions |
212 \item an action of the operad of $\Obj(\cC)$-labeled cell decompositions |
201 \item and a compatible action of $\CD{[0,1]}$. |
213 \item and a compatible action of $\CD{[0,1]}$. |
202 \end{itemize} |
214 \end{itemize} |
203 \end{defn} |
215 \end{defn} |
204 Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals. An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$. Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. In the $X$-labeled case, we insist that the appropriate labels match up. Saying we have an action of this operad means that for each labeled cell decomposition $0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC(a_0,a_{k+1})$$ and these chain maps compose exactly as the cell decompositions. |
216 Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of |
205 An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which is supported on the subintervals determined by $\pi$, then the two possible operations (glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy). |
217 points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals. |
206 |
218 An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$. |
207 Translating between this notion and the usual definition of an $A_\infty$ category is now straightforward. To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$. Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels) |
219 Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose |
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220 them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points |
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221 of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. |
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222 In the $X$-labeled case, we insist that the appropriate labels match up. |
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223 Saying we have an action of this operad means that for each labeled cell decomposition |
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224 $0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain |
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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 |
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226 chain maps compose exactly as the cell decompositions. |
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227 An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad |
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228 if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which |
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229 is supported on the subintervals determined by $\pi$, then the two possible operations |
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230 (glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms |
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231 separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy). |
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232 |
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233 Translating between this notion and the usual definition of an $A_\infty$ category is now straightforward. |
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234 To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$. |
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235 Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels) |
208 $$\cC([0,1])^{\tensor k+1} \to \cC([0,x_1]) \tensor \cdots \tensor \cC[x_k,1] \to \cC([0,1])$$ |
236 $$\cC([0,1])^{\tensor k+1} \to \cC([0,x_1]) \tensor \cdots \tensor \cC[x_k,1] \to \cC([0,1])$$ |
209 where the factors of the first map are induced by the linear isometries $[0,1] \to [x_i, x_{i+1}]$, and the second map is just gluing. The action of $\CD{[0,1]}$ carries across, and is automatically compatible. Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism $\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map $\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$ given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$. You can readily check that this gluing map is associative on the nose. \todo{really?} |
237 where the factors of the first map are induced by the linear isometries $[0,1] \to [x_i, x_{i+1}]$, and the second map is just gluing. |
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238 The action of $\CD{[0,1]}$ carries across, and is automatically compatible. |
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239 Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism |
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240 $\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map |
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241 $\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying |
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242 the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$ |
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243 given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$. |
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244 You can readily check that this gluing map is associative on the nose. \todo{really?} |
210 |
245 |
211 %First recall the \emph{coloured little intervals operad}. Given a set of labels $\cL$, the operations are indexed by \emph{decompositions of the interval}, each of which is a collection of disjoint subintervals $\{(a_i,b_i)\}_{i=1}^k$ of $[0,1]$, along with a labeling of the complementary regions by $\cL$, $\{l_0, \ldots, l_k\}$. Given two decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$ such that $l^{(1)}_{m-1} = l^{(2)}_0$ and $l^{(1)}_{m} = l^{(2)}_{k^{(2)}}$, we can form a new decomposition by inserting the intervals of $\cJ^{(2)}$ linearly inside the $m$-th interval of $\cJ^{(1)}$. We call the resulting decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$. |
246 %First recall the \emph{coloured little intervals operad}. Given a set of labels $\cL$, the operations are indexed by \emph{decompositions of the interval}, each of which is a collection of disjoint subintervals $\{(a_i,b_i)\}_{i=1}^k$ of $[0,1]$, along with a labeling of the complementary regions by $\cL$, $\{l_0, \ldots, l_k\}$. Given two decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$ such that $l^{(1)}_{m-1} = l^{(2)}_0$ and $l^{(1)}_{m} = l^{(2)}_{k^{(2)}}$, we can form a new decomposition by inserting the intervals of $\cJ^{(2)}$ linearly inside the $m$-th interval of $\cJ^{(1)}$. We call the resulting decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$. |
212 |
247 |
213 %\begin{defn} |
248 %\begin{defn} |
214 %A \emph{topological $A_\infty$ category} $\cC$ has a set of objects $\Obj(\cC)$ and for each $a,b \in \Obj(\cC)$ a chain complex $\cC_{a,b}$, along with a compatible `composition map' and an `action of families of diffeomorphisms'. |
249 %A \emph{topological $A_\infty$ category} $\cC$ has a set of objects $\Obj(\cC)$ and for each $a,b \in \Obj(\cC)$ a chain complex $\cC_{a,b}$, along with a compatible `composition map' and an `action of families of diffeomorphisms'. |
233 %\phi(f_\cJ(a_1, \cdots, a_k)) = f_{\phi(\cJ)}(\phi_1(a_1), \cdots, \phi_k(a_k)). |
268 %\phi(f_\cJ(a_1, \cdots, a_k)) = f_{\phi(\cJ)}(\phi_1(a_1), \cdots, \phi_k(a_k)). |
234 %\end{equation*} |
269 %\end{equation*} |
235 %\end{enumerate} |
270 %\end{enumerate} |
236 %\end{defn} |
271 %\end{defn} |
237 |
272 |
238 From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional' $A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. We'll just describe the algebra case (that is, a category with only one object), as the modifications required to deal with multiple objects are trivial. Define $A = \cC$ as a chain complex (so $m_1 = d$). Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms of $[0,1]$ that interpolates linearly between the identity and the piecewise linear diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define |
273 From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional' |
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274 $A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. |
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275 We'll just describe the algebra case (that is, a category with only one object), |
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276 as the modifications required to deal with multiple objects are trivial. |
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277 Define $A = \cC$ as a chain complex (so $m_1 = d$). |
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278 Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. |
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279 To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms |
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280 of $[0,1]$ that interpolates linearly between the identity and the piecewise linear |
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281 diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define |
239 \begin{equation*} |
282 \begin{equation*} |
240 m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)). |
283 m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)). |
241 \end{equation*} |
284 \end{equation*} |
242 |
285 |
243 It's then easy to calculate that |
286 It's then easy to calculate that |