268 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
269 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
269 |
270 |
270 \begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
271 \begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
271 \label{property:blobs-ainfty} |
272 \label{property:blobs-ainfty} |
272 Let $\cC$ be a topological $n$-category. Let $Y$ be an $n{-}k$-manifold. |
273 Let $\cC$ be a topological $n$-category. Let $Y$ be an $n{-}k$-manifold. |
273 There is an $A_\infty$ $k$-category $A_*(Y, \cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, to be the set $$A_*(Y,\cC)(D) = A^\cC(Y \times D)$$ and on $k$-balls $D$ to be the set $$A_*(Y, \cC)(D) = \bc_*(Y \times D, \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Property \ref{property:evaluation}. |
274 There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Property \ref{property:evaluation}. |
274 \end{property} |
275 \end{property} |
275 \begin{rem} |
276 \begin{rem} |
276 Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category. We think of this $A_\infty$ $n$-category as a free resolution. |
277 Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category. We think of this $A_\infty$ $n$-category as a free resolution. |
277 \end{rem} |
278 \end{rem} |
278 |
279 |
279 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
280 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
280 instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. |
281 instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
281 |
282 |
282 \begin{property}[Product formula] |
283 \begin{property}[Product formula] |
283 \label{property:product} |
284 \label{property:product} |
284 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. Let $\cC$ be an $n$-category. |
285 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. Let $\cC$ be an $n$-category. |
285 Let $A_*(Y,\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}). |
286 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}). |
286 Then |
287 Then |
287 \[ |
288 \[ |
288 \bc_*(Y\times W, \cC) \simeq \bc_*(W, A_*(Y,\cC)) . |
289 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
289 \] |
290 \] |
290 Note on the right hand side we have the version of the blob complex for $A_\infty$ $n$-categories. |
291 \end{property} |
291 \end{property} |
292 We also give a generalization of this statement for arbitrary fibre bundles, in \S \ref{moddecss}, and a sketch of a statement for arbitrary maps. |
292 It seems reasonable to expect a generalization describing an arbitrary fibre bundle. See in particular \S \ref{moddecss} for the framework for such a statement. |
293 |
|
294 Fix a topological $n$-category $\cC$, which we'll omit from the notation. Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. (See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) |
293 |
295 |
294 \begin{property}[Gluing formula] |
296 \begin{property}[Gluing formula] |
295 \label{property:gluing}% |
297 \label{property:gluing}% |
296 \mbox{}% <-- gets the indenting right |
298 \mbox{}% <-- gets the indenting right |
297 \begin{itemize} |
299 \begin{itemize} |
298 \item For any $(n-1)$-manifold $Y$, the blob complex of $Y \times I$ is |
|
299 naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
|
300 |
|
301 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
300 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
302 $A_\infty$ module for $\bc_*(Y \times I)$. |
301 $A_\infty$ module for $\bc_*(Y)$. |
303 |
302 |
304 \item For any $n$-manifold $X_\text{glued} = X_\text{cut} \bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{glued})$ is the $A_\infty$ self-tensor product of |
303 \item For any $n$-manifold $X_\text{glued} = X_\text{cut} \bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{glued})$ is the $A_\infty$ self-tensor product of |
305 $\bc_*(X_\text{cut})$ as an $\bc_*(Y \times I)$-bimodule: |
304 $\bc_*(X_\text{cut})$ as an $\bc_*(Y)$-bimodule: |
306 \begin{equation*} |
305 \begin{equation*} |
307 \bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \selfarrow |
306 \bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
308 \end{equation*} |
307 \end{equation*} |
309 \end{itemize} |
308 \end{itemize} |
310 \end{property} |
309 \end{property} |
311 |
310 |
312 Finally, we state two more properties, which we will not prove in this paper. |
311 Finally, we prove two theorems which we consider as applications. |
313 \nn{revise this; expect that we will prove these in the paper} |
312 |
314 |
313 \begin{thm}[Mapping spaces] |
315 \begin{property}[Mapping spaces] |
|
316 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
314 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
317 $B^n \to T$. |
315 $B^n \to T$. |
318 (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
316 (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
319 Then |
317 Then |
320 $$\bc_*(X, \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
318 $$\bc_*(X, \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
321 \end{property} |
319 \end{thm} |
322 |
320 |
323 This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. |
321 This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. |
324 |
322 |
325 \begin{property}[Higher dimensional Deligne conjecture] |
323 \begin{thm}[Higher dimensional Deligne conjecture] |
326 \label{property:deligne} |
324 \label{thm:deligne} |
327 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
325 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
328 \end{property} |
326 \end{thm} |
329 See \S \ref{sec:deligne} for an explanation of the terms appearing here. The proof will appear elsewhere. |
327 See \S \ref{sec:deligne} for a full explanation of the statement, and an outline of the proof. |
330 |
328 |
331 Properties \ref{property:functoriality} and \ref{property:skein-modules} will be immediate from the definition given in |
329 Properties \ref{property:functoriality} and \ref{property:skein-modules} will be immediate from the definition given in |
332 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. |
330 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. |
333 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
331 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
334 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, Property \ref{property:blobs-ainfty} as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}, |
332 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, Property \ref{property:blobs-ainfty} as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}, |