8 \item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. |
8 \item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. |
9 (See Theorem \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
9 (See Theorem \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
10 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), |
10 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), |
11 the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. |
11 the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. |
12 (See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.) |
12 (See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.) |
13 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have |
13 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category, we have |
14 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
14 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
15 on the configuration space of unlabeled points in $M$. |
15 on the configuration space of unlabeled points in $M$. (See \S \ref{sec:comm_alg}.) |
16 %$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ |
16 %$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ |
17 \end{itemize} |
17 \end{itemize} |
18 The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space |
18 The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space |
19 (replacing quotient of fields by local relations with some sort of resolution), |
19 (replacing quotient of fields by local relations with some sort of resolution), |
20 and for a generalization of Hochschild homology to higher $n$-categories. |
20 and for a generalization of Hochschild homology to higher $n$-categories. |
109 \caption{The main gadgets and constructions of the paper.} |
109 \caption{The main gadgets and constructions of the paper.} |
110 \label{fig:outline} |
110 \label{fig:outline} |
111 \end{figure} |
111 \end{figure} |
112 |
112 |
113 Finally, later sections address other topics. |
113 Finally, later sections address other topics. |
114 Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
114 Section \S \ref{sec:deligne} gives |
115 thought of as a topological $n$-category, in terms of the topology of $M$. |
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116 Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) |
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117 a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. |
115 a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. |
118 The appendixes prove technical results about $\CH{M}$ and the ``small blob complex", |
116 The appendixes prove technical results about $\CH{M}$ and the ``small blob complex", |
119 and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
117 and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
120 as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
118 as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
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119 thought of as a topological $n$-category, in terms of the topology of $M$. |
121 |
120 |
122 |
121 |
123 \nn{some more things to cover in the intro} |
122 \nn{some more things to cover in the intro} |
124 \begin{itemize} |
123 \begin{itemize} |
125 \item related: we are being unsophisticated from a homotopy theory point of |
124 \item related: we are being unsophisticated from a homotopy theory point of |
148 Our main motivating example (though we will not develop it in this paper) |
147 Our main motivating example (though we will not develop it in this paper) |
149 is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology. |
148 is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology. |
150 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
149 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
151 with a link $L \subset \bd W$. |
150 with a link $L \subset \bd W$. |
152 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
151 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
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152 \todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S} |
153 |
153 |
154 How would we go about computing $A_{Kh}(W^4, L)$? |
154 How would we go about computing $A_{Kh}(W^4, L)$? |
155 For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence) |
155 For the Khovanov homology of a link in $S^3$ the main tool is the exact triangle (long exact sequence) |
156 relating resolutions of a crossing. |
156 relating resolutions of a crossing. |
157 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
157 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
158 to compute $A_{Kh}(S^1\times B^3, L)$. |
158 to compute $A_{Kh}(S^1\times B^3, L)$. |
159 According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$ |
159 According to the gluing theorem for TQFTs, gluing along $B^3 \subset \bd B^4$ |
160 corresponds to taking a coend (self tensor product) over the cylinder category |
160 corresponds to taking a coend (self tensor product) over the cylinder category |
161 associated to $B^3$ (with appropriate boundary conditions). |
161 associated to $B^3$ (with appropriate boundary conditions). |
162 The coend is not an exact functor, so the exactness of the triangle breaks. |
162 The coend is not an exact functor, so the exactness of the triangle breaks. |
163 |
163 |
164 |
164 |
226 \begin{equation*} |
226 \begin{equation*} |
227 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
227 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
228 \end{equation*} |
228 \end{equation*} |
229 \end{property} |
229 \end{property} |
230 |
230 |
231 If an $n$-manifold $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, |
231 If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, |
232 write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
232 write $X_\text{gl} = X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
233 Note that this includes the case of gluing two disjoint manifolds together. |
233 Note that this includes the case of gluing two disjoint manifolds together. |
234 \begin{property}[Gluing map] |
234 \begin{property}[Gluing map] |
235 \label{property:gluing-map}% |
235 \label{property:gluing-map}% |
236 %If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map |
236 %If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map |
237 %\begin{equation*} |
237 %\begin{equation*} |
238 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
238 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
239 %\end{equation*} |
239 %\end{equation*} |
240 Given a gluing $X_\mathrm{cut} \to X_\mathrm{glued}$, there is |
240 Given a gluing $X \to X_\mathrm{gl}$, there is |
241 a natural map |
241 a natural map |
242 \[ |
242 \[ |
243 \bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{glued}) |
243 \bc_*(X) \to \bc_*(X_\mathrm{gl}) |
244 \] |
244 \] |
245 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
245 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
246 \end{property} |
246 \end{property} |
247 |
247 |
248 \begin{property}[Contractibility] |
248 \begin{property}[Contractibility] |
249 \label{property:contractibility}% |
249 \label{property:contractibility}% |
250 With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology. |
250 With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology. |
251 Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls. |
251 Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls. |
252 \begin{equation} |
252 \begin{equation*} |
253 \xymatrix{\bc_*^{\cC}(B^n) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*^{\cC}(B^n)) \ar[r]^(0.6)\iso & \cC(B^n)} |
253 \xymatrix{\bc_*^{\cC}(B^n) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*^{\cC}(B^n)) \ar[r]^(0.6)\iso & \cC(B^n)} |
254 \end{equation} |
254 \end{equation*} |
255 \end{property} |
255 \end{property} |
256 |
256 |
257 Properties \ref{property:functoriality} will be immediate from the definition given in |
257 Properties \ref{property:functoriality} will be immediate from the definition given in |
258 \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
258 \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
259 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
259 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
261 \subsection{Specializations} |
261 \subsection{Specializations} |
262 \label{sec:specializations} |
262 \label{sec:specializations} |
263 |
263 |
264 The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology. |
264 The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology. |
265 |
265 |
266 \begin{thm}[Skein modules] |
266 \newtheorem*{thm:skein-modules}{Theorem \ref{thm:skein-modules}} |
267 \label{thm:skein-modules}% |
267 |
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268 \begin{thm:skein-modules}[Skein modules] |
268 The $0$-th blob homology of $X$ is the usual |
269 The $0$-th blob homology of $X$ is the usual |
269 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
270 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
270 by $\cC$. |
271 by $\cC$. |
271 (See \S \ref{sec:local-relations}.) |
272 (See \S \ref{sec:local-relations}.) |
272 \begin{equation*} |
273 \begin{equation*} |
273 H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X) |
274 H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X) |
274 \end{equation*} |
275 \end{equation*} |
275 \end{thm} |
276 \end{thm:skein-modules} |
276 |
277 |
277 \newtheorem*{thm:hochschild}{Theorem \ref{thm:hochschild}} |
278 \newtheorem*{thm:hochschild}{Theorem \ref{thm:hochschild}} |
278 |
279 |
279 \begin{thm:hochschild}[Hochschild homology when $X=S^1$] |
280 \begin{thm:hochschild}[Hochschild homology when $X=S^1$] |
280 The blob complex for a $1$-category $\cC$ on the circle is |
281 The blob complex for a $1$-category $\cC$ on the circle is |
284 \end{equation*} |
285 \end{equation*} |
285 \end{thm:hochschild} |
286 \end{thm:hochschild} |
286 |
287 |
287 Theorem \ref{thm:skein-modules} is immediate from the definition, and |
288 Theorem \ref{thm:skein-modules} is immediate from the definition, and |
288 Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}. |
289 Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}. |
289 We also note Appendix \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of certain commutative algebras thought of as an $n$-category. |
290 We also note \S \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of certain commutative algebras thought of as an $n$-category. |
290 |
291 |
291 |
292 |
292 \subsection{Structure of the blob complex} |
293 \subsection{Structure of the blob complex} |
293 \label{sec:structure} |
294 \label{sec:structure} |
294 |
295 |
295 In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
296 In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
296 \begin{thm}[$C_*(\Homeo(-))$ action]\mbox{}\\ |
297 |
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298 \newtheorem*{thm:CH}{Theorem \ref{thm:CH}} |
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299 |
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300 \begin{thm:CH}[$C_*(\Homeo(-))$ action]\mbox{}\\ |
297 \vspace{-0.5cm} |
301 \vspace{-0.5cm} |
298 \label{thm:evaluation}% |
302 \label{thm:evaluation}% |
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303 There is a chain map |
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304 \begin{equation*} |
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305 \ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
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306 \end{equation*} |
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307 such that |
299 \begin{enumerate} |
308 \begin{enumerate} |
300 \item There is a chain map |
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301 \begin{equation*} |
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302 \ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
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303 \end{equation*} |
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304 |
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305 \item Restricted to $C_0(\Homeo(X))$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. |
309 \item Restricted to $C_0(\Homeo(X))$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. |
306 |
310 |
307 \item For |
311 \item For |
308 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
312 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
309 (using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
313 (using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
313 \CH{X} \otimes \bc_*(X) |
317 \CH{X} \otimes \bc_*(X) |
314 \ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
318 \ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
315 \bc_*(X) \ar[u]_{\gl_Y} |
319 \bc_*(X) \ar[u]_{\gl_Y} |
316 } |
320 } |
317 \end{equation*} |
321 \end{equation*} |
318 \item Any such chain map satisfying points 2. and 3. above is unique, up to an iterated homotopy. |
322 \end{enumerate} |
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323 Moreover any such chain map is unique, up to an iterated homotopy. |
319 (That is, any pair of homotopies have a homotopy between them, and so on.) |
324 (That is, any pair of homotopies have a homotopy between them, and so on.) |
320 \item This map is associative, in the sense that the following diagram commutes (up to homotopy). |
325 \end{thm:CH} |
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326 |
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327 \newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}} |
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328 |
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329 |
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330 Further, |
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331 \begin{thm:CH-associativity} |
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332 \item The chain map of Theorem \ref{thm:CH} is associative, in the sense that the following diagram commutes (up to homotopy). |
321 \begin{equation*} |
333 \begin{equation*} |
322 \xymatrix{ |
334 \xymatrix{ |
323 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor \ev_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{\ev_X} \\ |
335 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor \ev_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{\ev_X} \\ |
324 \CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) |
336 \CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) |
325 } |
337 } |
326 \end{equation*} |
338 \end{equation*} |
327 \end{enumerate} |
339 \end{thm:CH-associativity} |
328 \end{thm} |
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329 |
340 |
330 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
341 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
331 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
342 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
332 for any homeomorphic pair $X$ and $Y$, |
343 for any homeomorphic pair $X$ and $Y$, |
333 satisfying corresponding conditions. |
344 satisfying corresponding conditions. |
334 |
345 |
335 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. |
346 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. |
336 Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. |
347 Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. |
337 Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
348 Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
338 |
349 |
339 \begin{thm}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
350 \todo{Give this a number inside the text} |
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351 \begin{thm}[Blob complexes of products with balls form an $A_\infty$ $n$-category] |
340 \label{thm:blobs-ainfty} |
352 \label{thm:blobs-ainfty} |
341 Let $\cC$ be a topological $n$-category. |
353 Let $\cC$ be a topological $n$-category. |
342 Let $Y$ be an $n{-}k$-manifold. |
354 Let $Y$ be an $n{-}k$-manifold. |
343 There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
355 There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
344 to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set |
356 to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set |
349 \end{thm} |
361 \end{thm} |
350 \begin{rem} |
362 \begin{rem} |
351 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. |
363 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. |
352 We think of this $A_\infty$ $n$-category as a free resolution. |
364 We think of this $A_\infty$ $n$-category as a free resolution. |
353 \end{rem} |
365 \end{rem} |
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366 Theorem \ref{thm:blobs-ainfty} appears as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats} |
354 |
367 |
355 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
368 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
356 instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. |
369 instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. |
357 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
370 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
358 |
371 |
359 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
372 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
360 |
373 |
361 \begin{thm:product}[Product formula] |
374 \begin{thm:product}[Product formula] |
362 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
375 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
379 \mbox{}% <-- gets the indenting right |
392 \mbox{}% <-- gets the indenting right |
380 \begin{itemize} |
393 \begin{itemize} |
381 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
394 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
382 $A_\infty$ module for $\bc_*(Y)$. |
395 $A_\infty$ module for $\bc_*(Y)$. |
383 |
396 |
384 \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 |
397 \item For any $n$-manifold $X_\text{gl} = X\bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{gl})$ is the $A_\infty$ self-tensor product of |
385 $\bc_*(X_\text{cut})$ as an $\bc_*(Y)$-bimodule: |
398 $\bc_*(X)$ as an $\bc_*(Y)$-bimodule: |
386 \begin{equation*} |
399 \begin{equation*} |
387 \bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
400 \bc_*(X_\text{gl}) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
388 \end{equation*} |
401 \end{equation*} |
389 \end{itemize} |
402 \end{itemize} |
390 \end{thm:gluing} |
403 \end{thm:gluing} |
391 |
404 |
392 Theorem \ref{thm:evaluation} is proved in |
405 Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, with Theorem \ref{thm:gluing} then a relatively straightforward consequence of the proof, explained in \S \ref{sec:gluing}. |
393 in \S \ref{sec:evaluation}, Theorem \ref{thm:blobs-ainfty} appears as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}, |
|
394 and Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, with Theorem \ref{thm:gluing} then a relatively straightforward consequence of the proof, explained in \S \ref{sec:gluing}. |
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395 |
406 |
396 \subsection{Applications} |
407 \subsection{Applications} |
397 \label{sec:applications} |
408 \label{sec:applications} |
398 Finally, we give two theorems which we consider as applications. |
409 Finally, we give two theorems which we consider as applications. |
399 |
410 |
424 Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). |
435 Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). |
425 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. |
436 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. |
426 We could presumably also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories), |
437 We could presumably also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories), |
427 and likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories. |
438 and likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories. |
428 More could be said about finite characteristic |
439 More could be said about finite characteristic |
429 (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$, for example). |
440 (there appears in be $2$-torsion in $\bc_1(S^2; \cC)$ for any spherical $2$-category $\cC$, for example). |
430 Much more could be said about other types of manifolds, in particular oriented, |
441 Much more could be said about other types of manifolds, in particular oriented, |
431 $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. |
442 $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. |
432 (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) |
443 (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) |
433 We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; |
444 We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; |
434 there may be some differences for topological manifolds and smooth manifolds. |
445 there may be some differences for topological manifolds and smooth manifolds. |
436 The paper ``Skein homology'' \cite{MR1624157} has similar motivations, and it may be |
447 The paper ``Skein homology'' \cite{MR1624157} has similar motivations, and it may be |
437 interesting to investigate if there is a connection with the material here. |
448 interesting to investigate if there is a connection with the material here. |
438 |
449 |
439 Many results in Hochschild homology can be understood ``topologically" via the blob complex. |
450 Many results in Hochschild homology can be understood ``topologically" via the blob complex. |
440 For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ |
451 For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ |
441 (see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, |
452 (see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1]; A)$, |
442 but haven't investigated the details. |
453 but haven't investigated the details. |
443 |
454 |
444 Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} |
455 Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} \nn{stabilization} \nn{stable categories, generalized cohomology theories} |
445 |
456 |
446 |
457 |
447 \subsection{Thanks and acknowledgements} |
458 \subsection{Thanks and acknowledgements} |
448 % attempting to make this chronological rather than alphabetical |
459 % attempting to make this chronological rather than alphabetical |
449 We'd like to thank |
460 We'd like to thank |