109 |
109 |
110 \draw[->] (FU) -- node[right=10pt] {$\cF(M)/\cU$} (A); |
110 \draw[->] (FU) -- node[right=10pt] {$\cF(M)/\cU$} (A); |
111 |
111 |
112 \draw[->] (C) -- node[left=10pt] { |
112 \draw[->] (C) -- node[left=10pt] { |
113 Example \ref{ex:traditional-n-categories(fields)} \\ and \S \ref{ss:ncat_fields} |
113 Example \ref{ex:traditional-n-categories(fields)} \\ and \S \ref{ss:ncat_fields} |
114 %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ |
114 %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker e: \cC(c) \to \cC(B)$ |
115 } (FU); |
115 } (FU); |
116 \draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A); |
116 \draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A); |
117 |
117 |
118 \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
118 \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
119 \draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
119 \draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
132 and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
132 and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
133 as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
133 as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
134 Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
134 Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
135 thought of as a topological $n$-category, in terms of the topology of $M$. |
135 thought of as a topological $n$-category, in terms of the topology of $M$. |
136 |
136 |
137 Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes) |
137 %%%% this is said later in the intro |
138 \nn{...} |
138 %Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes) |
139 |
139 %even when we could work in greater generality (symmetric monoidal categories, model categories, etc.). |
140 |
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141 %\item related: we are being unsophisticated from a homotopy theory point of |
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142 %view and using chain complexes in many places where we could get by with spaces |
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143 |
140 |
144 %\item ? one of the points we make (far) below is that there is not really much |
141 %\item ? one of the points we make (far) below is that there is not really much |
145 %difference between (a) systems of fields and local relations and (b) $n$-cats; |
142 %difference between (a) systems of fields and local relations and (b) $n$-cats; |
146 %thus we tend to switch between talking in terms of one or the other |
143 %thus we tend to switch between talking in terms of one or the other |
147 |
144 |
149 |
146 |
150 \subsection{Motivations} |
147 \subsection{Motivations} |
151 \label{sec:motivations} |
148 \label{sec:motivations} |
152 |
149 |
153 We will briefly sketch our original motivation for defining the blob complex. |
150 We will briefly sketch our original motivation for defining the blob complex. |
154 \nn{this is adapted from an old draft of the intro; it needs further modification |
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155 in order to better integrate it into the current intro.} |
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156 |
151 |
157 As a starting point, consider TQFTs constructed via fields and local relations. |
152 As a starting point, consider TQFTs constructed via fields and local relations. |
158 (See \S\ref{sec:tqftsviafields} or \cite{kw:tqft}.) |
153 (See \S\ref{sec:tqftsviafields} or \cite{kw:tqft}.) |
159 This gives a satisfactory treatment for semisimple TQFTs |
154 This gives a satisfactory treatment for semisimple TQFTs |
160 (i.e.\ TQFTs for which the cylinder 1-category associated to an |
155 (i.e.\ TQFTs for which the cylinder 1-category associated to an |
164 Our main motivating example (though we will not develop it in this paper) |
159 Our main motivating example (though we will not develop it in this paper) |
165 is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology. |
160 is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology. |
166 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
161 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
167 with a link $L \subset \bd W$. |
162 with a link $L \subset \bd W$. |
168 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
163 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
169 \todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S} |
164 %\todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S} |
170 |
165 |
171 How would we go about computing $A_{Kh}(W^4, L)$? |
166 How would we go about computing $A_{Kh}(W^4, L)$? |
172 For the Khovanov homology of a link in $S^3$ the main tool is the exact triangle (long exact sequence) |
167 For the Khovanov homology of a link in $S^3$ the main tool is the exact triangle (long exact sequence) |
173 relating resolutions of a crossing. |
168 relating resolutions of a crossing. |
174 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
169 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
176 According to the gluing theorem for TQFTs, gluing along $B^3 \subset \bd B^4$ |
171 According to the gluing theorem for TQFTs, gluing along $B^3 \subset \bd B^4$ |
177 corresponds to taking a coend (self tensor product) over the cylinder category |
172 corresponds to taking a coend (self tensor product) over the cylinder category |
178 associated to $B^3$ (with appropriate boundary conditions). |
173 associated to $B^3$ (with appropriate boundary conditions). |
179 The coend is not an exact functor, so the exactness of the triangle breaks. |
174 The coend is not an exact functor, so the exactness of the triangle breaks. |
180 |
175 |
181 |
176 The obvious solution to this problem is to replace the coend with its derived counterpart, |
182 The obvious solution to this problem is to replace the coend with its derived counterpart. |
177 Hochschild homology. |
183 This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology |
178 This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology |
184 of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired. |
179 of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired. |
185 If we build our manifold up via a handle decomposition, the computation |
180 If we build our manifold up via a handle decomposition, the computation |
186 would be a sequence of derived coends. |
181 would be a sequence of derived coends. |
187 A different handle decomposition of the same manifold would yield a different |
182 A different handle decomposition of the same manifold would yield a different |
188 sequence of derived coends. |
183 sequence of derived coends. |
189 To show that our definition in terms of derived coends is well-defined, we |
184 To show that our definition in terms of derived coends is well-defined, we |
190 would need to show that the above two sequences of derived coends yield the same answer. |
185 would need to show that the above two sequences of derived coends yield |
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186 isomorphic answers, and that the isomorphism does not depend on any |
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187 choices we made along the way. |
191 This is probably not easy to do. |
188 This is probably not easy to do. |
192 |
189 |
193 Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$ |
190 Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$ |
194 which is manifestly invariant. |
191 which is manifestly invariant. |
195 (That is, a definition that does not |
192 (That is, a definition that does not |
199 |
196 |
200 The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient |
197 The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient |
201 \[ |
198 \[ |
202 \text{linear combinations of fields} \;\big/\; \text{local relations} , |
199 \text{linear combinations of fields} \;\big/\; \text{local relations} , |
203 \] |
200 \] |
204 with an appropriately free resolution (the ``blob complex") |
201 with an appropriately free resolution (the blob complex) |
205 \[ |
202 \[ |
206 \cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
203 \cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
207 \] |
204 \] |
208 Here $\bc_0$ is linear combinations of fields on $W$, |
205 Here $\bc_0$ is linear combinations of fields on $W$, |
209 $\bc_1$ is linear combinations of local relations on $W$, |
206 $\bc_1$ is linear combinations of local relations on $W$, |
210 $\bc_2$ is linear combinations of relations amongst relations on $W$, |
207 $\bc_2$ is linear combinations of relations amongst relations on $W$, |
211 and so on. |
208 and so on. |
212 |
209 |
213 None of the above ideas depend on the details of the Khovanov homology example, |
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214 so we develop the general theory in this paper and postpone specific applications |
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215 to later papers. |
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216 |
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217 |
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218 |
210 |
219 \subsection{Formal properties} |
211 \subsection{Formal properties} |
220 \label{sec:properties} |
212 \label{sec:properties} |
221 The blob complex enjoys the following list of formal properties. |
213 The blob complex enjoys the following list of formal properties. |
222 |
214 |
234 As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cF)$; |
226 As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cF)$; |
235 this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below. |
227 this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below. |
236 |
228 |
237 The blob complex is also functorial (indeed, exact) with respect to $\cF$, |
229 The blob complex is also functorial (indeed, exact) with respect to $\cF$, |
238 although we will not address this in detail here. |
230 although we will not address this in detail here. |
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231 \nn{KW: what exactly does ``exact in $\cF$" mean? |
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232 Do we mean a similar statement for module labels?} |
239 |
233 |
240 \begin{property}[Disjoint union] |
234 \begin{property}[Disjoint union] |
241 \label{property:disjoint-union} |
235 \label{property:disjoint-union} |
242 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
236 The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes. |
243 \begin{equation*} |
237 \begin{equation*} |
244 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
238 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
245 \end{equation*} |
239 \end{equation*} |
246 \end{property} |
240 \end{property} |
247 |
241 |
262 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
256 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
263 \end{property} |
257 \end{property} |
264 |
258 |
265 \begin{property}[Contractibility] |
259 \begin{property}[Contractibility] |
266 \label{property:contractibility}% |
260 \label{property:contractibility}% |
267 With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology. |
261 With field coefficients, the blob complex on an $n$-ball is contractible in the sense |
268 Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cF$ to balls. |
262 that it is homotopic to its $0$-th homology. |
269 \begin{equation*} |
263 Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces |
270 \xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & \cF(B^n)} |
264 associated by the system of fields $\cF$ to balls. |
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265 \begin{equation*} |
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266 \xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & A_\cF(B^n)} |
271 \end{equation*} |
267 \end{equation*} |
272 \end{property} |
268 \end{property} |
273 |
269 |
274 Properties \ref{property:functoriality} will be immediate from the definition given in |
270 Properties \ref{property:functoriality} will be immediate from the definition given in |
275 \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
271 \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
276 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
272 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and |
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273 \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
277 |
274 |
278 \subsection{Specializations} |
275 \subsection{Specializations} |
279 \label{sec:specializations} |
276 \label{sec:specializations} |
280 |
277 |
281 The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology. |
278 The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology. |
296 |
293 |
297 \begin{thm:hochschild}[Hochschild homology when $X=S^1$] |
294 \begin{thm:hochschild}[Hochschild homology when $X=S^1$] |
298 The blob complex for a $1$-category $\cC$ on the circle is |
295 The blob complex for a $1$-category $\cC$ on the circle is |
299 quasi-isomorphic to the Hochschild complex. |
296 quasi-isomorphic to the Hochschild complex. |
300 \begin{equation*} |
297 \begin{equation*} |
301 \xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} |
298 \xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
302 \end{equation*} |
299 \end{equation*} |
303 \end{thm:hochschild} |
300 \end{thm:hochschild} |
304 |
301 |
305 Theorem \ref{thm:skein-modules} is immediate from the definition, and |
302 Theorem \ref{thm:skein-modules} is immediate from the definition, and |
306 Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}. |
303 Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}. |
307 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. |
304 We also note \S \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of |
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305 certain commutative algebras thought of as $n$-categories. |
308 |
306 |
309 |
307 |
310 \subsection{Structure of the blob complex} |
308 \subsection{Structure of the blob complex} |
311 \label{sec:structure} |
309 \label{sec:structure} |
312 |
310 |
328 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
326 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
329 (using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
327 (using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
330 \begin{equation*} |
328 \begin{equation*} |
331 \xymatrix@C+2cm{ |
329 \xymatrix@C+2cm{ |
332 \CH{X} \otimes \bc_*(X) |
330 \CH{X} \otimes \bc_*(X) |
333 \ar[r]_{\ev_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
331 \ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
334 \bc_*(X) \ar[d]_{\gl_Y} \\ |
332 \bc_*(X) \ar[d]_{\gl_Y} \\ |
335 \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) |
333 \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) |
336 } |
334 } |
337 \end{equation*} |
335 \end{equation*} |
338 \end{enumerate} |
336 \end{enumerate} |
339 Moreover any such chain map is unique, up to an iterated homotopy. |
337 Moreover any such chain map is unique, up to an iterated homotopy. |
340 (That is, any pair of homotopies have a homotopy between them, and so on.) |
338 (That is, any pair of homotopies have a homotopy between them, and so on.) |
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339 \nn{revisit this after proof below has stabilized} |
341 \end{thm:CH} |
340 \end{thm:CH} |
342 |
341 |
343 \newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}} |
342 \newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}} |
344 |
343 |
345 |
344 |
346 Further, |
345 Further, |
347 \begin{thm:CH-associativity} |
346 \begin{thm:CH-associativity} |
348 The chain map of Theorem \ref{thm:CH} is associative, in the sense that the following diagram commutes (up to homotopy). |
347 The chain map of Theorem \ref{thm:CH} is associative, in the sense that the following diagram commutes (up to homotopy). |
349 \begin{equation*} |
348 \begin{equation*} |
350 \xymatrix{ |
349 \xymatrix{ |
351 \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} \\ |
350 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor e_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{e_X} \\ |
352 \CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) |
351 \CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X) |
353 } |
352 } |
354 \end{equation*} |
353 \end{equation*} |
355 \end{thm:CH-associativity} |
354 \end{thm:CH-associativity} |
356 |
355 |
357 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
356 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |