121 %% \author{Roberta Graff\affil{1}{University of Cambridge, Cambridge, |
121 %% \author{Roberta Graff\affil{1}{University of Cambridge, Cambridge, |
122 %% United Kingdom}, |
122 %% United Kingdom}, |
123 %% Javier de Ruiz Garcia\affil{2}{Universidad de Murcia, Bioquimica y Biologia |
123 %% Javier de Ruiz Garcia\affil{2}{Universidad de Murcia, Bioquimica y Biologia |
124 %% Molecular, Murcia, Spain}, \and Franklin Sonnery\affil{2}{}} |
124 %% Molecular, Murcia, Spain}, \and Franklin Sonnery\affil{2}{}} |
125 |
125 |
126 \author{Scott Morrison\affil{1}{Miller Institute for Basic Research, UC Berkeley, CA 94704, USA} \and Kevin Walker\affil{2}{Microsoft Station Q, 2243 CNSI Building, UC Santa Barbara, CA 93106, USA}} |
126 \author{Scott Morrison\affil{1}{Miller Institute for Basic Research, UC Berkeley, CA 94704, USA} |
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127 \and Kevin Walker\affil{2}{Microsoft Station Q, 2243 CNSI Building, UC Santa Barbara, CA 93106, USA}} |
127 |
128 |
128 \contributor{Submitted to Proceedings of the National Academy of Sciences |
129 \contributor{Submitted to Proceedings of the National Academy of Sciences |
129 of the United States of America} |
130 of the United States of America} |
130 |
131 |
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135 \begin{article} |
136 \begin{article} |
136 |
137 |
137 \begin{abstract} |
138 \begin{abstract} |
138 We explain the need for new axioms for topological quantum field theories that include ideas from derived categories and homotopy theory. We summarize our axioms for higher categories, and describe the `blob complex'. Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$. The $0$-th homology of this chain complex recovers the usual TQFT invariants of $W$. The higher homology groups should be viewed as generalizations of Hochschild homology. The blob complex has a very natural definition in terms of homotopy colimits along decompositions of the manifold $W$. We outline the important properties of the blob complex, and sketch the proof of a generalization of Deligne's conjecture on Hochschild cohomology and the little discs operad to higher dimensions. |
139 We explain the need for new axioms for topological quantum field theories that include ideas from derived |
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140 categories and homotopy theory. We summarize our axioms for higher categories, and describe the `blob complex'. |
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141 Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$. |
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142 The $0$-th homology of this chain complex recovers the usual TQFT invariants of $W$. |
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143 The higher homology groups should be viewed as generalizations of Hochschild homology. |
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144 The blob complex has a very natural definition in terms of homotopy colimits along decompositions of the manifold $W$. |
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145 We outline the important properties of the blob complex, and sketch the proof of a generalization of |
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146 Deligne's conjecture on Hochschild cohomology and the little discs operad to higher dimensions. |
139 \end{abstract} |
147 \end{abstract} |
140 |
148 |
141 |
149 |
142 %% When adding keywords, separate each term with a straight line: | |
150 %% When adding keywords, separate each term with a straight line: | |
143 \keywords{n-categories | topological quantum field theory | hochschild homology} |
151 \keywords{n-categories | topological quantum field theory | hochschild homology} |
174 A linear 0-category is a vector space, and a representation |
182 A linear 0-category is a vector space, and a representation |
175 of a vector space is an element of the dual space. |
183 of a vector space is an element of the dual space. |
176 Thus a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$, |
184 Thus a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$, |
177 and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$. |
185 and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$. |
178 For the remainder of this paper we will in fact be interested in so-called $(n{+}\epsilon)$-dimensional |
186 For the remainder of this paper we will in fact be interested in so-called $(n{+}\epsilon)$-dimensional |
179 TQFTs, which are slightly weaker structures and do not assign anything to general $(n{+}1)$-manifolds, but only to mapping cylinders. |
187 TQFTs, which are slightly weaker structures and do not assign anything to general $(n{+}1)$-manifolds, |
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188 but only to mapping cylinders. |
180 |
189 |
181 When $k=n-1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$, |
190 When $k=n-1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$, |
182 and a representation of $A(\bd Y)$ for each $n$-manifold $Y$. |
191 and a representation of $A(\bd Y)$ for each $n$-manifold $Y$. |
183 The TQFT gluing rule in dimension $n$ states that |
192 The TQFT gluing rule in dimension $n$ states that |
184 $A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$, |
193 $A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$, |
191 We call a TQFT semisimple if $A(S)$ is a semisimple 1-category for all $(n{-}1)$-manifolds $S$ |
200 We call a TQFT semisimple if $A(S)$ is a semisimple 1-category for all $(n{-}1)$-manifolds $S$ |
192 and $A(Y)$ is a finite-dimensional vector space for all $n$-manifolds $Y$. |
201 and $A(Y)$ is a finite-dimensional vector space for all $n$-manifolds $Y$. |
193 Examples of semisimple TQFTs include Witten-Reshetikhin-Turaev theories, |
202 Examples of semisimple TQFTs include Witten-Reshetikhin-Turaev theories, |
194 Turaev-Viro theories, and Dijkgraaf-Witten theories. |
203 Turaev-Viro theories, and Dijkgraaf-Witten theories. |
195 These can all be given satisfactory accounts in the framework outlined above. |
204 These can all be given satisfactory accounts in the framework outlined above. |
196 (The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories with only a weak dependence on interiors in order to be |
205 (The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories with only a weak |
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206 dependence on interiors in order to be |
197 extended all the way down to dimension 0.) |
207 extended all the way down to dimension 0.) |
198 |
208 |
199 For other non-semisimple TQFT-like invariants, however, the above framework seems to be inadequate. |
209 For other non-semisimple TQFT-like invariants, however, the above framework seems to be inadequate. |
200 For example, the gluing rule for 3-manifolds in Ozsv\'{a}th-Szab\'{o}/Seiberg-Witten theory |
210 For example, the gluing rule for 3-manifolds in Ozsv\'{a}th-Szab\'{o}/Seiberg-Witten theory |
201 involves a tensor product over an $A_\infty$ 1-category associated to 2-manifolds \cite{1003.0598,1005.1248}. |
211 involves a tensor product over an $A_\infty$ 1-category associated to 2-manifolds \cite{1003.0598,1005.1248}. |
257 %of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
269 %of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
258 %We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
270 %We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
259 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
271 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
260 |
272 |
261 We will define two variations simultaneously, as all but one of the axioms are identical |
273 We will define two variations simultaneously, as all but one of the axioms are identical |
262 in the two cases. These variations are `linear $n$-categories', where the sets associated to $n$-balls with specified boundary conditions are in fact vector spaces, and `$A_\infty$ $n$-categories', where these sets are chain complexes. |
274 in the two cases. These variations are `linear $n$-categories', where the sets associated to |
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275 $n$-balls with specified boundary conditions are in fact vector spaces, and `$A_\infty$ $n$-categories', |
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276 where these sets are chain complexes. |
263 |
277 |
264 |
278 |
265 There are five basic ingredients |
279 There are five basic ingredients |
266 \cite{life-of-brian} of an $n$-category definition: |
280 \cite{life-of-brian} of an $n$-category definition: |
267 $k$-morphisms (for $0\le k \le n$), domain and range, composition, |
281 $k$-morphisms (for $0\le k \le n$), domain and range, composition, |
279 In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
293 In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
280 {\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
294 {\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
281 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
295 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
282 We wish to imitate this strategy in higher categories. |
296 We wish to imitate this strategy in higher categories. |
283 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
297 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
284 a product of $k$ intervals (c.f. \cite{0909.2212}) but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic |
298 a product of $k$ intervals (c.f. \cite{0909.2212}) but rather with any $k$-ball, that is, |
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299 any $k$-manifold which is homeomorphic |
285 to the standard $k$-ball $B^k$. |
300 to the standard $k$-ball $B^k$. |
286 |
301 |
287 By default our balls are unoriented, |
302 By default our balls are unoriented, |
288 but it is useful at times to vary this, |
303 but it is useful at times to vary this, |
289 for example by considering oriented or Spin balls. |
304 for example by considering oriented or Spin balls. |
302 homeomorphisms which are not the identity on the boundary of the $k$-ball. |
317 homeomorphisms which are not the identity on the boundary of the $k$-ball. |
303 The action of these homeomorphisms gives the ``strong duality" structure. |
318 The action of these homeomorphisms gives the ``strong duality" structure. |
304 As such, we don't subdivide the boundary of a morphism |
319 As such, we don't subdivide the boundary of a morphism |
305 into domain and range --- the duality operations can convert between domain and range. |
320 into domain and range --- the duality operations can convert between domain and range. |
306 |
321 |
307 Later we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom. |
322 Later we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ |
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323 from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, |
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324 for $k<n$, for the next axiom. |
308 |
325 |
309 \begin{axiom}[Boundaries]\label{nca-boundary} |
326 \begin{axiom}[Boundaries]\label{nca-boundary} |
310 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
327 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
311 These maps, for various $X$, comprise a natural transformation of functors. |
328 These maps, for various $X$, comprise a natural transformation of functors. |
312 \end{axiom} |
329 \end{axiom} |
318 This means that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure |
335 This means that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure |
319 of an object of $\cS$, and all of the structure maps of the category (above and below) are |
336 of an object of $\cS$, and all of the structure maps of the category (above and below) are |
320 compatible with the $\cS$ structure on $\cC_n(X; c)$. |
337 compatible with the $\cS$ structure on $\cC_n(X; c)$. |
321 |
338 |
322 |
339 |
323 Given two hemispheres (a `domain' and `range') that agree on the equator, we need to be able to assemble them into a boundary value of the entire sphere. |
340 Given two hemispheres (a `domain' and `range') that agree on the equator, we need to be able to |
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341 assemble them into a boundary value of the entire sphere. |
324 |
342 |
325 \begin{lem} |
343 \begin{lem} |
326 \label{lem:domain-and-range} |
344 \label{lem:domain-and-range} |
327 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
345 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
328 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
346 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
490 When $X$ is a $k$-ball with $k<n$, $\Bord^n(X)$ is the set of all $k$-dimensional |
508 When $X$ is a $k$-ball with $k<n$, $\Bord^n(X)$ is the set of all $k$-dimensional |
491 submanifolds $W$ in $X\times \bbR^\infty$ which project to $X$ transversely |
509 submanifolds $W$ in $X\times \bbR^\infty$ which project to $X$ transversely |
492 to $\bd X$. |
510 to $\bd X$. |
493 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes rel boundary of such $n$-dimensional submanifolds. |
511 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes rel boundary of such $n$-dimensional submanifolds. |
494 |
512 |
495 There is an $A_\infty$ analogue enriched in topological spaces, where at the top level we take all such submanifolds, rather than homeomorphism classes. For each fixed $\bdy W \subset \bdy X \times \bbR^\infty$, we can topologize the set of submanifolds by ambient isotopy rel boundary. |
513 There is an $A_\infty$ analogue enriched in topological spaces, where at the top level we take |
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514 all such submanifolds, rather than homeomorphism classes. |
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515 For each fixed $\bdy W \subset \bdy X \times \bbR^\infty$, we can |
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516 topologize the set of submanifolds by ambient isotopy rel boundary. |
496 |
517 |
497 \subsection{The blob complex} |
518 \subsection{The blob complex} |
498 \subsubsection{Decompositions of manifolds} |
519 \subsubsection{Decompositions of manifolds} |
499 |
520 |
500 A \emph{ball decomposition} of a $k$-manifold $W$ is a |
521 A \emph{ball decomposition} of a $k$-manifold $W$ is a |
517 The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
538 The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
518 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
539 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
519 See Figure \ref{partofJfig} for an example. |
540 See Figure \ref{partofJfig} for an example. |
520 \end{defn} |
541 \end{defn} |
521 |
542 |
522 This poset in fact has more structure, since we can glue together permissible decompositions of $W_1$ and $W_2$ to obtain a permissible decomposition of $W_1 \sqcup W_2$. |
543 This poset in fact has more structure, since we can glue together permissible decompositions of |
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544 $W_1$ and $W_2$ to obtain a permissible decomposition of $W_1 \sqcup W_2$. |
523 |
545 |
524 An $n$-category $\cC$ determines |
546 An $n$-category $\cC$ determines |
525 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
547 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
526 (possibly with additional structure if $k=n$). |
548 (possibly with additional structure if $k=n$). |
527 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
549 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
534 \begin{equation*} |
556 \begin{equation*} |
535 %\label{eq:psi-C} |
557 %\label{eq:psi-C} |
536 \psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl |
558 \psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl |
537 \end{equation*} |
559 \end{equation*} |
538 where the restrictions to the various pieces of shared boundaries amongst the cells |
560 where the restrictions to the various pieces of shared boundaries amongst the cells |
539 $X_a$ all agree (this is a fibered product of all the labels of $k$-cells over the labels of $k-1$-cells). When $k=n$, the `subset' and `product' in the above formula should be interpreted in the appropriate enriching category. |
561 $X_a$ all agree (this is a fibered product of all the labels of $k$-cells over the labels of $k-1$-cells). |
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562 When $k=n$, the `subset' and `product' in the above formula should be |
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563 interpreted in the appropriate enriching category. |
540 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
564 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
541 \end{defn} |
565 \end{defn} |
542 |
566 |
543 We will use the term `field on $W$' to refer to a point of this functor, |
567 We will use the term `field on $W$' to refer to a point of this functor, |
544 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
568 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
545 |
569 |
546 |
570 |
547 \subsubsection{Colimits} |
571 \subsubsection{Colimits} |
548 Our definition of an $n$-category is essentially a collection of functors defined on $k$-balls (and homeomorphisms) for $k \leq n$ satisfying certain axioms. It is natural to consider extending such functors to the larger categories of all $k$-manifolds (again, with homeomorphisms). In fact, the axioms stated above explicitly require such an extension to $k$-spheres for $k<n$. |
572 Our definition of an $n$-category is essentially a collection of functors defined on $k$-balls (and homeomorphisms) |
549 |
573 for $k \leq n$ satisfying certain axioms. |
550 The natural construction achieving this is the colimit. For a linear $n$-category $\cC$, we denote the extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, this is defined to be the colimit of the function $\psi_{\cC;W}$. Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. Recall that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, the set $\cC(X;c)$ is a vector space. Using this, we note that for $c \in \cl{\cC}(\bdy X)$, for $X$ an arbitrary $n$-manifold, the set $\cl{\cC}(X;c) = \bdy^{-1} (c)$ inherits the structure of a vector space. These are the usual TQFT skein module invariants on $n$-manifolds. |
574 It is natural to consider extending such functors to the |
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575 larger categories of all $k$-manifolds (again, with homeomorphisms). |
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576 In fact, the axioms stated above explicitly require such an extension to $k$-spheres for $k<n$. |
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577 |
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578 The natural construction achieving this is the colimit. For a linear $n$-category $\cC$, |
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579 we denote the extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, |
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580 this is defined to be the colimit of the function $\psi_{\cC;W}$. |
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581 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} |
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582 imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. |
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583 Recall that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, |
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584 the set $\cC(X;c)$ is a vector space. Using this, we note that for $c \in \cl{\cC}(\bdy X)$, |
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585 for $X$ an arbitrary $n$-manifold, the set $\cl{\cC}(X;c) = \bdy^{-1} (c)$ inherits the structure of a vector space. |
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586 These are the usual TQFT skein module invariants on $n$-manifolds. |
551 |
587 |
552 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
588 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
553 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ |
589 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ |
554 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
590 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
555 |
591 |
556 An explicit realization of the homotopy colimit is provided by the simplices of the functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{\bar{x}} \psi_{\cC; W}(x_0)[m],$$ where $\bar{x} = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. The differential acts on $(\bar{x},a)$ (here $a \in \psi_{\cC; W}(x_0)$) as |
592 An explicit realization of the homotopy colimit is provided by the simplices of the |
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593 functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{\bar{x}} \psi_{\cC; W}(x_0)[m],$$ |
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594 where $\bar{x} = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. |
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595 The differential acts on $(\bar{x},a)$ (here $a \in \psi_{\cC; W}(x_0)$) as |
557 $$\bdy (\bar{x},a) = (\bar{x}, \bdy a) + (-1)^{\deg a} \left( (d_0 \bar{x}, g(a)) + \sum_{i=1}^m (-1)^i (d_i \bar{x}, a) \right)$$ |
596 $$\bdy (\bar{x},a) = (\bar{x}, \bdy a) + (-1)^{\deg a} \left( (d_0 \bar{x}, g(a)) + \sum_{i=1}^m (-1)^i (d_i \bar{x}, a) \right)$$ |
558 where $g$ is the gluing map from $x_0$ to $x_1$, and $d_i \bar{x}$ denotes the $i$-th face of the simplex $\bar{x}$. |
597 where $g$ is the gluing map from $x_0$ to $x_1$, and $d_i \bar{x}$ denotes the $i$-th face of the simplex $\bar{x}$. |
559 |
598 |
560 Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the homotopy colimit via the cone-product polyhedra in $\cell(W)$. A cone-product polyhedra is obtained from a point by successively taking the cone or taking the product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, and taking product identifies the roots of several trees. The `local homotopy colimit' is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required. |
599 Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the |
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600 homotopy colimit via the cone-product polyhedra in $\cell(W)$. |
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601 A cone-product polyhedra is obtained from a point by successively taking the cone or taking the |
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602 product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, |
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603 cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, |
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604 and taking product identifies the roots of several trees. |
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605 The `local homotopy colimit' is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. |
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606 The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required. |
561 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
607 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
562 |
608 |
563 %When $\cC$ is a topological $n$-category, |
609 %When $\cC$ is a topological $n$-category, |
564 %the flexibility available in the construction of a homotopy colimit allows |
610 %the flexibility available in the construction of a homotopy colimit allows |
565 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
611 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
574 it evaluates to a zero $n$-morphism of $C$. |
620 it evaluates to a zero $n$-morphism of $C$. |
575 The next few paragraphs describe this in more detail. |
621 The next few paragraphs describe this in more detail. |
576 |
622 |
577 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
623 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
578 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
624 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
579 each $B_i$ appears as a connected component of one of the $M_j$. Note that this allows the balls to be pairwise either disjoint or nested. Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. These pieces need not be manifolds, but they do automatically have permissible decompositions. |
625 each $B_i$ appears as a connected component of one of the $M_j$. |
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626 Note that this allows the balls to be pairwise either disjoint or nested. |
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627 Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. |
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628 These pieces need not be manifolds, but they do automatically have permissible decompositions. |
580 |
629 |
581 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of |
630 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of |
582 \begin{itemize} |
631 \begin{itemize} |
583 \item a permissible collection of $k$ embedded balls, and |
632 \item a permissible collection of $k$ embedded balls, and |
584 \item for each resulting piece of $W$, a field, |
633 \item for each resulting piece of $W$, a field, |
585 \end{itemize} |
634 \end{itemize} |
586 such that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. We call such a field a `null field on $B$'. |
635 such that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. |
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636 We call such a field a `null field on $B$'. |
587 |
637 |
588 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs. |
638 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs. |
589 |
639 |
590 We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$. For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field to that ball is a null field. The differential simply forgets the ball. Thus we see that $H_0$ of the blob complex is the quotient of fields by fields which are null on some ball. |
640 We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$. |
591 |
641 For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field to that ball is a null field. |
592 For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. In the first case, the balls are disjoint, and $f$ restricted to either of the $B_i$ is a null field. In the second case, the balls are properly nested, say $B_1 \subset B_2$, and $f$ restricted to $B_1$ is null. Note that this implies that $f$ restricted to $B_2$ is also null, by the associativity of the gluing operation. This ensures that the differential is well-defined. |
642 The differential simply forgets the ball. |
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643 Thus we see that $H_0$ of the blob complex is the quotient of fields by fields which are null on some ball. |
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644 |
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645 For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. |
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646 In the first case, the balls are disjoint, and $f$ restricted to either of the $B_i$ is a null field. |
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647 In the second case, the balls are properly nested, say $B_1 \subset B_2$, and $f$ restricted to $B_1$ is null. |
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648 Note that this implies that $f$ restricted to $B_2$ is also null, by the associativity of the gluing operation. |
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649 This ensures that the differential is well-defined. |
593 |
650 |
594 \section{Properties of the blob complex} |
651 \section{Properties of the blob complex} |
595 \subsection{Formal properties} |
652 \subsection{Formal properties} |
596 \label{sec:properties} |
653 \label{sec:properties} |
597 The blob complex enjoys the following list of formal properties. The first three are immediate from the definitions. |
654 The blob complex enjoys the following list of formal properties. The first three are immediate from the definitions. |
667 by $\cC$. |
724 by $\cC$. |
668 \begin{equation*} |
725 \begin{equation*} |
669 H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X) |
726 H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X) |
670 \end{equation*} |
727 \end{equation*} |
671 \end{thm} |
728 \end{thm} |
672 This follows from the fact that the $0$-th homology of a homotopy colimit is the usual colimit, or directly from the explicit description of the blob complex. |
729 This follows from the fact that the $0$-th homology of a homotopy colimit is the usual colimit, |
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730 or directly from the explicit description of the blob complex. |
673 |
731 |
674 \begin{thm}[Hochschild homology when $X=S^1$] |
732 \begin{thm}[Hochschild homology when $X=S^1$] |
675 \label{thm:hochschild} |
733 \label{thm:hochschild} |
676 The blob complex for a $1$-category $\cC$ on the circle is |
734 The blob complex for a $1$-category $\cC$ on the circle is |
677 quasi-isomorphic to the Hochschild complex. |
735 quasi-isomorphic to the Hochschild complex. |
678 \begin{equation*} |
736 \begin{equation*} |
679 \xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
737 \xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
680 \end{equation*} |
738 \end{equation*} |
681 \end{thm} |
739 \end{thm} |
682 This theorem is established by extending the statement to bimodules as well as categories, then verifying that the universal properties of Hochschild homology also hold for $\bc_*(S^1; -)$. |
740 This theorem is established by extending the statement to bimodules as well as categories, |
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741 then verifying that the universal properties of Hochschild homology also hold for $\bc_*(S^1; -)$. |
683 |
742 |
684 \begin{thm}[Mapping spaces] |
743 \begin{thm}[Mapping spaces] |
685 \label{thm:map-recon} |
744 \label{thm:map-recon} |
686 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
745 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
687 $B^n \to T$. |
746 $B^n \to T$. |
734 We introduce yet another homotopy equivalent version of |
796 We introduce yet another homotopy equivalent version of |
735 the blob complex, $\cB\cT_*(X)$. |
797 the blob complex, $\cB\cT_*(X)$. |
736 Blob diagrams have a natural topology, which is ignored by $\bc_*(X)$. |
798 Blob diagrams have a natural topology, which is ignored by $\bc_*(X)$. |
737 In $\cB\cT_*(X)$ we take this topology into account, treating the blob diagrams as something |
799 In $\cB\cT_*(X)$ we take this topology into account, treating the blob diagrams as something |
738 analogous to a simplicial space (but with cone-product polyhedra replacing simplices). |
800 analogous to a simplicial space (but with cone-product polyhedra replacing simplices). |
739 More specifically, a generator of $\cB\cT_k(X)$ is an $i$-parameter family of $j$-blob diagrams, with $i+j=k$. An essential step in the proof of this equivalence is a result to the effect that a $k$-parameter family of homeomorphism can be localized to at most $k$ small sets. |
801 More specifically, a generator of $\cB\cT_k(X)$ is an $i$-parameter family of $j$-blob diagrams, with $i+j=k$. |
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802 An essential step in the proof of this equivalence is a result to the effect that a $k$-parameter |
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803 family of homeomorphism can be localized to at most $k$ small sets. |
740 |
804 |
741 With this alternate version in hand, it is straightforward to prove the theorem. |
805 With this alternate version in hand, it is straightforward to prove the theorem. |
742 The evaluation map $\Homeo(X)\times BD_j(X)\to BD_j(X)$ |
806 The evaluation map $\Homeo(X)\times BD_j(X)\to BD_j(X)$ |
743 induces a chain map $\CH{X}\tensor C_*(BD_j(X))\to C_*(BD_j(X))$ |
807 induces a chain map $\CH{X}\tensor C_*(BD_j(X))\to C_*(BD_j(X))$ |
744 and hence a map $e_X: \CH{X} \tensor \cB\cT_*(X) \to \cB\cT_*(X)$. |
808 and hence a map $e_X: \CH{X} \tensor \cB\cT_*(X) \to \cB\cT_*(X)$. |