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162 \nn{ |
162 \dropcap{T}opological quantum field theories (TQFTs) provide local invariants of manifolds, which are determined by the algebraic data of a higher category. |
163 background: TQFTs are important, historically, semisimple categories well-understood. |
163 |
164 Many new examples arising recently which do not fit this framework, e.g. SW and OS theory. |
164 An $n+1$-dimensional TQFT $\cA$ associates a vector space $\cA(M)$ |
165 These have more complicated gluing formulas (\cite{1003.0598,1005.1248}, etc); |
165 (or more generally, some object in a specified symmetric monoidal category) |
166 it would be nice to give generalized TQFT axioms that encompass these. |
166 to each $n$-dimensional manifold $M$, and a linear map |
167 Triangulated categories are important; often calculations are via exact sequences, |
167 $\cA(W): \cA(M_0) \to \cA(M_1)$ to each $n+1$-dimensional manifold $W$ |
168 and the standard TQFT constructions are quotients, which destroy exactness. |
168 with incoming boundary $M_0$ and outgoing boundary $M_1$. |
169 A first attempt to deal with this might be to replace all the tensor products in gluing formulas |
169 An $n+\epsilon$-dimensional TQFT provides slightly less; |
170 with derived tensor products (cite Kh?). |
170 it only assigns linear maps to mapping cylinders. |
171 However, in this approach it's probably difficult to prove invariance of constructions, |
171 |
172 because they depend on explicit presentations of the manifold. |
172 There is a standard formalism for constructing an $n+\epsilon$-dimensional |
173 We'll give a manifestly invariant construction, |
173 TQFT from any $n$-category with sufficiently strong duality, |
174 and deduce gluing formulas based on derived (actually, $A_\infty$) tensor products.} |
174 and with a further finiteness condition this TQFT is in fact $n+1$-dimensional. |
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175 \nn{not so standard, err} |
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176 |
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177 These invariants are local in the following sense. |
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178 The vector space $\cA(Y \times I)$, for $Y$ an $n-1$-manifold, |
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179 naturally has the structure of a category, with composition given by the gluing map |
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180 $I \sqcup I \to I$. Moreover, the vector space $\cA(Y \times I^k)$, |
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181 for $Y$ and $n-k$-manifold, has the structure of a $k$-category. |
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182 The original $n$-category can be recovered as $\cA(I^n)$. |
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183 For the rest of the paragraph, we implicitly drop the factors of $I$. |
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184 (So for example the original $n$-category is associated to the point.) |
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185 If $Y$ contains $Z$ as a codimension $0$ submanifold of its boundary, |
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186 then $\cA(Y)$ is natually a module over $\cA(Z)$. For any $k$-manifold |
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187 $Y = Y_1 \cup_Z Y_2$, where $Z$ is a $k-1$-manifold, the category |
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188 $\cA(Y)$ can be calculated via a gluing formula, |
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189 $$\cA(Y) = \cA(Y_1) \Tensor_{\cA(Z)} \cA(Y_2).$$ |
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190 |
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191 In fact, recent work of Lurie on the `cobordism hypothesis' \cite{0905.0465} |
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192 shows that all invariants of $n$-manifolds satisfying a certain related locality property |
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193 are in a sense TQFT invariants, and in particular determined by |
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194 a `fully dualizable object' in some $n+1$-category. |
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195 (The discussion above begins with an object in the $n+1$-category of $n$-categories. |
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196 The `sufficiently strong duality' mentioned above corresponds roughly to `fully dualizable'.) |
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197 |
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198 This formalism successfully captures Turaev-Viro and Reshetikhin-Turaev invariants |
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199 (and indeed invariants based on semisimple categories). |
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200 However new invariants on manifolds, particularly those coming from |
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201 Seiberg-Witten theory and Ozsv\'{a}th-Szab\'{o} theory, do not fit the framework well. |
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202 In particular, they have more complicated gluing formulas, involving derived or |
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203 $A_\infty$ tensor products \cite{1003.0598,1005.1248}. |
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204 It seems worthwhile to find a more general notion of TQFT that explain these. |
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205 While we don't claim to fulfill that goal here, our notions of $n$-category and |
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206 of the blob complex are hopefully a step in the right direction, |
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207 and provide similar gluing formulas. |
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208 |
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209 One approach to such a generalization might be simply to define a |
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210 TQFT invariant via its gluing formulas, replacing tensor products with |
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211 derived tensor products. However, it is probably difficult to prove |
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212 the invariance of such a definition, as the object associated to a manifold |
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213 will a priori depend on the explicit presentation used to apply the gluing formulas. |
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214 We instead give a manifestly invariant construction, and |
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215 deduce gluing formulas based on $A_\infty$ tensor products. |
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216 |
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217 \nn{Triangulated categories are important; often calculations are via exact sequences, |
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218 and the standard TQFT constructions are quotients, which destroy exactness.} |
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219 |
175 |
220 |
176 \section{Definitions} |
221 \section{Definitions} |
177 \subsection{$n$-categories} \mbox{} |
222 \subsection{$n$-categories} \mbox{} |
178 |
223 |
179 \nn{rough draft of n-cat stuff...} |
224 \nn{rough draft of n-cat stuff...} |
382 These action maps are required to be associative up to homotopy, |
427 These action maps are required to be associative up to homotopy, |
383 and also compatible with composition (gluing) in the sense that |
428 and also compatible with composition (gluing) in the sense that |
384 a diagram like the one in Theorem \ref{thm:CH} commutes. |
429 a diagram like the one in Theorem \ref{thm:CH} commutes. |
385 \end{axiom} |
430 \end{axiom} |
386 |
431 |
387 |
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388 \todo{ |
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389 Decide if we need a friendlier, skein-module version. |
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390 } |
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391 |
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392 \subsection{Example (the fundamental $n$-groupoid)} |
432 \subsection{Example (the fundamental $n$-groupoid)} |
393 We will define $\pi_{\le n}(T)$, the fundamental $n$-groupoid of a topological space $T$. |
433 We will define $\pi_{\le n}(T)$, the fundamental $n$-groupoid of a topological space $T$. |
394 When $X$ is a $k$-ball with $k<n$, define $\pi_{\le n}(T)(X)$ |
434 When $X$ is a $k$-ball with $k<n$, define $\pi_{\le n}(T)(X)$ |
395 to be the set of continuous maps from $X$ to $T$. |
435 to be the set of continuous maps from $X$ to $T$. |
396 When $X$ is an $n$-ball, define $\pi_{\le n}(T)(X)$ to be homotopy classes (rel boundary) of such maps. |
436 When $X$ is an $n$-ball, define $\pi_{\le n}(T)(X)$ to be homotopy classes (rel boundary) of such maps. |
465 where the restrictions to the various pieces of shared boundaries amongst the cells |
505 where the restrictions to the various pieces of shared boundaries amongst the cells |
466 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). When $k=n$, the `subset' and `product' in the above formula should be interpreted in the appropriate enriching category. |
506 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). When $k=n$, the `subset' and `product' in the above formula should be interpreted in the appropriate enriching category. |
467 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 $\cC$. |
507 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 $\cC$. |
468 \end{defn} |
508 \end{defn} |
469 |
509 |
470 We will use the term `field on $W$' to refer to \nn{a point} of this functor, |
510 We will use the term `field on $W$' to refer to a point of this functor, |
471 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
511 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
472 |
512 |
473 \todo{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?} |
513 \todo{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?} |
474 |
514 |
475 \subsubsection{Homotopy colimits} |
515 \subsubsection{Homotopy colimits} |
481 |
521 |
482 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 |
522 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 |
483 $$\bdy (\bar{x},a) = (\bar{x}, \bdy a) + (-1)^{\deg a} \left( (d_0 \bar{x}, g(a)) + \sum_{i=1}^m (-1)^i (d_i \bar{x}, a) \right)$$ |
523 $$\bdy (\bar{x},a) = (\bar{x}, \bdy a) + (-1)^{\deg a} \left( (d_0 \bar{x}, g(a)) + \sum_{i=1}^m (-1)^i (d_i \bar{x}, a) \right)$$ |
484 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 $\bar{x}$. |
524 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 $\bar{x}$. |
485 |
525 |
486 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 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 $x$ a cone-product polyhedron in $\cell(W)$. |
526 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 $x$ a cone-product polyhedron in $\cell(W)$. |
487 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
527 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
488 |
528 |
489 When $\cC$ is a topological $n$-category, |
529 When $\cC$ is a topological $n$-category, |
490 the flexibility available in the construction of a homotopy colimit allows |
530 the flexibility available in the construction of a homotopy colimit allows |
491 us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
531 us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. \todo{either need to explain why this is the same, or significantly rewrite this section} |
492 |
532 |
493 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
533 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
494 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
534 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
495 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. |
535 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. |
496 |
536 |
509 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. |
549 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. |
510 |
550 |
511 \section{Properties of the blob complex} |
551 \section{Properties of the blob complex} |
512 \subsection{Formal properties} |
552 \subsection{Formal properties} |
513 \label{sec:properties} |
553 \label{sec:properties} |
514 The blob complex enjoys the following list of formal properties. The first three properties are immediate from the definitions. |
554 The blob complex enjoys the following list of formal properties. The first three are immediate from the definitions. |
515 |
555 |
516 \begin{property}[Functoriality] |
556 \begin{property}[Functoriality] |
517 \label{property:functoriality}% |
557 \label{property:functoriality}% |
518 The blob complex is functorial with respect to homeomorphisms. |
558 The blob complex is functorial with respect to homeomorphisms. |
519 That is, |
559 That is, |