135 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
135 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
136 \begin{article} |
136 \begin{article} |
137 |
137 |
138 \begin{abstract} |
138 \begin{abstract} |
139 We explain the need for new axioms for topological quantum field theories that include ideas from derived |
139 We explain the need for new axioms for topological quantum field theories that include ideas from derived |
140 categories and homotopy theory. We summarize our axioms for higher categories, and describe the `blob complex'. |
140 categories and homotopy theory. We summarize our axioms for higher categories, and describe the ``blob complex". |
141 Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$. |
141 Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$. |
142 The $0$-th homology of this chain complex recovers the usual TQFT invariants of $W$. |
142 The $0$-th homology of this chain complex recovers the usual TQFT invariants of $W$. |
143 The higher homology groups should be viewed as generalizations of Hochschild homology. |
143 The higher homology groups should be viewed as generalizations of Hochschild homology. |
144 The blob complex has a very natural definition in terms of homotopy colimits along decompositions of the manifold $W$. |
144 The blob complex has a very natural definition in terms of homotopy colimits along decompositions of the manifold $W$. |
145 We outline the important properties of the blob complex, and sketch the proof of a generalization of |
145 We outline the important properties of the blob complex, and sketch the proof of a generalization of |
269 %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. |
270 %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. |
271 %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.} |
272 |
272 |
273 We will define two variations simultaneously, as all but one of the axioms are identical in the two cases. |
273 We will define two variations simultaneously, as all but one of the axioms are identical in the two cases. |
274 These variations are `isotopy $n$-categories', where homeomorphisms fixing the boundary |
274 These variations are ``isotopy $n$-categories", where homeomorphisms fixing the boundary |
275 act trivially on the sets associated to $n$-balls |
275 act trivially on the sets associated to $n$-balls |
276 (and these sets are usually vector spaces or more generally modules over a commutative ring) |
276 (and these sets are usually vector spaces or more generally modules over a commutative ring) |
277 and `$A_\infty$ $n$-categories', where there is a homotopy action of |
277 and ``$A_\infty$ $n$-categories", where there is a homotopy action of |
278 $k$-parameter families of homeomorphisms on these sets |
278 $k$-parameter families of homeomorphisms on these sets |
279 (which are usually chain complexes or topological spaces). |
279 (which are usually chain complexes or topological spaces). |
280 |
280 |
281 There are five basic ingredients |
281 There are five basic ingredients |
282 \cite{life-of-brian} of an $n$-category definition: |
282 \cite{life-of-brian} of an $n$-category definition: |
337 This means that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure |
337 This means that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure |
338 of an object of $\cS$, and all of the structure maps of the category (above and below) are |
338 of an object of $\cS$, and all of the structure maps of the category (above and below) are |
339 compatible with the $\cS$ structure on $\cC_n(X; c)$. |
339 compatible with the $\cS$ structure on $\cC_n(X; c)$. |
340 |
340 |
341 |
341 |
342 Given two hemispheres (a `domain' and `range') that agree on the equator, we need to be able to |
342 Given two hemispheres (a ``domain" and ``range") that agree on the equator, we need to be able to |
343 assemble them into a boundary value of the entire sphere. |
343 assemble them into a boundary value of the entire sphere. |
344 |
344 |
345 \begin{lem} |
345 \begin{lem} |
346 \label{lem:domain-and-range} |
346 \label{lem:domain-and-range} |
347 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
347 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
383 The gluing maps above are strictly associative. |
383 The gluing maps above are strictly associative. |
384 Given any decomposition of a ball $B$ into smaller balls |
384 Given any decomposition of a ball $B$ into smaller balls |
385 $$\bigsqcup B_i \to B,$$ |
385 $$\bigsqcup B_i \to B,$$ |
386 any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result. |
386 any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result. |
387 \end{axiom} |
387 \end{axiom} |
388 This axiom is only reasonable because the definition assigns a set to every ball; any identifications would limit the extent to which we can demand associativity. |
388 This axiom is only reasonable because the definition assigns a set to every ball; |
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389 any identifications would limit the extent to which we can demand associativity. |
389 For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices. |
390 For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices. |
390 \begin{axiom}[Product (identity) morphisms] |
391 \begin{axiom}[Product (identity) morphisms] |
391 \label{axiom:product} |
392 \label{axiom:product} |
392 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
393 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
393 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
394 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
498 For $X$ an $n$-ball define $\pi_{\le n}^\infty(T)(X)$ to be the space of all maps from $X$ to $T$ |
499 For $X$ an $n$-ball define $\pi_{\le n}^\infty(T)(X)$ to be the space of all maps from $X$ to $T$ |
499 (if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes). |
500 (if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes). |
500 |
501 |
501 |
502 |
502 \subsection{Example (string diagrams)} |
503 \subsection{Example (string diagrams)} |
503 Fix a `traditional' $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category). |
504 Fix a ``traditional" $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category). |
504 Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$; |
505 Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$; |
505 that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
506 that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
506 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
507 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
507 Boundary restrictions and gluing are again straightforward to define. |
508 Boundary restrictions and gluing are again straightforward to define. |
508 Define product morphisms via product cell decompositions. |
509 Define product morphisms via product cell decompositions. |
560 %\label{eq:psi-C} |
561 %\label{eq:psi-C} |
561 \psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl |
562 \psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl |
562 \end{equation*} |
563 \end{equation*} |
563 where the restrictions to the various pieces of shared boundaries amongst the balls |
564 where the restrictions to the various pieces of shared boundaries amongst the balls |
564 $X_a$ all agree (this is a fibered product of all the labels of $k$-balls over the labels of $k-1$-manifolds). |
565 $X_a$ all agree (this is a fibered product of all the labels of $k$-balls over the labels of $k-1$-manifolds). |
565 When $k=n$, the `subset' and `product' in the above formula should be |
566 When $k=n$, the ``subset" and ``product" in the above formula should be |
566 interpreted in the appropriate enriching category. |
567 interpreted in the appropriate enriching category. |
567 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 $\cC$. |
568 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 $\cC$. |
568 \end{defn} |
569 \end{defn} |
569 |
570 |
570 We will use the term `field on $W$' to refer to a point of this functor, |
571 We will use the term ``field on $W$" to refer to a point of this functor, |
571 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
572 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
572 |
573 |
573 |
574 |
574 \subsubsection{Colimits} |
575 \subsubsection{Colimits} |
575 Recall that our definition of an $n$-category is essentially a collection of functors |
576 Recall that our definition of an $n$-category is essentially a collection of functors |
605 homotopy colimit via the cone-product polyhedra in $\cell(W)$. |
606 homotopy colimit via the cone-product polyhedra in $\cell(W)$. |
606 A cone-product polyhedra is obtained from a point by successively taking the cone or taking the |
607 A cone-product polyhedra is obtained from a point by successively taking the cone or taking the |
607 product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, |
608 product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, |
608 cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, |
609 cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, |
609 and taking product identifies the roots of several trees. |
610 and taking product identifies the roots of several trees. |
610 The `local homotopy colimit' is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. |
611 The ``local homotopy colimit" is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. |
611 The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required. |
612 The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required. |
612 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. In the local homotopy colimit we can further require that any composition of morphisms in a directed tree is not expressible as a product. |
613 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
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614 In the local homotopy colimit we can further require that any composition of morphisms in a directed tree is not expressible as a product. |
613 |
615 |
614 %When $\cC$ is a topological $n$-category, |
616 %When $\cC$ is a topological $n$-category, |
615 %the flexibility available in the construction of a homotopy colimit allows |
617 %the flexibility available in the construction of a homotopy colimit allows |
616 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
618 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
617 %\todo{either need to explain why this is the same, or significantly rewrite this section} |
619 %\todo{either need to explain why this is the same, or significantly rewrite this section} |
636 \begin{itemize} |
638 \begin{itemize} |
637 \item a permissible collection of $k$ embedded balls, and |
639 \item a permissible collection of $k$ embedded balls, and |
638 \item for each resulting piece of $W$, a field, |
640 \item for each resulting piece of $W$, a field, |
639 \end{itemize} |
641 \end{itemize} |
640 such that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. |
642 such that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. |
641 We call such a field a `null field on $B$'. |
643 We call such a field a ``null field on $B$". |
642 |
644 |
643 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs. |
645 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs. |
644 |
646 |
645 We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$. |
647 We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$. |
646 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. |
648 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. |
806 More specifically, a generator of $\cB\cT_k(X)$ is an $i$-parameter family of $j$-blob diagrams, with $i+j=k$. |
808 More specifically, a generator of $\cB\cT_k(X)$ is an $i$-parameter family of $j$-blob diagrams, with $i+j=k$. |
807 An essential step in the proof of this equivalence is a result to the effect that a $k$-parameter |
809 An essential step in the proof of this equivalence is a result to the effect that a $k$-parameter |
808 family of homeomorphisms can be localized to at most $k$ small sets. |
810 family of homeomorphisms can be localized to at most $k$ small sets. |
809 |
811 |
810 With this alternate version in hand, the theorem is straightforward. |
812 With this alternate version in hand, the theorem is straightforward. |
811 By functoriality (Property \cite{property:functoriality}) $Homeo(X)$ acts on the set $BD_j(X)$ of $j$-blob diagrams, and this |
813 By functoriality (Property \ref{property:functoriality}) $\Homeo(X)$ acts on the set $BD_j(X)$ of $j$-blob diagrams, and this |
812 induces a chain map $\CH{X}\tensor C_*(BD_j(X))\to C_*(BD_j(X))$ |
814 induces a chain map $\CH{X}\tensor C_*(BD_j(X))\to C_*(BD_j(X))$ |
813 and hence a map $e_X: \CH{X} \tensor \cB\cT_*(X) \to \cB\cT_*(X)$. |
815 and hence a map $e_X: \CH{X} \tensor \cB\cT_*(X) \to \cB\cT_*(X)$. |
814 It is easy to check that $e_X$ thus defined has the desired properties. |
816 It is easy to check that $e_X$ thus defined has the desired properties. |
815 \end{proof} |
817 \end{proof} |
816 |
818 |