22 For examples of a more purely algebraic origin, one would typically need the combinatorial |
22 For examples of a more purely algebraic origin, one would typically need the combinatorial |
23 results that we have avoided here. |
23 results that we have avoided here. |
24 |
24 |
25 \medskip |
25 \medskip |
26 |
26 |
27 There are many existing definitions of $n$-categories, with various intended uses. In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. Generally, these sets are indexed by instances of a certain typical shape. |
27 There are many existing definitions of $n$-categories, with various intended uses. |
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28 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
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29 Generally, these sets are indexed by instances of a certain typical shape. |
28 Some $n$-category definitions model $k$-morphisms on the standard bihedrons (interval, bigon, and so on). |
30 Some $n$-category definitions model $k$-morphisms on the standard bihedrons (interval, bigon, and so on). |
29 Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
31 Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
30 a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
32 a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
31 and so on. |
33 and so on. |
32 (This allows for strict associativity.) |
34 (This allows for strict associativity.) |
33 Still other definitions (see, for example, \cite{MR2094071}) |
35 Still other definitions (see, for example, \cite{MR2094071}) |
34 model the $k$-morphisms on more complicated combinatorial polyhedra. |
36 model the $k$-morphisms on more complicated combinatorial polyhedra. |
35 |
37 |
36 For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball. Thus we expect to associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
38 For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball. |
37 to the standard $k$-ball. By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
39 Thus we expect to associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
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40 to the standard $k$-ball. |
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41 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
38 standard $k$-ball. |
42 standard $k$-ball. |
39 We {\it do not} assume that it is equipped with a |
43 We {\it do not} assume that it is equipped with a |
40 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
44 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
41 |
45 |
42 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
46 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
77 of morphisms). |
81 of morphisms). |
78 The 0-sphere is unusual among spheres in that it is disconnected. |
82 The 0-sphere is unusual among spheres in that it is disconnected. |
79 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
83 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
80 (Actually, this is only true in the oriented case, with 1-morphisms parameterized |
84 (Actually, this is only true in the oriented case, with 1-morphisms parameterized |
81 by oriented 1-balls.) |
85 by oriented 1-balls.) |
82 For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. We prefer to not make the distinction in the first place. |
86 For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. |
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87 For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. |
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88 (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. |
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89 We prefer to not make the distinction in the first place. |
83 |
90 |
84 Instead, we will combine the domain and range into a single entity which we call the |
91 Instead, we will combine the domain and range into a single entity which we call the |
85 boundary of a morphism. |
92 boundary of a morphism. |
86 Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
93 Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
87 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
94 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
96 For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
103 For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
97 the category of $k{-}1$-spheres and |
104 the category of $k{-}1$-spheres and |
98 homeomorphisms to the category of sets and bijections. |
105 homeomorphisms to the category of sets and bijections. |
99 \end{lem} |
106 \end{lem} |
100 |
107 |
101 We postpone the proof \todo{} of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in the other Axioms at lower levels. |
108 We postpone the proof \todo{} of this result until after we've actually given all the axioms. |
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109 Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, |
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110 along with the data described in the other Axioms at lower levels. |
102 |
111 |
103 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
112 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
104 |
113 |
105 \begin{axiom}[Boundaries]\label{nca-boundary} |
114 \begin{axiom}[Boundaries]\label{nca-boundary} |
106 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
115 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
282 \caption{Operad composition and associativity}\label{blah7}\end{figure} |
291 \caption{Operad composition and associativity}\label{blah7}\end{figure} |
283 |
292 |
284 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
293 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
285 |
294 |
286 \begin{axiom}[Product (identity) morphisms] |
295 \begin{axiom}[Product (identity) morphisms] |
287 For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. These maps must satisfy the following conditions. |
296 For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, |
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297 usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
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298 These maps must satisfy the following conditions. |
288 \begin{enumerate} |
299 \begin{enumerate} |
289 \item |
300 \item |
290 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
301 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
291 \[ \xymatrix{ |
302 \[ \xymatrix{ |
292 X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
303 X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
476 There are two essential differences. |
487 There are two essential differences. |
477 First, for the $n$-category definition we restrict our attention to balls |
488 First, for the $n$-category definition we restrict our attention to balls |
478 (and their boundaries), while for fields we consider all manifolds. |
489 (and their boundaries), while for fields we consider all manifolds. |
479 Second, in category definition we directly impose isotopy |
490 Second, in category definition we directly impose isotopy |
480 invariance in dimension $n$, while in the fields definition we have do not expect isotopy invariance on fields |
491 invariance in dimension $n$, while in the fields definition we have do not expect isotopy invariance on fields |
481 but instead remember a subspace of local relations which contain differences of isotopic fields. (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
492 but instead remember a subspace of local relations which contain differences of isotopic fields. |
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493 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
482 Thus a system of fields and local relations $(\cF,\cU)$ determines an $n$-category $\cC_ {\cF,\cU}$ simply by restricting our attention to |
494 Thus a system of fields and local relations $(\cF,\cU)$ determines an $n$-category $\cC_ {\cF,\cU}$ simply by restricting our attention to |
483 balls and, at level $n$, quotienting out by the local relations: |
495 balls and, at level $n$, quotienting out by the local relations: |
484 \begin{align*} |
496 \begin{align*} |
485 \cC_{\cF,\cU}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / \cU(B) & \text{when $k=n$.}\end{cases} |
497 \cC_{\cF,\cU}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / \cU(B) & \text{when $k=n$.}\end{cases} |
486 \end{align*} |
498 \end{align*} |
495 We now describe several classes of examples of $n$-categories satisfying our axioms. |
507 We now describe several classes of examples of $n$-categories satisfying our axioms. |
496 |
508 |
497 \begin{example}[Maps to a space] |
509 \begin{example}[Maps to a space] |
498 \rm |
510 \rm |
499 \label{ex:maps-to-a-space}% |
511 \label{ex:maps-to-a-space}% |
500 Fix a `target space' $T$, any topological space. We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
512 Fix a `target space' $T$, any topological space. |
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513 We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
501 For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
514 For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
502 all continuous maps from $X$ to $T$. |
515 all continuous maps from $X$ to $T$. |
503 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
516 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
504 homotopies fixed on $\bd X$. |
517 homotopies fixed on $\bd X$. |
505 (Note that homotopy invariance implies isotopy invariance.) |
518 (Note that homotopy invariance implies isotopy invariance.) |
506 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
519 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
507 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
520 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
508 |
521 |
509 Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above. Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. |
522 Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above. |
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523 Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. |
510 \end{example} |
524 \end{example} |
511 |
525 |
512 \begin{example}[Maps to a space, with a fiber] |
526 \begin{example}[Maps to a space, with a fiber] |
513 \rm |
527 \rm |
514 \label{ex:maps-to-a-space-with-a-fiber}% |
528 \label{ex:maps-to-a-space-with-a-fiber}% |
515 We can modify the example above, by fixing a |
529 We can modify the example above, by fixing a |
516 closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. Taking $F$ to be a point recovers the previous case. |
530 closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, |
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531 otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. |
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532 Taking $F$ to be a point recovers the previous case. |
517 \end{example} |
533 \end{example} |
518 |
534 |
519 \begin{example}[Linearized, twisted, maps to a space] |
535 \begin{example}[Linearized, twisted, maps to a space] |
520 \rm |
536 \rm |
521 \label{ex:linearized-maps-to-a-space}% |
537 \label{ex:linearized-maps-to-a-space}% |
528 modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
544 modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
529 $h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
545 $h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
530 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
546 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
531 \end{example} |
547 \end{example} |
532 |
548 |
533 The next example is only intended to be illustrative, as we don't specify which definition of a `traditional $n$-category' we intend. Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here. |
549 The next example is only intended to be illustrative, as we don't specify which definition of a `traditional $n$-category' we intend. |
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550 Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here. |
534 \begin{example}[Traditional $n$-categories] |
551 \begin{example}[Traditional $n$-categories] |
535 \rm |
552 \rm |
536 \label{ex:traditional-n-categories} |
553 \label{ex:traditional-n-categories} |
537 Given a `traditional $n$-category with strong duality' $C$ |
554 Given a `traditional $n$-category with strong duality' $C$ |
538 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
555 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
548 Define $\cC(X; c)$, for $X$ an $n$-ball, |
565 Define $\cC(X; c)$, for $X$ an $n$-ball, |
549 to be the dual Hilbert space $A(X\times F; c)$. |
566 to be the dual Hilbert space $A(X\times F; c)$. |
550 \nn{refer elsewhere for details?} |
567 \nn{refer elsewhere for details?} |
551 |
568 |
552 |
569 |
553 Recall we described a system of fields and local relations based on a `traditional $n$-category' $C$ in Example \ref{ex:traditional-n-categories(fields)} above. Constructing a system of fields from $\cC$ recovers that example. \todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} |
570 Recall we described a system of fields and local relations based on a `traditional $n$-category' |
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571 $C$ in Example \ref{ex:traditional-n-categories(fields)} above. |
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572 Constructing a system of fields from $\cC$ recovers that example. |
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573 \todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} |
554 \end{example} |
574 \end{example} |
555 |
575 |
556 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
576 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
557 |
577 |
558 \nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
578 \nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
591 $$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
611 $$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
592 and $C_*$ denotes singular chains. |
612 and $C_*$ denotes singular chains. |
593 \nn{maybe should also mention version where we enrich over spaces rather than chain complexes} |
613 \nn{maybe should also mention version where we enrich over spaces rather than chain complexes} |
594 \end{example} |
614 \end{example} |
595 |
615 |
596 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
616 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
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617 homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
597 |
618 |
598 \begin{example}[Blob complexes of balls (with a fiber)] |
619 \begin{example}[Blob complexes of balls (with a fiber)] |
599 \rm |
620 \rm |
600 \label{ex:blob-complexes-of-balls} |
621 \label{ex:blob-complexes-of-balls} |
601 Fix an $n-k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
622 Fix an $n-k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
604 When $X$ is an $k$-ball, |
625 When $X$ is an $k$-ball, |
605 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
626 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
606 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
627 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
607 \end{example} |
628 \end{example} |
608 |
629 |
609 This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. Notice that with $F$ a point, the above example is a construction turning a topological $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. We think of this as providing a `free resolution' of the topological $n$-category. \todo{Say more here!} In fact, there is also a trivial, but mostly uninteresting, way to do this: we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, and take $\CD{B}$ to act trivially. |
630 This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. |
610 |
631 Notice that with $F$ a point, the above example is a construction turning a topological |
611 Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. It's easy to see that with $n=0$, the corresponding system of fields is just linear combinations of connected components of $T$, and the local relations are trivial. There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
632 $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. |
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633 We think of this as providing a `free resolution' of the topological $n$-category. |
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634 \todo{Say more here!} |
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635 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
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636 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
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637 and take $\CD{B}$ to act trivially. |
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638 |
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639 Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
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640 It's easy to see that with $n=0$, the corresponding system of fields is just |
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641 linear combinations of connected components of $T$, and the local relations are trivial. |
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642 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
612 |
643 |
613 \begin{example}[The bordism $n$-category, $A_\infty$ version] |
644 \begin{example}[The bordism $n$-category, $A_\infty$ version] |
614 \rm |
645 \rm |
615 \label{ex:bordism-category-ainf} |
646 \label{ex:bordism-category-ainf} |
616 blah blah \nn{to do...} |
647 blah blah \nn{to do...} |
637 |
668 |
638 |
669 |
639 %\subsection{From $n$-categories to systems of fields} |
670 %\subsection{From $n$-categories to systems of fields} |
640 \subsection{From balls to manifolds} |
671 \subsection{From balls to manifolds} |
641 \label{ss:ncat_fields} \label{ss:ncat-coend} |
672 \label{ss:ncat_fields} \label{ss:ncat-coend} |
642 In this section we describe how to extend an $n$-category $\cC$ as described above (of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. This extension is a certain colimit, and we've chosen the notation to remind you of this. |
673 In this section we describe how to extend an $n$-category $\cC$ as described above |
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674 (of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
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675 This extension is a certain colimit, and we've chosen the notation to remind you of this. |
643 That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension |
676 That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension |
644 from $k$-balls to arbitrary $k$-manifolds. Recall that we've already anticipated this construction in the previous section, inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
677 from $k$-balls to arbitrary $k$-manifolds. |
645 In the case of plain $n$-categories, this construction factors into a construction of a system of fields and local relations, followed by the usual TQFT definition of a vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
678 Recall that we've already anticipated this construction in the previous section, |
646 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. Recall that we can take a plain $n$-category $\cC$ and pass to the `free resolution', an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls (recall Example \ref{ex:blob-complexes-of-balls} above). We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the same as the original blob complex for $M$ with coefficients in $\cC$. |
679 inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
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680 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
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681 In the case of plain $n$-categories, this construction factors into a construction of a |
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682 system of fields and local relations, followed by the usual TQFT definition of a |
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683 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
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684 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
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685 Recall that we can take a plain $n$-category $\cC$ and pass to the `free resolution', |
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686 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls (recall Example \ref{ex:blob-complexes-of-balls} above). |
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687 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
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688 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the same as the original blob complex for $M$ with coefficients in $\cC$. |
647 |
689 |
648 We will first define the `cell-decomposition' poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
690 We will first define the `cell-decomposition' poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
649 An $n$-category $\cC$ provides a functor from this poset to the category of sets, and we will define $\cC(W)$ as a suitable colimit (or homotopy colimit in the $A_\infty$ case) of this functor. |
691 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
650 We'll later give a more explicit description of this colimit. In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). |
692 and we will define $\cC(W)$ as a suitable colimit |
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693 (or homotopy colimit in the $A_\infty$ case) of this functor. |
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694 We'll later give a more explicit description of this colimit. |
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695 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), |
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696 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). |
651 |
697 |
652 \begin{defn} |
698 \begin{defn} |
653 Say that a `permissible decomposition' of $W$ is a cell decomposition |
699 Say that a `permissible decomposition' of $W$ is a cell decomposition |
654 \[ |
700 \[ |
655 W = \bigcup_a X_a , |
701 W = \bigcup_a X_a , |
657 where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
703 where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
658 |
704 |
659 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
705 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
660 of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$. |
706 of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$. |
661 |
707 |
662 The category $\cell(W)$ has objects the permissible decompositions of $W$, and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
708 The category $\cell(W)$ has objects the permissible decompositions of $W$, |
|
709 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
663 See Figure \ref{partofJfig} for an example. |
710 See Figure \ref{partofJfig} for an example. |
664 \end{defn} |
711 \end{defn} |
665 |
712 |
666 \begin{figure}[!ht] |
713 \begin{figure}[!ht] |
667 \begin{equation*} |
714 \begin{equation*} |
693 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 $\cC$. |
740 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 $\cC$. |
694 \end{defn} |
741 \end{defn} |
695 |
742 |
696 When the $n$-category $\cC$ is enriched in some symmetric monoidal category $(A,\boxtimes)$, and $W$ is a |
743 When the $n$-category $\cC$ is enriched in some symmetric monoidal category $(A,\boxtimes)$, and $W$ is a |
697 closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and |
744 closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and |
698 we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$. (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.) |
745 we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$. |
|
746 (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.) |
699 Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we |
747 Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we |
700 fix a field on $\bd W$ |
748 fix a field on $\bd W$ |
701 (i.e. fix an element of the colimit associated to $\bd W$). |
749 (i.e. fix an element of the colimit associated to $\bd W$). |
702 |
750 |
703 Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
751 Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
708 $\psi_{\cC;W}(x)\to \cC(W)$, these maps are compatible with the refinement maps |
756 $\psi_{\cC;W}(x)\to \cC(W)$, these maps are compatible with the refinement maps |
709 above, and $\cC(W)$ is universal with respect to these properties. |
757 above, and $\cC(W)$ is universal with respect to these properties. |
710 \end{defn} |
758 \end{defn} |
711 |
759 |
712 \begin{defn}[System of fields functor, $A_\infty$ case] |
760 \begin{defn}[System of fields functor, $A_\infty$ case] |
713 When $\cC$ is an $A_\infty$ $n$-category, $\cC(W)$ for $W$ a $k$-manifold with $k < n$ is defined as above, as the colimit of $\psi_{\cC;W}$. When $W$ is an $n$-manifold, the chain complex $\cC(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
761 When $\cC$ is an $A_\infty$ $n$-category, $\cC(W)$ for $W$ a $k$-manifold with $k < n$ |
|
762 is defined as above, as the colimit of $\psi_{\cC;W}$. |
|
763 When $W$ is an $n$-manifold, the chain complex $\cC(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
714 \end{defn} |
764 \end{defn} |
715 |
765 |
716 We can specify boundary data $c \in \cC(\bdy W)$, and define functors $\psi_{\cC;W,c}$ with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
766 We can specify boundary data $c \in \cC(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
717 |
767 with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
718 We now give a more concrete description of the colimit in each case. If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold, we can take the vector space $\cC(W,c)$ to be the direct sum over all permissible decompositions of $W$ |
768 |
|
769 We now give a more concrete description of the colimit in each case. |
|
770 If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold, |
|
771 we can take the vector space $\cC(W,c)$ to be the direct sum over all permissible decompositions of $W$ |
719 \begin{equation*} |
772 \begin{equation*} |
720 \cC(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K |
773 \cC(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K |
721 \end{equation*} |
774 \end{equation*} |
722 where $K$ is the vector space spanned by elements $a - g(a)$, with |
775 where $K$ is the vector space spanned by elements $a - g(a)$, with |
723 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
776 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
730 Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
783 Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
731 Define $V$ as a vector space via |
784 Define $V$ as a vector space via |
732 \[ |
785 \[ |
733 V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
786 V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
734 \] |
787 \] |
735 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. (Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, the complex $U[m]$ is concentrated in degree $m$.) |
788 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. |
|
789 (Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, |
|
790 the complex $U[m]$ is concentrated in degree $m$.) |
736 We endow $V$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
791 We endow $V$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
737 summands plus another term using the differential of the simplicial set of $m$-sequences. |
792 summands plus another term using the differential of the simplicial set of $m$-sequences. |
738 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
793 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
739 summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
794 summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
740 \[ |
795 \[ |
750 We will call $m$ the filtration degree of the complex. |
805 We will call $m$ the filtration degree of the complex. |
751 We can think of this construction as starting with a disjoint copy of a complex for each |
806 We can think of this construction as starting with a disjoint copy of a complex for each |
752 permissible decomposition (filtration degree 0). |
807 permissible decomposition (filtration degree 0). |
753 Then we glue these together with mapping cylinders coming from gluing maps |
808 Then we glue these together with mapping cylinders coming from gluing maps |
754 (filtration degree 1). |
809 (filtration degree 1). |
755 Then we kill the extra homology we just introduced with mapping cylinders between the mapping cylinders (filtration degree 2), and so on. |
810 Then we kill the extra homology we just introduced with mapping |
|
811 cylinders between the mapping cylinders (filtration degree 2), and so on. |
756 |
812 |
757 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
813 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
758 |
814 |
759 It is easy to see that |
815 It is easy to see that |
760 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
816 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
779 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
835 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
780 This will be explained in more detail as we present the axioms. |
836 This will be explained in more detail as we present the axioms. |
781 |
837 |
782 \nn{should also develop $\pi_{\le n}(T, S)$ as a module for $\pi_{\le n}(T)$, where $S\sub T$.} |
838 \nn{should also develop $\pi_{\le n}(T, S)$ as a module for $\pi_{\le n}(T)$, where $S\sub T$.} |
783 |
839 |
784 Throughout, we fix an $n$-category $\cC$. For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. We state the final axiom, on actions of homeomorphisms, differently in the two cases. |
840 Throughout, we fix an $n$-category $\cC$. |
|
841 For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. |
|
842 We state the final axiom, on actions of homeomorphisms, differently in the two cases. |
785 |
843 |
786 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
844 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
787 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
845 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
788 We call $B$ the ball and $N$ the marking. |
846 We call $B$ the ball and $N$ the marking. |
789 A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
847 A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
817 \label{lem:hemispheres} |
875 \label{lem:hemispheres} |
818 {For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
876 {For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
819 the category of marked $k$-hemispheres and |
877 the category of marked $k$-hemispheres and |
820 homeomorphisms to the category of sets and bijections.} |
878 homeomorphisms to the category of sets and bijections.} |
821 \end{lem} |
879 \end{lem} |
822 The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. We use the same type of colimit construction. |
880 The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. |
|
881 We use the same type of colimit construction. |
823 |
882 |
824 In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$. |
883 In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$. |
825 |
884 |
826 \begin{module-axiom}[Module boundaries (maps)] |
885 \begin{module-axiom}[Module boundaries (maps)] |
827 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
886 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
1038 \begin{example}[Examples from TQFTs] |
1097 \begin{example}[Examples from TQFTs] |
1039 \todo{} |
1098 \todo{} |
1040 \end{example} |
1099 \end{example} |
1041 |
1100 |
1042 \begin{example} |
1101 \begin{example} |
1043 Suppose $S$ is a topological space, with a subspace $T$. We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all such maps modulo homotopies fixed on $\bdy B \setminus N$. This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains. |
1102 Suppose $S$ is a topological space, with a subspace $T$. |
|
1103 We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ |
|
1104 for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs |
|
1105 $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all |
|
1106 such maps modulo homotopies fixed on $\bdy B \setminus N$. |
|
1107 This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. |
|
1108 Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and |
|
1109 \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to |
|
1110 Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains. |
1044 \end{example} |
1111 \end{example} |
1045 |
1112 |
1046 \subsection{Modules as boundary labels (colimits for decorated manifolds)} |
1113 \subsection{Modules as boundary labels (colimits for decorated manifolds)} |
1047 \label{moddecss} |
1114 \label{moddecss} |
1048 |
1115 |
1049 Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. Let $W$ be a $k$-manifold ($k\le n$), |
1116 Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. |
|
1117 Let $W$ be a $k$-manifold ($k\le n$), |
1050 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
1118 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
1051 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
1119 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
1052 |
1120 |
1053 %Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
1121 %Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
1054 %and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
1122 %and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
1055 %component $\bd_i W$ of $W$. |
1123 %component $\bd_i W$ of $W$. |
1056 %(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.) |
1124 %(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.) |
1057 |
1125 |
1058 We will define a set $\cC(W, \cN)$ using a colimit construction similar to the one appearing in \S \ref{ss:ncat_fields} above. |
1126 We will define a set $\cC(W, \cN)$ using a colimit construction similar to |
|
1127 the one appearing in \S \ref{ss:ncat_fields} above. |
1059 (If $k = n$ and our $n$-categories are enriched, then |
1128 (If $k = n$ and our $n$-categories are enriched, then |
1060 $\cC(W, \cN)$ will have additional structure; see below.) |
1129 $\cC(W, \cN)$ will have additional structure; see below.) |
1061 |
1130 |
1062 Define a permissible decomposition of $W$ to be a decomposition |
1131 Define a permissible decomposition of $W$ to be a decomposition |
1063 \[ |
1132 \[ |
1068 with $M_{ib}\cap Y_i$ being the marking. |
1137 with $M_{ib}\cap Y_i$ being the marking. |
1069 (See Figure \ref{mblabel}.) |
1138 (See Figure \ref{mblabel}.) |
1070 \begin{figure}[!ht]\begin{equation*} |
1139 \begin{figure}[!ht]\begin{equation*} |
1071 \mathfig{.4}{ncat/mblabel} |
1140 \mathfig{.4}{ncat/mblabel} |
1072 \end{equation*}\caption{A permissible decomposition of a manifold |
1141 \end{equation*}\caption{A permissible decomposition of a manifold |
1073 whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure} |
1142 whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. |
|
1143 Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure} |
1074 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1144 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1075 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
1145 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
1076 This defines a partial ordering $\cell(W)$, which we will think of as a category. |
1146 This defines a partial ordering $\cell(W)$, which we will think of as a category. |
1077 (The objects of $\cell(D)$ are permissible decompositions of $W$, and there is a unique |
1147 (The objects of $\cell(D)$ are permissible decompositions of $W$, and there is a unique |
1078 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
1148 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
1094 (As usual, if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means |
1164 (As usual, if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means |
1095 homotopy colimit.) |
1165 homotopy colimit.) |
1096 |
1166 |
1097 If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
1167 If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
1098 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
1168 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
1099 $D\times Y_i \sub \bd(D\times W)$. It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ |
1169 $D\times Y_i \sub \bd(D\times W)$. |
|
1170 It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ |
1100 has the structure of an $n{-}k$-category, which we call $\cT(W, \cN)(D)$. |
1171 has the structure of an $n{-}k$-category, which we call $\cT(W, \cN)(D)$. |
1101 |
1172 |
1102 \medskip |
1173 \medskip |
1103 |
1174 |
1104 |
1175 |
1108 Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
1179 Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
1109 (If $k=1$ and our manifolds are oriented, then one should be |
1180 (If $k=1$ and our manifolds are oriented, then one should be |
1110 a left module and the other a right module.) |
1181 a left module and the other a right module.) |
1111 Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
1182 Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
1112 Define the tensor product $\cM_1 \tensor \cM_2$ to be the |
1183 Define the tensor product $\cM_1 \tensor \cM_2$ to be the |
1113 $n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$. This of course depends (functorially) |
1184 $n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$. |
|
1185 This of course depends (functorially) |
1114 on the choice of 1-ball $J$. |
1186 on the choice of 1-ball $J$. |
1115 |
1187 |
1116 We will define a more general self tensor product (categorified coend) below. |
1188 We will define a more general self tensor product (categorified coend) below. |
1117 |
1189 |
1118 %\nn{what about self tensor products /coends ?} |
1190 %\nn{what about self tensor products /coends ?} |
1130 In order to state and prove our version of the higher dimensional Deligne conjecture |
1202 In order to state and prove our version of the higher dimensional Deligne conjecture |
1131 (Section \ref{sec:deligne}), |
1203 (Section \ref{sec:deligne}), |
1132 we need to define morphisms of $A_\infty$ $1$-category modules and establish |
1204 we need to define morphisms of $A_\infty$ $1$-category modules and establish |
1133 some of their elementary properties. |
1205 some of their elementary properties. |
1134 |
1206 |
1135 To motivate the definitions which follow, consider algebras $A$ and $B$, right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction |
1207 To motivate the definitions which follow, consider algebras $A$ and $B$, |
|
1208 right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction |
1136 \begin{eqnarray*} |
1209 \begin{eqnarray*} |
1137 \hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\ |
1210 \hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\ |
1138 f &\mapsto& [x \mapsto f(x\ot -)] \\ |
1211 f &\mapsto& [x \mapsto f(x\ot -)] \\ |
1139 {}[x\ot y \mapsto g(x)(y)] & \mapsfrom & g . |
1212 {}[x\ot y \mapsto g(x)(y)] & \mapsfrom & g . |
1140 \end{eqnarray*} |
1213 \end{eqnarray*} |
1258 |
1331 |
1259 Abusing notation slightly, we will denote elements of the above space by $g$, with |
1332 Abusing notation slightly, we will denote elements of the above space by $g$, with |
1260 \[ |
1333 \[ |
1261 \olD\ot x \ot \cbar \mapsto g(\olD\ot x \ot \cbar) \in \cY(I_1\cup\cdots\cup I_{p-1}) . |
1334 \olD\ot x \ot \cbar \mapsto g(\olD\ot x \ot \cbar) \in \cY(I_1\cup\cdots\cup I_{p-1}) . |
1262 \] |
1335 \] |
1263 For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and $\cbar''$ corresponds to the subintervals |
1336 For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, |
|
1337 where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and |
|
1338 $\cbar''$ corresponds to the subintervals |
1264 which are dropped off the right side. |
1339 which are dropped off the right side. |
1265 (Either $\cbar'$ or $\cbar''$ might be empty.) |
1340 (Either $\cbar'$ or $\cbar''$ might be empty.) |
1266 \nn{surely $\cbar'$ can't be empy: we don't allow $D_1$ to be empty.} |
1341 \nn{surely $\cbar'$ can't be empy: we don't allow $D_1$ to be empty.} |
1267 Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, |
1342 Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, |
1268 we have |
1343 we have |
1387 \end{equation*} |
1462 \end{equation*} |
1388 \caption{0-marked 1-ball and 0-marked 2-ball} |
1463 \caption{0-marked 1-ball and 0-marked 2-ball} |
1389 \label{feb21a} |
1464 \label{feb21a} |
1390 \end{figure} |
1465 \end{figure} |
1391 |
1466 |
1392 The $0$-marked balls can be cut into smaller balls in various ways. We only consider those decompositions in which the smaller balls are either |
1467 The $0$-marked balls can be cut into smaller balls in various ways. |
1393 $0$-marked (i.e. intersect the $0$-marking of the large ball in a disc) or plain (don't intersect the $0$-marking of the large ball). |
1468 We only consider those decompositions in which the smaller balls are either |
|
1469 $0$-marked (i.e. intersect the $0$-marking of the large ball in a disc) |
|
1470 or plain (don't intersect the $0$-marking of the large ball). |
1394 We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere. |
1471 We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere. |
1395 |
1472 |
1396 Fix $n$-categories $\cA$ and $\cB$. |
1473 Fix $n$-categories $\cA$ and $\cB$. |
1397 These will label the two halves of a $0$-marked $k$-ball. |
1474 These will label the two halves of a $0$-marked $k$-ball. |
1398 The $0$-sphere module we define next will depend on $\cA$ and $\cB$ |
1475 The $0$-sphere module we define next will depend on $\cA$ and $\cB$ |