79 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
79 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-morphsims parameterized |
80 (Actually, this is only true in the oriented case, with 1-morphsims parameterized |
81 by oriented 1-balls.) |
81 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. |
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. |
83 |
83 |
84 Instead, we combine the domain and range into a single entity which we call the |
84 Instead, we will combine the domain and range into a single entity which we call the |
85 boundary of a morphism. |
85 boundary of a morphism. |
86 Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
86 Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
87 |
87 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
88 \begin{axiom}[Boundaries (spheres)] |
88 $1\le k \le n$. |
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89 At first might seem that we need another axiom for this, but in fact once we have |
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90 all the axioms in the subsection for $0$ through $k-1$ we can use a coend |
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91 construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
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92 to spheres (and any other manifolds): |
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93 |
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94 \begin{prop} |
89 \label{axiom:spheres} |
95 \label{axiom:spheres} |
90 For each $0 \le k \le n-1$, we have a functor $\cC_k$ from |
96 For each $1 \le k \le n$, we have a functor $\cC_{k-1}$ from |
91 the category of $k$-spheres and |
97 the category of $k{-}1$-spheres and |
92 homeomorphisms to the category of sets and bijections. |
98 homeomorphisms to the category of sets and bijections. |
93 \end{axiom} |
99 \end{prop} |
94 |
100 |
95 In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries, and again often omit the subscript. |
101 |
96 |
102 %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. |
97 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. |
103 |
98 |
104 \begin{axiom}[Boundaries]\label{nca-boundary} |
99 \begin{axiom}[Boundaries (maps)]\label{nca-boundary} |
105 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cC_{k-1}(\bd X)$. |
100 For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. |
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101 These maps, for various $X$, comprise a natural transformation of functors. |
106 These maps, for various $X$, comprise a natural transformation of functors. |
102 \end{axiom} |
107 \end{axiom} |
103 |
108 |
104 (Note that the first ``$\bd$" above is part of the data for the category, |
109 (Note that the first ``$\bd$" above is part of the data for the category, |
105 while the second is the ordinary boundary of manifolds.) |
110 while the second is the ordinary boundary of manifolds.) |
134 \medskip |
139 \medskip |
135 |
140 |
136 We have just argued that the boundary of a morphism has no preferred splitting into |
141 We have just argued that the boundary of a morphism has no preferred splitting into |
137 domain and range, but the converse meets with our approval. |
142 domain and range, but the converse meets with our approval. |
138 That is, given compatible domain and range, we should be able to combine them into |
143 That is, given compatible domain and range, we should be able to combine them into |
139 the full boundary of a morphism: |
144 the full boundary of a morphism. |
140 |
145 The following proposition follows from the coend construction used to define $\cC_{k-1}$ |
141 \begin{axiom}[Boundary from domain and range] |
146 on spheres. |
142 Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere $(0\le k\le n-1)$, |
147 |
143 $B_i$ is a $k$-ball, and $E = B_1\cap B_2$ is a $k{-}1$-sphere (Figure \ref{blah3}). |
148 \begin{prop}[Boundary from domain and range] |
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149 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
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150 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
144 Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the |
151 Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the |
145 two maps $\bd: \cC(B_i)\to \cC(E)$. |
152 two maps $\bd: \cC(B_i)\to \cC(E)$. |
146 Then we have an injective map |
153 Then we have an injective map |
147 \[ |
154 \[ |
148 \gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \into \cC(S) |
155 \gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \into \cC(S) |
149 \] |
156 \] |
150 which is natural with respect to the actions of homeomorphisms. |
157 which is natural with respect to the actions of homeomorphisms. |
151 \end{axiom} |
158 (When $k=1$ we stipulate that $\cC(E)$ is a point, so that the above fibered product |
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159 becomes a normal product.) |
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160 \end{prop} |
152 |
161 |
153 \begin{figure}[!ht] |
162 \begin{figure}[!ht] |
154 $$ |
163 $$ |
155 \begin{tikzpicture}[%every label/.style={green} |
164 \begin{tikzpicture}[%every label/.style={green} |
156 ] |
165 ] |
242 |
251 |
243 We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
252 We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
244 We will call elements of $\cC(B)_Y$ morphisms which are `splittable along $Y$' or `transverse to $Y$'. |
253 We will call elements of $\cC(B)_Y$ morphisms which are `splittable along $Y$' or `transverse to $Y$'. |
245 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
254 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
246 |
255 |
247 More generally, let $\alpha$ be a subdivision of a ball (or sphere) $X$ into smaller balls. |
256 More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls. |
248 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
257 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
249 the smaller balls to $X$. |
258 the smaller balls to $X$. |
250 We say that elements of $\cC(X)_\alpha$ are morphisms which are `splittable along $\alpha$'. |
259 We say that elements of $\cC(X)_\alpha$ are morphisms which are `splittable along $\alpha$'. |
251 In situations where the subdivision is notationally anonymous, we will write |
260 In situations where the subdivision is notationally anonymous, we will write |
252 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
261 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
488 |
497 |
489 \begin{example}[Maps to a space] |
498 \begin{example}[Maps to a space] |
490 \rm |
499 \rm |
491 \label{ex:maps-to-a-space}% |
500 \label{ex:maps-to-a-space}% |
492 Fix a `target space' $T$, any topological space. We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
501 Fix a `target space' $T$, any topological space. We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
493 For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
502 For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
494 all continuous maps from $X$ to $T$. |
503 all continuous maps from $X$ to $T$. |
495 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
504 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
496 homotopies fixed on $\bd X$. |
505 homotopies fixed on $\bd X$. |
497 (Note that homotopy invariance implies isotopy invariance.) |
506 (Note that homotopy invariance implies isotopy invariance.) |
498 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
507 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
523 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. |
532 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. |
524 \begin{example}[Traditional $n$-categories] |
533 \begin{example}[Traditional $n$-categories] |
525 \rm |
534 \rm |
526 \label{ex:traditional-n-categories} |
535 \label{ex:traditional-n-categories} |
527 Given a `traditional $n$-category with strong duality' $C$ |
536 Given a `traditional $n$-category with strong duality' $C$ |
528 define $\cC(X)$, for $X$ a $k$-ball or $k$-sphere with $k < n$, |
537 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
529 to be the set of all $C$-labeled sub cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
538 to be the set of all $C$-labeled sub cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
530 For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
539 For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
531 combinations of $C$-labeled sub cell complexes of $X$ |
540 combinations of $C$-labeled sub cell complexes of $X$ |
532 modulo the kernel of the evaluation map. |
541 modulo the kernel of the evaluation map. |
533 Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
542 Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
546 |
555 |
547 \newcommand{\Bord}{\operatorname{Bord}} |
556 \newcommand{\Bord}{\operatorname{Bord}} |
548 \begin{example}[The bordism $n$-category, plain version] |
557 \begin{example}[The bordism $n$-category, plain version] |
549 \rm |
558 \rm |
550 \label{ex:bordism-category} |
559 \label{ex:bordism-category} |
551 For a $k$-ball or $k$-sphere $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
560 For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
552 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
561 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
553 to $\bd X$. |
562 to $\bd X$. |
554 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such $n$-dimensional submanifolds; |
563 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such $n$-dimensional submanifolds; |
555 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
564 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
556 $W \to W'$ which restricts to the identity on the boundary. |
565 $W \to W'$ which restricts to the identity on the boundary. |
571 |
580 |
572 \begin{example}[Chains of maps to a space] |
581 \begin{example}[Chains of maps to a space] |
573 \rm |
582 \rm |
574 \label{ex:chains-of-maps-to-a-space} |
583 \label{ex:chains-of-maps-to-a-space} |
575 We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$. |
584 We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$. |
576 For $k$-balls and $k$-spheres $X$, with $k < n$, the sets $\pi^\infty_{\leq n}(T)(X)$ are just $\Maps(X \to T)$. |
585 For a $k$-ball $X$, with $k < n$, the set $\pi^\infty_{\leq n}(T)(X)$ is just $\Maps(X \to T)$. |
577 Define $\pi^\infty_{\leq n}(T)(X; c)$ for an $n$-ball $X$ and $c \in \pi^\infty_{\leq n}(T)(\bdy X)$ to be the chain complex |
586 Define $\pi^\infty_{\leq n}(T)(X; c)$ for an $n$-ball $X$ and $c \in \pi^\infty_{\leq n}(T)(\bdy X)$ to be the chain complex |
578 $$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
587 $$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
579 and $C_*$ denotes singular chains. |
588 and $C_*$ denotes singular chains. |
580 \nn{maybe should also mention version where we enrich over spaces rather than chain complexes} |
589 \nn{maybe should also mention version where we enrich over spaces rather than chain complexes} |
581 \end{example} |
590 \end{example} |
585 \begin{example}[Blob complexes of balls (with a fiber)] |
594 \begin{example}[Blob complexes of balls (with a fiber)] |
586 \rm |
595 \rm |
587 \label{ex:blob-complexes-of-balls} |
596 \label{ex:blob-complexes-of-balls} |
588 Fix an $n-k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
597 Fix an $n-k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
589 We will define an $A_\infty$ $k$-category $\cC$. |
598 We will define an $A_\infty$ $k$-category $\cC$. |
590 When $X$ is a $m$-ball or $m$-sphere, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
599 When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
591 When $X$ is an $k$-ball, |
600 When $X$ is an $k$-ball, |
592 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
601 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
593 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
602 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
594 \end{example} |
603 \end{example} |
595 |
604 |
613 (By shrining the little balls, we see that both are homotopic to the space of $k$ framed points |
622 (By shrining the little balls, we see that both are homotopic to the space of $k$ framed points |
614 in $B^n$.) |
623 in $B^n$.) |
615 |
624 |
616 Let $A$ be an $\cE\cB_n$-algebra. |
625 Let $A$ be an $\cE\cB_n$-algebra. |
617 We will define an $A_\infty$ $n$-category $\cC^A$. |
626 We will define an $A_\infty$ $n$-category $\cC^A$. |
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627 \nn{...} |
618 \end{example} |
628 \end{example} |
619 |
629 |
620 |
630 |
621 |
631 |
622 |
632 |
623 |
633 |
624 |
634 |
625 \subsection{From $n$-categories to systems of fields} |
635 %\subsection{From $n$-categories to systems of fields} |
626 \label{ss:ncat_fields} |
636 \subsection{From balls to manifolds} |
627 In this section we describe how to extend an $n$-category as described above (of either the plain or $A_\infty$ variety) to a system of fields. That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension, from $k$-balls and $k$-spheres to arbitrary $k$-manifolds. |
637 \label{ss:ncat_fields} \label{ss:ncat-coend} |
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638 In this section we describe how to extend an $n$-category as described above (of either the plain or $A_\infty$ variety) to a system of fields. |
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639 That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension |
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640 from $k$-balls to arbitrary $k$-manifolds. |
628 In the case of plain $n$-categories, this is just the usual construction of a TQFT |
641 In the case of plain $n$-categories, this is just the usual construction of a TQFT |
629 from an $n$-category. |
642 from an $n$-category. |
630 For $A_\infty$ $n$-categories, this gives an alternate (and |
643 For $A_\infty$ $n$-categories, this gives an alternate (and |
631 somewhat more canonical/tautological) construction of the blob complex. |
644 somewhat more canonical/tautological) construction of the blob complex. |
632 \nn{though from this point of view it seems more natural to just add some |
645 \nn{though from this point of view it seems more natural to just add some |
750 \nn{need to finish explaining why we have a system of fields; |
763 \nn{need to finish explaining why we have a system of fields; |
751 need to say more about ``homological" fields? |
764 need to say more about ``homological" fields? |
752 (actions of homeomorphisms); |
765 (actions of homeomorphisms); |
753 define $k$-cat $\cC(\cdot\times W)$} |
766 define $k$-cat $\cC(\cdot\times W)$} |
754 |
767 |
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768 \nn{need to revise stuff below, since we no longer have the sphere axiom} |
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769 |
755 Recall that Axiom \ref{axiom:spheres} for an $n$-category provided functors $\cC$ from $k$-spheres to sets for $0 \leq k < n$. We claim now that these functors automatically agree with the colimits we have associated to spheres in this section. \todo{} \todo{In fact, we probably should do this for balls as well!} For the remainder of this section we will write $\underrightarrow{\cC}(W)$ for the colimit associated to an arbitary manifold $W$, to distinguish it, in the case that $W$ is a ball or a sphere, from $\cC(W)$, which is part of the definition of the $n$-category. After the next three lemmas, there will be no further need for this notational distinction. |
770 Recall that Axiom \ref{axiom:spheres} for an $n$-category provided functors $\cC$ from $k$-spheres to sets for $0 \leq k < n$. We claim now that these functors automatically agree with the colimits we have associated to spheres in this section. \todo{} \todo{In fact, we probably should do this for balls as well!} For the remainder of this section we will write $\underrightarrow{\cC}(W)$ for the colimit associated to an arbitary manifold $W$, to distinguish it, in the case that $W$ is a ball or a sphere, from $\cC(W)$, which is part of the definition of the $n$-category. After the next three lemmas, there will be no further need for this notational distinction. |
756 |
771 |
757 \begin{lem} |
772 \begin{lem} |
758 For a $k$-ball or $k$-sphere $W$, with $0\leq k < n$, $$\underrightarrow{\cC}(W) = \cC(W).$$ |
773 For a $k$-ball or $k$-sphere $W$, with $0\leq k < n$, $$\underrightarrow{\cC}(W) = \cC(W).$$ |
759 \end{lem} |
774 \end{lem} |
771 |
786 |
772 Next we define plain and $A_\infty$ $n$-category modules. |
787 Next we define plain and $A_\infty$ $n$-category modules. |
773 The definition will be very similar to that of $n$-categories, |
788 The definition will be very similar to that of $n$-categories, |
774 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
789 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
775 \nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
790 \nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
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791 \nn{in particular, need to to get rid of the ``hemisphere axiom"} |
776 %\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
792 %\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
777 |
793 |
778 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
794 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
779 in the context of an $m{+}1$-dimensional TQFT. |
795 in the context of an $m{+}1$-dimensional TQFT. |
780 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
796 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |