66 We call these ``disk-like $n$-categories'', to differentiate them from previous versions. |
66 We call these ``disk-like $n$-categories'', to differentiate them from previous versions. |
67 Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms. |
67 Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms. |
68 (See \S \ref{n-cat-names} below for a discussion of $n$-category terminology.) |
68 (See \S \ref{n-cat-names} below for a discussion of $n$-category terminology.) |
69 |
69 |
70 The basic idea is that each potential definition of an $n$-category makes a choice about the ``shape" of morphisms. |
70 The basic idea is that each potential definition of an $n$-category makes a choice about the ``shape" of morphisms. |
71 We try to be as lax as possible: a disk-like $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. |
71 We try to be as lax as possible: a disk-like $n$-category associates a |
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72 vector space to every $B$ homeomorphic to the $n$-ball. |
72 These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid. |
73 These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid. |
73 For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of |
74 For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to |
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75 each such $B$ and ask that the action of |
74 homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. |
76 homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. |
75 The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a |
77 The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: |
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78 the blob complexes of $n$-balls labelled by a |
76 disk-like $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
79 disk-like $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
77 |
80 |
78 In \S \ref{ssec:spherecat} we explain how $n$-categories can be viewed as objects in an $n{+}1$-category |
81 In \S \ref{ssec:spherecat} we explain how $n$-categories can be viewed as objects in an $n{+}1$-category |
79 of sphere modules. |
82 of sphere modules. |
80 When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwinors. |
83 When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwinors. |
148 The appendices prove technical results about $\CH{M}$ and |
151 The appendices prove technical results about $\CH{M}$ and |
149 make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
152 make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
150 as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
153 as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
151 %Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
154 %Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
152 %thought of as a disk-like $n$-category, in terms of the topology of $M$. |
155 %thought of as a disk-like $n$-category, in terms of the topology of $M$. |
153 |
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154 %%%% this is said later in the intro |
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155 %Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes) |
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156 %even when we could work in greater generality (symmetric monoidal categories, model categories, etc.). |
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157 |
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158 %\item ? one of the points we make (far) below is that there is not really much |
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159 %difference between (a) systems of fields and local relations and (b) $n$-cats; |
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160 %thus we tend to switch between talking in terms of one or the other |
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161 |
156 |
162 |
157 |
163 |
158 |
164 \subsection{Motivations} |
159 \subsection{Motivations} |
165 \label{sec:motivations} |
160 \label{sec:motivations} |
257 If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, |
252 If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, |
258 write $X_\text{gl} = X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
253 write $X_\text{gl} = X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
259 Note that this includes the case of gluing two disjoint manifolds together. |
254 Note that this includes the case of gluing two disjoint manifolds together. |
260 \begin{property}[Gluing map] |
255 \begin{property}[Gluing map] |
261 \label{property:gluing-map}% |
256 \label{property:gluing-map}% |
262 %If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map |
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263 %\begin{equation*} |
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264 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
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265 %\end{equation*} |
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266 Given a gluing $X \to X_\mathrm{gl}$, there is |
257 Given a gluing $X \to X_\mathrm{gl}$, there is |
267 a natural map |
258 a natural map |
268 \[ |
259 \[ |
269 \bc_*(X) \to \bc_*(X_\mathrm{gl}) |
260 \bc_*(X) \to \bc_*(X_\mathrm{gl}) |
270 \] |
261 \] |
370 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
361 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
371 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
362 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
372 for any homeomorphic pair $X$ and $Y$, |
363 for any homeomorphic pair $X$ and $Y$, |
373 satisfying corresponding conditions. |
364 satisfying corresponding conditions. |
374 |
365 |
375 In \S \ref{sec:ncats} we introduce the notion of disk-like $n$-categories, from which we can construct systems of fields. |
366 In \S \ref{sec:ncats} we introduce the notion of disk-like $n$-categories, |
376 Below, when we talk about the blob complex for a disk-like $n$-category, we are implicitly passing first to this associated system of fields. |
367 from which we can construct systems of fields. |
377 Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. In that section we describe how to use the blob complex to construct $A_\infty$ $n$-categories from ordinary $n$-categories: |
368 Below, when we talk about the blob complex for a disk-like $n$-category, |
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369 we are implicitly passing first to this associated system of fields. |
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370 Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
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371 In that section we describe how to use the blob complex to |
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372 construct $A_\infty$ $n$-categories from ordinary $n$-categories: |
378 |
373 |
379 \newtheorem*{ex:blob-complexes-of-balls}{Example \ref{ex:blob-complexes-of-balls}} |
374 \newtheorem*{ex:blob-complexes-of-balls}{Example \ref{ex:blob-complexes-of-balls}} |
380 |
375 |
381 \begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form an $A_\infty$ $n$-category] |
376 \begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form an $A_\infty$ $n$-category] |
382 %\label{thm:blobs-ainfty} |
377 %\label{thm:blobs-ainfty} |
388 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
383 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
389 These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
384 These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
390 Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
385 Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
391 \end{ex:blob-complexes-of-balls} |
386 \end{ex:blob-complexes-of-balls} |
392 \begin{rem} |
387 \begin{rem} |
393 Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from an ordinary $n$-category. |
388 Perhaps the most interesting case is when $Y$ is just a point; |
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389 then we have a way of building an $A_\infty$ $n$-category from an ordinary $n$-category. |
394 We think of this $A_\infty$ $n$-category as a free resolution. |
390 We think of this $A_\infty$ $n$-category as a free resolution. |
395 \end{rem} |
391 \end{rem} |
396 |
392 |
397 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
393 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
398 instead of an ordinary $n$-category; this is described in \S \ref{sec:ainfblob}. |
394 instead of an ordinary $n$-category; this is described in \S \ref{sec:ainfblob}. |
399 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. The next theorem describes the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example. |
395 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
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396 The next theorem describes the blob complex for product manifolds, |
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397 in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example. |
400 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
398 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
401 |
399 |
402 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
400 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
403 |
401 |
404 \begin{thm:product}[Product formula] |
402 \begin{thm:product}[Product formula] |
405 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
403 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
406 Let $\cC$ be an $n$-category. |
404 Let $\cC$ be an $n$-category. |
407 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Example \ref{ex:blob-complexes-of-balls}). |
405 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology |
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406 (see Example \ref{ex:blob-complexes-of-balls}). |
408 Then |
407 Then |
409 \[ |
408 \[ |
410 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
409 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
411 \] |
410 \] |
412 \end{thm:product} |
411 \end{thm:product} |
530 rotating $k$-morphisms correspond to all the ways of rotating $k$-balls. |
529 rotating $k$-morphisms correspond to all the ways of rotating $k$-balls. |
531 We sometimes refer to this as ``strong duality", and sometimes we consider it to be implied |
530 We sometimes refer to this as ``strong duality", and sometimes we consider it to be implied |
532 by ``disk-like". |
531 by ``disk-like". |
533 (But beware: disks can come in various flavors, and some of them, such as framed disks, |
532 (But beware: disks can come in various flavors, and some of them, such as framed disks, |
534 don't actually imply much duality.) |
533 don't actually imply much duality.) |
535 Another possibility considered here was ``pivotal $n$-category", but we prefer to preserve pivotal for its usual sense. It will thus be a theorem that our disk-like 2-categories are equivalent to pivotal 2-categories, c.f. \S \ref{ssec:2-cats}. |
534 Another possibility considered here was ``pivotal $n$-category", but we prefer to preserve pivotal for its usual sense. |
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535 It will thus be a theorem that our disk-like 2-categories |
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536 are equivalent to pivotal 2-categories, c.f. \S \ref{ssec:2-cats}. |
536 |
537 |
537 Finally, we need a general name for isomorphisms between balls, where the balls could be |
538 Finally, we need a general name for isomorphisms between balls, where the balls could be |
538 piecewise linear or smooth or topological or Spin or framed or etc., or some combination thereof. |
539 piecewise linear or smooth or topological or Spin or framed or etc., or some combination thereof. |
539 We have chosen to use ``homeomorphism" for the appropriate sort of isomorphism, so the reader should |
540 We have chosen to use ``homeomorphism" for the appropriate sort of isomorphism, so the reader should |
540 keep in mind that ``homeomorphism" could mean PL homeomorphism or diffeomorphism (and so on) |
541 keep in mind that ``homeomorphism" could mean PL homeomorphism or diffeomorphism (and so on) |
564 Alan Wilder, |
565 Alan Wilder, |
565 Dmitri Pavlov, |
566 Dmitri Pavlov, |
566 Ansgar Schneider, |
567 Ansgar Schneider, |
567 and |
568 and |
568 Dan Berwick-Evans. |
569 Dan Berwick-Evans. |
569 \nn{need to double-check this list once the reading course is over} |
570 During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at |
570 During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. We'd like to thank the Aspen Center for Physics for the pleasant and productive |
571 Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. |
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572 We'd like to thank the Aspen Center for Physics for the pleasant and productive |
571 environment provided there during the final preparation of this manuscript. |
573 environment provided there during the final preparation of this manuscript. |
572 |
574 |