62 we find this situation unsatisfactory. |
62 we find this situation unsatisfactory. |
63 Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another |
63 Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another |
64 definition of an $n$-category, or rather a definition of an $n$-category with strong duality. |
64 definition of an $n$-category, or rather a definition of an $n$-category with strong duality. |
65 (Removing the duality conditions from our definition would make it more complicated rather than less.) |
65 (Removing the duality conditions from our definition would make it more complicated rather than less.) |
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 $A_\infty$ disk-like $n$-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 |
71 We try to be as lax as possible: a disk-like $n$-category associates a |
72 vector space to every $B$ homeomorphic to the $n$-ball. |
72 vector space to every $B$ homeomorphic to the $n$-ball. |
73 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. |
74 For an $A_\infty$ disk-like $n$-category, we associate a chain complex instead of a vector space to |
74 For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to |
75 each such $B$ and ask that the action of |
75 each such $B$ and ask that the action of |
76 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. |
77 The axioms for an $A_\infty$ disk-like $n$-category are designed to capture two main examples: |
77 The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: |
78 the blob complexes of $n$-balls labelled by a |
78 the blob complexes of $n$-balls labelled by a |
79 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 $T$. |
80 |
80 |
81 In \S \ref{ssec:spherecat} we explain how disk-like $n$-categories can be viewed as objects in a disk-like $n{+}1$-category |
81 In \S \ref{ssec:spherecat} we explain how $n$-categories can be viewed as objects in an $n{+}1$-category |
82 of sphere modules. |
82 of sphere modules. |
83 When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwiners. |
83 When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwiners. |
84 |
84 |
85 In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a disk-like $n$-category |
85 In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a disk-like $n$-category |
86 (using a colimit along certain decompositions of a manifold into balls). |
86 (using a colimit along certain decompositions of a manifold into balls). |
87 With this in hand, we write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ |
87 With this in hand, we write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ |
88 with the system of fields constructed from the disk-like $n$-category $\cC$. |
88 with the system of fields constructed from the $n$-category $\cC$. |
89 %\nn{KW: I don't think we use this notational convention any more, right?} |
89 %\nn{KW: I don't think we use this notational convention any more, right?} |
90 In \S \ref{sec:ainfblob} we give an alternative definition |
90 In \S \ref{sec:ainfblob} we give an alternative definition |
91 of the blob complex for an $A_\infty$ disk-like $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
91 of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
92 Using these definitions, we show how to use the blob complex to ``resolve" any ordinary disk-like $n$-category as an |
92 Using these definitions, we show how to use the blob complex to ``resolve" any ordinary $n$-category as an |
93 $A_\infty$ disk-like $n$-category, and relate the first and second definitions of the blob complex. |
93 $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. |
94 We use the blob complex for $A_\infty$ disk-like $n$-categories to establish important properties of the blob complex (in both variants), |
94 We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), |
95 in particular the ``gluing formula" of Theorem \ref{thm:gluing} below. |
95 in particular the ``gluing formula" of Theorem \ref{thm:gluing} below. |
96 |
96 |
97 The relationship between all these ideas is sketched in Figure \ref{fig:outline}. |
97 The relationship between all these ideas is sketched in Figure \ref{fig:outline}. |
98 |
98 |
99 % NB: the following tikz requires a *more recent* version of PGF than is distributed with MacTex 2010. |
99 % NB: the following tikz requires a *more recent* version of PGF than is distributed with MacTex 2010. |
153 Later sections address other topics. |
153 Later sections address other topics. |
154 Section \S \ref{sec:deligne} gives |
154 Section \S \ref{sec:deligne} gives |
155 a higher dimensional generalization of the Deligne conjecture |
155 a higher dimensional generalization of the Deligne conjecture |
156 (that the little discs operad acts on Hochschild cochains) in terms of the blob complex. |
156 (that the little discs operad acts on Hochschild cochains) in terms of the blob complex. |
157 The appendices prove technical results about $\CH{M}$ and |
157 The appendices prove technical results about $\CH{M}$ and |
158 make connections between our definitions of disk-like $n$-categories and familiar definitions for $n=1$ and $n=2$, |
158 make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
159 as well as relating the $n=1$ case of our $A_\infty$ disk-like $n$-categories with usual $A_\infty$ algebras. |
159 as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
160 %Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
160 %Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
161 %thought of as a disk-like $n$-category, in terms of the topology of $M$. |
161 %thought of as a disk-like $n$-category, in terms of the topology of $M$. |
162 |
162 |
163 |
163 |
164 |
164 |
370 |
370 |
371 In \S \ref{sec:ncats} we introduce the notion of disk-like $n$-categories, |
371 In \S \ref{sec:ncats} we introduce the notion of disk-like $n$-categories, |
372 from which we can construct systems of fields. |
372 from which we can construct systems of fields. |
373 Below, when we talk about the blob complex for a disk-like $n$-category, |
373 Below, when we talk about the blob complex for a disk-like $n$-category, |
374 we are implicitly passing first to this associated system of fields. |
374 we are implicitly passing first to this associated system of fields. |
375 Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ disk-like $n$-category. |
375 Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
376 In that section we describe how to use the blob complex to |
376 In that section we describe how to use the blob complex to |
377 construct $A_\infty$ disk-like $n$-categories from ordinary disk-like $n$-categories: |
377 construct $A_\infty$ $n$-categories from ordinary $n$-categories: |
378 |
378 |
379 \newtheorem*{ex:blob-complexes-of-balls}{Example \ref{ex:blob-complexes-of-balls}} |
379 \newtheorem*{ex:blob-complexes-of-balls}{Example \ref{ex:blob-complexes-of-balls}} |
380 |
380 |
381 \begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form an $A_\infty$ disk-like $n$-category] |
381 \begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form an $A_\infty$ $n$-category] |
382 %\label{thm:blobs-ainfty} |
382 %\label{thm:blobs-ainfty} |
383 Let $\cC$ be an ordinary disk-like $n$-category. |
383 Let $\cC$ be an ordinary $n$-category. |
384 Let $Y$ be an $n{-}k$-manifold. |
384 Let $Y$ be an $n{-}k$-manifold. |
385 There is an $A_\infty$ disk-like $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
385 There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
386 to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set |
386 to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set |
387 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
387 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
388 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
388 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
389 These sets have the structure of an $A_\infty$ disk-like $k$-category, with compositions coming from the gluing map in |
389 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}. |
390 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} |
391 \end{ex:blob-complexes-of-balls} |
392 \begin{rem} |
392 \begin{rem} |
393 Perhaps the most interesting case is when $Y$ is just a point; |
393 Perhaps the most interesting case is when $Y$ is just a point; |
394 then we have a way of building an $A_\infty$ disk-like $n$-category from an ordinary disk-like $n$-category. |
394 then we have a way of building an $A_\infty$ $n$-category from an ordinary $n$-category. |
395 We think of this $A_\infty$ disk-like $n$-category as a free resolution. |
395 We think of this $A_\infty$ $n$-category as a free resolution. |
396 \end{rem} |
396 \end{rem} |
397 |
397 |
398 There is a version of the blob complex for $\cC$ an $A_\infty$ disk-like $n$-category |
398 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
399 instead of an ordinary disk-like $n$-category; this is described in \S \ref{sec:ainfblob}. |
399 instead of an ordinary $n$-category; this is described in \S \ref{sec:ainfblob}. |
400 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
400 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
401 The next theorem describes the blob complex for product manifolds, |
401 The next theorem describes the blob complex for product manifolds, |
402 in terms of the $A_\infty$ blob complex of the $A_\infty$ disk-like $n$-categories constructed as in the previous example. |
402 in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example. |
403 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
403 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
404 |
404 |
405 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
405 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
406 |
406 |
407 \begin{thm:product}[Product formula] |
407 \begin{thm:product}[Product formula] |
408 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
408 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
409 Let $\cC$ be a disk-like $n$-category. |
409 Let $\cC$ be an $n$-category. |
410 Let $\bc_*(Y;\cC)$ be the $A_\infty$ disk-like $k$-category associated to $Y$ via blob homology |
410 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology |
411 (see Example \ref{ex:blob-complexes-of-balls}). |
411 (see Example \ref{ex:blob-complexes-of-balls}). |
412 Then |
412 Then |
413 \[ |
413 \[ |
414 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
414 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
415 \] |
415 \] |
417 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
417 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
418 (see \S \ref{ss:product-formula}). |
418 (see \S \ref{ss:product-formula}). |
419 |
419 |
420 Fix a disk-like $n$-category $\cC$, which we'll omit from the notation. |
420 Fix a disk-like $n$-category $\cC$, which we'll omit from the notation. |
421 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
421 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
422 (See Appendix \ref{sec:comparing-A-infty} for the translation between $A_\infty$ disk-like $1$-categories and the usual algebraic notion of an $A_\infty$ category.) |
422 (See Appendix \ref{sec:comparing-A-infty} for the translation between disk-like $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) |
423 |
423 |
424 \newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}} |
424 \newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}} |
425 |
425 |
426 \begin{thm:gluing}[Gluing formula] |
426 \begin{thm:gluing}[Gluing formula] |
427 \mbox{}% <-- gets the indenting right |
427 \mbox{}% <-- gets the indenting right |
509 The latter are closely related to $(\infty, n)$-categories (i.e.\ $\infty$-categories where all morphisms |
509 The latter are closely related to $(\infty, n)$-categories (i.e.\ $\infty$-categories where all morphisms |
510 of dimension greater than $n$ are invertible), but we don't want to use that name |
510 of dimension greater than $n$ are invertible), but we don't want to use that name |
511 since we think of the higher homotopies not as morphisms of the $n$-category but |
511 since we think of the higher homotopies not as morphisms of the $n$-category but |
512 rather as belonging to some auxiliary category (like chain complexes) |
512 rather as belonging to some auxiliary category (like chain complexes) |
513 that we are enriching in. |
513 that we are enriching in. |
514 We have decided to call them ``$A_\infty$ disk-like $n$-categories", since they are a natural generalization |
514 We have decided to call them ``$A_\infty$ $n$-categories", since they are a natural generalization |
515 of the familiar $A_\infty$ 1-categories. |
515 of the familiar $A_\infty$ 1-categories. |
516 We also considered the names ``homotopy $n$-categories" and ``infinity $n$-categories". |
516 We also considered the names ``homotopy $n$-categories" and ``infinity $n$-categories". |
517 When we need to emphasize that we are talking about an $n$-category which is not $A_\infty$ in this sense |
517 When we need to emphasize that we are talking about an $n$-category which is not $A_\infty$ in this sense |
518 we will say ``ordinary disk-like $n$-category". |
518 we will say ``ordinary $n$-category". |
519 % small problem: our n-cats are of course strictly associative, since we have more morphisms. |
519 % small problem: our n-cats are of course strictly associative, since we have more morphisms. |
520 % when we say ``associative only up to homotopy" above we are thinking about |
520 % when we say ``associative only up to homotopy" above we are thinking about |
521 % what would happen we we tried to convert to a more traditional n-cat with fewer morphisms |
521 % what would happen we we tried to convert to a more traditional n-cat with fewer morphisms |
522 |
522 |
523 Another distinction we need to make is between our style of definition of $n$-categories and |
523 Another distinction we need to make is between our style of definition of $n$-categories and |