61 Nevertheless, when we attempt to establish all of the observed properties of the blob complex, |
61 Nevertheless, when we attempt to establish all of the observed properties of the blob complex, |
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 ``topological $n$-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 |
69 |
69 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. |
70 We try to be as lax as possible: a topological $n$-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 vector space to every $B$ homeomorphic to the $n$-ball. |
71 These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid. |
72 These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid. |
72 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 |
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 |
73 homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. |
74 homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. |
74 The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a |
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 |
75 topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
76 disk-like $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
76 |
77 |
77 In \S \ref{ssec:spherecat} we explain how $n$-categories can be viewed as objects in an $n{+}1$-category |
78 In \S \ref{ssec:spherecat} we explain how $n$-categories can be viewed as objects in an $n{+}1$-category |
78 of sphere modules. |
79 of sphere modules. |
79 When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwinors. |
80 When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwinors. |
80 |
81 |
81 In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category |
82 In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a disk-like $n$-category |
82 (using a colimit along certain decompositions of a manifold into balls). |
83 (using a colimit along certain decompositions of a manifold into balls). |
83 With this in hand, we write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ |
84 With this in hand, we write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ |
84 with the system of fields constructed from the $n$-category $\cC$. |
85 with the system of fields constructed from the $n$-category $\cC$. |
85 %\nn{KW: I don't think we use this notational convention any more, right?} |
86 %\nn{KW: I don't think we use this notational convention any more, right?} |
86 In \S \ref{sec:ainfblob} we give an alternative definition |
87 In \S \ref{sec:ainfblob} we give an alternative definition |
87 of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
88 of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
88 Using these definitions, we show how to use the blob complex to ``resolve" any topological $n$-category as an |
89 Using these definitions, we show how to use the blob complex to ``resolve" any ordinary $n$-category as an |
89 $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. |
90 $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. |
90 We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), |
91 We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), |
91 in particular the ``gluing formula" of Theorem \ref{thm:gluing} below. |
92 in particular the ``gluing formula" of Theorem \ref{thm:gluing} below. |
92 |
93 |
93 The relationship between all these ideas is sketched in Figure \ref{fig:outline}. |
94 The relationship between all these ideas is sketched in Figure \ref{fig:outline}. |
103 \newcommand{\yya}{14} |
104 \newcommand{\yya}{14} |
104 \newcommand{\yyb}{10} |
105 \newcommand{\yyb}{10} |
105 \newcommand{\yyc}{6} |
106 \newcommand{\yyc}{6} |
106 |
107 |
107 \node[box] at (-4,\yyb) (tC) {$C$ \\ a `traditional' \\ weak $n$-category}; |
108 \node[box] at (-4,\yyb) (tC) {$C$ \\ a `traditional' \\ weak $n$-category}; |
108 \node[box] at (\xxa,\yya) (C) {$\cC$ \\ a topological \\ $n$-category}; |
109 \node[box] at (\xxa,\yya) (C) {$\cC$ \\ a disk-like \\ $n$-category}; |
109 \node[box] at (\xxb,\yya) (A) {$\underrightarrow{\cC}(M)$ \\ the (dual) TQFT \\ Hilbert space}; |
110 \node[box] at (\xxb,\yya) (A) {$\underrightarrow{\cC}(M)$ \\ the (dual) TQFT \\ Hilbert space}; |
110 \node[box] at (\xxa,\yyb) (FU) {$(\cF, U)$ \\ fields and\\ local relations}; |
111 \node[box] at (\xxa,\yyb) (FU) {$(\cF, U)$ \\ fields and\\ local relations}; |
111 \node[box] at (\xxb,\yyb) (BC) {$\bc_*(M; \cF)$ \\ the blob complex}; |
112 \node[box] at (\xxb,\yyb) (BC) {$\bc_*(M; \cF)$ \\ the blob complex}; |
112 \node[box] at (\xxa,\yyc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category}; |
113 \node[box] at (\xxa,\yyc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category}; |
113 \node[box] at (\xxb,\yyc) (BCs) {$\underrightarrow{\cC_*}(M)$}; |
114 \node[box] at (\xxb,\yyc) (BCs) {$\underrightarrow{\cC_*}(M)$}; |
146 (that the little discs operad acts on Hochschild cochains) in terms of the blob complex. |
147 (that the little discs operad acts on Hochschild cochains) in terms of the blob complex. |
147 The appendices prove technical results about $\CH{M}$ and |
148 The appendices prove technical results about $\CH{M}$ and |
148 make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
149 make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
149 as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
150 as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
150 %Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
151 %Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
151 %thought of as a topological $n$-category, in terms of the topology of $M$. |
152 %thought of as a disk-like $n$-category, in terms of the topology of $M$. |
152 |
153 |
153 %%%% this is said later in the intro |
154 %%%% this is said later in the intro |
154 %Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes) |
155 %Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes) |
155 %even when we could work in greater generality (symmetric monoidal categories, model categories, etc.). |
156 %even when we could work in greater generality (symmetric monoidal categories, model categories, etc.). |
156 |
157 |
369 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
370 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
370 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
371 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
371 for any homeomorphic pair $X$ and $Y$, |
372 for any homeomorphic pair $X$ and $Y$, |
372 satisfying corresponding conditions. |
373 satisfying corresponding conditions. |
373 |
374 |
374 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. |
375 In \S \ref{sec:ncats} we introduce the notion of disk-like $n$-categories, from which we can construct systems of fields. |
375 Below, when we talk about the blob complex for a topological $n$-category, we are implicitly passing first to this associated system of fields. |
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. |
376 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 topological $n$-categories: |
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: |
377 |
378 |
378 \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}} |
379 |
380 |
380 \begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form an $A_\infty$ $n$-category] |
381 \begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form an $A_\infty$ $n$-category] |
381 %\label{thm:blobs-ainfty} |
382 %\label{thm:blobs-ainfty} |
382 Let $\cC$ be a topological $n$-category. |
383 Let $\cC$ be an ordinary $n$-category. |
383 Let $Y$ be an $n{-}k$-manifold. |
384 Let $Y$ be an $n{-}k$-manifold. |
384 There is an $A_\infty$ $k$-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 $0 \leq m < k$, |
385 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 |
386 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
387 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
387 (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.) |
388 These sets have the structure of an $A_\infty$ $k$-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 |
389 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}. |
390 \end{ex:blob-complexes-of-balls} |
391 \end{ex:blob-complexes-of-balls} |
391 \begin{rem} |
392 \begin{rem} |
392 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 a topological $n$-category. |
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. |
393 We think of this $A_\infty$ $n$-category as a free resolution. |
394 We think of this $A_\infty$ $n$-category as a free resolution. |
394 \end{rem} |
395 \end{rem} |
395 |
396 |
396 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
397 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
397 instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. |
398 instead of an ordinary $n$-category; this is described in \S \ref{sec:ainfblob}. |
398 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. |
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. |
399 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
400 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
400 |
401 |
401 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
402 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
402 |
403 |
410 \] |
411 \] |
411 \end{thm:product} |
412 \end{thm:product} |
412 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
413 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
413 (see \S \ref{ss:product-formula}). |
414 (see \S \ref{ss:product-formula}). |
414 |
415 |
415 Fix a topological $n$-category $\cC$, which we'll omit from the notation. |
416 Fix a disk-like $n$-category $\cC$, which we'll omit from the notation. |
416 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
417 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
417 (See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) |
418 (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.) |
418 |
419 |
419 \newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}} |
420 \newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}} |
420 |
421 |
421 \begin{thm:gluing}[Gluing formula] |
422 \begin{thm:gluing}[Gluing formula] |
422 \mbox{}% <-- gets the indenting right |
423 \mbox{}% <-- gets the indenting right |