1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \section{Introduction} |
3 \section{Introduction} |
4 |
4 |
5 We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$. |
5 We construct a chain complex $\bc_*(M; \cC)$ --- the ``blob complex'' --- |
6 This blob complex provides a simultaneous generalization of several well-understood constructions: |
6 associated to an $n$-manifold $M$ and a linear $n$-category with strong duality $\cC$. |
|
7 This blob complex provides a simultaneous generalization of several well known constructions: |
7 \begin{itemize} |
8 \begin{itemize} |
8 \item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. |
9 \item The 0-th homology $H_0(\bc_*(M; \cC))$ is isomorphic to the usual |
|
10 topological quantum field theory invariant of $M$ associated to $\cC$. |
9 (See Theorem \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
11 (See Theorem \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
10 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), |
12 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), |
11 the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. |
13 the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. |
12 (See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.) |
14 (See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.) |
13 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category, we have |
15 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category, we have |
14 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
16 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
15 on the configuration space of unlabeled points in $M$. (See \S \ref{sec:comm_alg}.) |
17 on the configuration space of unlabeled points in $M$. (See \S \ref{sec:comm_alg}.) |
16 %$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ |
18 \item When $\cC$ is $\pi^\infty_{\leq n}(T)$, the $A_\infty$ version of the fundamental $n$-groupoid of |
|
19 the space $T$ (Example \ref{ex:chains-of-maps-to-a-space}), |
|
20 $\bc_*(M; \cC)$ is homotopy equivalent to $C_*(\Maps(M\to T))$, |
|
21 the singular chains on the space of maps from $M$ to $T$. |
|
22 (See Theorem \ref{thm:map-recon}.) |
17 \end{itemize} |
23 \end{itemize} |
|
24 |
18 The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space |
25 The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space |
19 (replacing quotient of fields by local relations with some sort of resolution), |
26 (replacing the quotient of fields by local relations with some sort of resolution), |
20 and for a generalization of Hochschild homology to higher $n$-categories. |
27 and for a generalization of Hochschild homology to higher $n$-categories. |
21 We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. |
28 One can think of it as the push-out of these two familiar constructions. |
22 The blob complex gives us all of these! More detailed motivations are described in \S \ref{sec:motivations}. |
29 More detailed motivations are described in \S \ref{sec:motivations}. |
23 |
30 |
24 The blob complex has good formal properties, summarized in \S \ref{sec:properties}. |
31 The blob complex has good formal properties, summarized in \S \ref{sec:properties}. |
25 These include an action of $\CH{M}$, |
32 These include an action of $\CH{M}$, |
26 extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Theorem \ref{thm:evaluation}) and a gluing |
33 extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (Theorem \ref{thm:evaluation}) and a gluing |
27 formula allowing calculations by cutting manifolds into smaller parts (see Theorem \ref{thm:gluing}). |
34 formula allowing calculations by cutting manifolds into smaller parts (Theorem \ref{thm:gluing}). |
28 |
35 |
29 We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper. |
36 We expect applications of the blob complex to contact topology and Khovanov homology |
|
37 but do not address these in this paper. |
30 See \S \ref{sec:future} for slightly more detail. |
38 See \S \ref{sec:future} for slightly more detail. |
|
39 |
31 |
40 |
32 \subsection{Structure of the paper} |
41 \subsection{Structure of the paper} |
33 The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), |
42 The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), |
34 summarise the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}) |
43 summarize the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}) |
35 and outline anticipated future directions (see \S \ref{sec:future}). |
44 and outline anticipated future directions (see \S \ref{sec:future}). |
|
45 \nn{recheck this list after done editing intro} |
36 |
46 |
37 The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, |
47 The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, |
38 and establishes some of its properties. |
48 and establishes some of its properties. |
39 There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is |
49 There are many alternative definitions of $n$-categories, and part of the challenge of defining the blob complex is |
40 simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. |
50 simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. |
41 At first we entirely avoid this problem by introducing the notion of a ``system of fields", and define the blob complex |
51 At first we entirely avoid this problem by introducing the notion of a ``system of fields", and define the blob complex |
42 associated to an $n$-manifold and an $n$-dimensional system of fields. |
52 associated to an $n$-manifold and an $n$-dimensional system of fields. |
43 We sketch the construction of a system of fields from a $1$-category or from a pivotal $2$-category. |
53 We sketch the construction of a system of fields from a *-$1$-category and from a pivotal $2$-category. |
44 |
54 |
45 Nevertheless, when we attempt to establish all of the observed properties of the blob complex, |
55 Nevertheless, when we attempt to establish all of the observed properties of the blob complex, |
46 we find this situation unsatisfactory. |
56 we find this situation unsatisfactory. |
47 Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another |
57 Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another |
48 definition of an $n$-category, or rather a definition of an $n$-category with strong duality. |
58 definition of an $n$-category, or rather a definition of an $n$-category with strong duality. |
49 (It appears that removing the duality conditions from our definition would make it more complicated rather than less.) |
59 (Removing the duality conditions from our definition would make it more complicated rather than less.) |
50 We call these ``topological $n$-categories'', to differentiate them from previous versions. |
60 We call these ``topological $n$-categories'', to differentiate them from previous versions. |
51 Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms. |
61 Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms. |
52 |
62 |
53 The basic idea is that each potential definition of an $n$-category makes a choice about the ``shape" of morphisms. |
63 The basic idea is that each potential definition of an $n$-category makes a choice about the ``shape" of morphisms. |
54 We try to be as lax as possible: a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. |
64 We try to be as lax as possible: a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. |
57 homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. |
67 homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. |
58 The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a |
68 The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a |
59 topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
69 topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
60 |
70 |
61 In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category |
71 In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category |
62 (using a colimit along certain decompositions of a manifold into balls). With this in hand, we freely write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ with the system of fields constructed from the $n$-category $\cC$. In \S \ref{sec:ainfblob} we give an alternative definition |
72 (using a colimit along certain decompositions of a manifold into balls). |
|
73 With this in hand, we freely write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ |
|
74 with the system of fields constructed from the $n$-category $\cC$. |
|
75 \nn{KW: I don't think we use this notational convention any more, right?} |
|
76 In \S \ref{sec:ainfblob} we give an alternative definition |
63 of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
77 of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
64 Using these definitions, we show how to use the blob complex to ``resolve" any topological $n$-category as an |
78 Using these definitions, we show how to use the blob complex to ``resolve" any topological $n$-category as an |
65 $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. |
79 $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. |
66 We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), |
80 We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), |
67 in particular the ``gluing formula" of Theorem \ref{thm:gluing} below. |
81 in particular the ``gluing formula" of Theorem \ref{thm:gluing} below. |
110 \label{fig:outline} |
124 \label{fig:outline} |
111 \end{figure} |
125 \end{figure} |
112 |
126 |
113 Finally, later sections address other topics. |
127 Finally, later sections address other topics. |
114 Section \S \ref{sec:deligne} gives |
128 Section \S \ref{sec:deligne} gives |
115 a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. |
129 a higher dimensional generalization of the Deligne conjecture |
|
130 (that the little discs operad acts on Hochschild cochains) in terms of the blob complex. |
116 The appendixes prove technical results about $\CH{M}$ and the ``small blob complex", |
131 The appendixes prove technical results about $\CH{M}$ and the ``small blob complex", |
117 and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
132 and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
118 as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
133 as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
|
134 Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
119 thought of as a topological $n$-category, in terms of the topology of $M$. |
135 thought of as a topological $n$-category, in terms of the topology of $M$. |
120 |
136 |
121 |
137 Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes) |
122 \nn{some more things to cover in the intro} |
138 \nn{...} |
123 \begin{itemize} |
139 |
124 \item related: we are being unsophisticated from a homotopy theory point of |
140 |
125 view and using chain complexes in many places where we could get by with spaces |
141 %\item related: we are being unsophisticated from a homotopy theory point of |
126 \item ? one of the points we make (far) below is that there is not really much |
142 %view and using chain complexes in many places where we could get by with spaces |
127 difference between (a) systems of fields and local relations and (b) $n$-cats; |
143 |
128 thus we tend to switch between talking in terms of one or the other |
144 %\item ? one of the points we make (far) below is that there is not really much |
129 \end{itemize} |
145 %difference between (a) systems of fields and local relations and (b) $n$-cats; |
130 |
146 %thus we tend to switch between talking in terms of one or the other |
131 \medskip\hrule\medskip |
147 |
|
148 |
132 |
149 |
133 \subsection{Motivations} |
150 \subsection{Motivations} |
134 \label{sec:motivations} |
151 \label{sec:motivations} |
135 |
152 |
136 We will briefly sketch our original motivation for defining the blob complex. |
153 We will briefly sketch our original motivation for defining the blob complex. |