36 |
36 |
37 The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, |
37 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. |
38 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 |
39 There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is |
40 simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. |
40 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 |
41 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. |
42 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. |
43 We sketch the construction of a system of fields from a $1$-category or from a pivotal $2$-category. |
44 |
44 |
45 Nevertheless, when we attempt to establish all of the observed properties of the blob complex, |
45 Nevertheless, when we attempt to establish all of the observed properties of the blob complex, |
46 we find this situation unsatisfactory. |
46 we find this situation unsatisfactory. |
48 definition of an $n$-category, or rather a definition of an $n$-category with strong duality. |
48 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.) |
49 (It appears that 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. |
50 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. |
51 Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms. |
52 |
52 |
53 The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms. |
53 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. |
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. |
55 These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid. |
55 These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid. |
56 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 |
56 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 |
57 homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. |
57 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 |
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 |
59 topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
59 topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
60 |
60 |
61 In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category |
61 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 cellulations of a manifold), and in \S \ref{sec:ainfblob} give an alternative definition |
62 (using a colimit along cellulations of a manifold), and in \S \ref{sec:ainfblob} give an alternative definition |
63 of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
63 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 |
64 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. |
65 $A_\infty$ $n$-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), |
66 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. |
67 in particular the ``gluing formula" of Theorem \ref{thm:gluing} below. |
68 |
68 |
69 The relationship between all these ideas is sketched in Figure \ref{fig:outline}. |
69 The relationship between all these ideas is sketched in Figure \ref{fig:outline}. |
70 |
70 |
71 \tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt] |
71 \tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt] |
72 |
72 |
113 Finally, later sections address other topics. |
113 Finally, later sections address other topics. |
114 Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
114 Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
115 thought of as a topological $n$-category, in terms of the topology of $M$. |
115 thought of as a topological $n$-category, in terms of the topology of $M$. |
116 Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) |
116 Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) |
117 a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. |
117 a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. |
118 The appendixes prove technical results about $\CH{M}$ and the `small blob complex', |
118 The appendixes prove technical results about $\CH{M}$ and the ``small blob complex", |
119 and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
119 and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
120 as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
120 as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
121 |
121 |
122 |
122 |
123 \nn{some more things to cover in the intro} |
123 \nn{some more things to cover in the intro} |
434 there may be some differences for topological manifolds and smooth manifolds. |
434 there may be some differences for topological manifolds and smooth manifolds. |
435 |
435 |
436 The paper ``Skein homology'' \cite{MR1624157} has similar motivations, and it may be |
436 The paper ``Skein homology'' \cite{MR1624157} has similar motivations, and it may be |
437 interesting to investigate if there is a connection with the material here. |
437 interesting to investigate if there is a connection with the material here. |
438 |
438 |
439 Many results in Hochschild homology can be understood `topologically' via the blob complex. |
439 Many results in Hochschild homology can be understood ``topologically" via the blob complex. |
440 For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ |
440 For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ |
441 (see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, |
441 (see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, |
442 but haven't investigated the details. |
442 but haven't investigated the details. |
443 |
443 |
444 Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} |
444 Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} |