40 |
40 |
41 |
41 |
42 \nn{some more things to cover in the intro} |
42 \nn{some more things to cover in the intro} |
43 \begin{itemize} |
43 \begin{itemize} |
44 \item related: we are being unsophisticated from a homotopy theory point of |
44 \item related: we are being unsophisticated from a homotopy theory point of |
45 view and using chain complexes in many places where we could be by with spaces |
45 view and using chain complexes in many places where we could get by with spaces |
46 \item ? one of the points we make (far) below is that there is not really much |
46 \item ? one of the points we make (far) below is that there is not really much |
47 difference between (a) systems of fields and local relations and (b) $n$-cats; |
47 difference between (a) systems of fields and local relations and (b) $n$-cats; |
48 thus we tend to switch between talking in terms of one or the other |
48 thus we tend to switch between talking in terms of one or the other |
49 \end{itemize} |
49 \end{itemize} |
50 |
50 |
51 \medskip\hrule\medskip |
51 \medskip\hrule\medskip |
52 |
52 |
53 \subsection{Motivations} |
53 \subsection{Motivations} |
54 \label{sec:motivations} |
54 \label{sec:motivations} |
55 |
55 |
56 [Old outline for intro] |
56 We will briefly sketch our original motivation for defining the blob complex. |
57 \begin{itemize} |
57 \nn{this is adapted from an old draft of the intro; it needs further modification |
58 \item Starting point: TQFTs via fields and local relations. |
58 in order to better integrate it into the current intro.} |
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59 |
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60 As a starting point, consider TQFTs constructed via fields and local relations. |
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61 (See Section \ref{sec:tqftsviafields} or \cite{kwtqft}.) |
59 This gives a satisfactory treatment for semisimple TQFTs |
62 This gives a satisfactory treatment for semisimple TQFTs |
60 (i.e.\ TQFTs for which the cylinder 1-category associated to an |
63 (i.e.\ TQFTs for which the cylinder 1-category associated to an |
61 $n{-}1$-manifold $Y$ is semisimple for all $Y$). |
64 $n{-}1$-manifold $Y$ is semisimple for all $Y$). |
62 \item For non-semiemple TQFTs, this approach is less satisfactory. |
65 |
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66 For non-semiemple TQFTs, this approach is less satisfactory. |
63 Our main motivating example (though we will not develop it in this paper) |
67 Our main motivating example (though we will not develop it in this paper) |
64 is the $4{+}1$-dimensional TQFT associated to Khovanov homology. |
68 is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology. |
65 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
69 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
66 with a link $L \subset \bd W$. |
70 with a link $L \subset \bd W$. |
67 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
71 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
68 \item How would we go about computing $A_{Kh}(W^4, L)$? |
72 |
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73 How would we go about computing $A_{Kh}(W^4, L)$? |
69 For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence) |
74 For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence) |
70 \nn{... $L_1, L_2, L_3$}. |
75 \nn{... $L_1, L_2, L_3$}. |
71 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
76 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
72 to compute $A_{Kh}(S^1\times B^3, L)$. |
77 to compute $A_{Kh}(S^1\times B^3, L)$. |
73 According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$ |
78 According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$ |
74 corresponds to taking a coend (self tensor product) over the cylinder category |
79 corresponds to taking a coend (self tensor product) over the cylinder category |
75 associated to $B^3$ (with appropriate boundary conditions). |
80 associated to $B^3$ (with appropriate boundary conditions). |
76 The coend is not an exact functor, so the exactness of the triangle breaks. |
81 The coend is not an exact functor, so the exactness of the triangle breaks. |
77 \item The obvious solution to this problem is to replace the coend with its derived counterpart. |
82 |
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83 |
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84 The obvious solution to this problem is to replace the coend with its derived counterpart. |
78 This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology |
85 This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology |
79 of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired. |
86 of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired. |
80 If we build our manifold up via a handle decomposition, the computation |
87 If we build our manifold up via a handle decomposition, the computation |
81 would be a sequence of derived coends. |
88 would be a sequence of derived coends. |
82 A different handle decomposition of the same manifold would yield a different |
89 A different handle decomposition of the same manifold would yield a different |
83 sequence of derived coends. |
90 sequence of derived coends. |
84 To show that our definition in terms of derived coends is well-defined, we |
91 To show that our definition in terms of derived coends is well-defined, we |
85 would need to show that the above two sequences of derived coends yield the same answer. |
92 would need to show that the above two sequences of derived coends yield the same answer. |
86 This is probably not easy to do. |
93 This is probably not easy to do. |
87 \item Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$ |
94 |
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95 Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$ |
88 which is manifestly invariant. |
96 which is manifestly invariant. |
89 (That is, a definition that does not |
97 (That is, a definition that does not |
90 involve choosing a decomposition of $W$. |
98 involve choosing a decomposition of $W$. |
91 After all, one of the virtues of our starting point --- TQFTs via field and local relations --- |
99 After all, one of the virtues of our starting point --- TQFTs via field and local relations --- |
92 is that it has just this sort of manifest invariance.) |
100 is that it has just this sort of manifest invariance.) |
93 \item The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient |
101 |
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102 The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient |
94 \[ |
103 \[ |
95 \text{linear combinations of fields} \;\big/\; \text{local relations} , |
104 \text{linear combinations of fields} \;\big/\; \text{local relations} , |
96 \] |
105 \] |
97 with an appropriately free resolution (the ``blob complex") |
106 with an appropriately free resolution (the ``blob complex") |
98 \[ |
107 \[ |