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   121 (motivation, summary/outline, etc.)

   120 [Outline for intro]

   122 

   121 \begin{itemize}

   123 (motivation:

   122 \item Starting point: TQFTs via fields and local relations.

   124 (1) restore exactness in pictures-mod-relations;

   123 This gives a satisfactory treatment for semisimple TQFTs

   125 (1') add relations-amongst-relations etc. to pictures-mod-relations;

   124 (i.e. TQFTs for which the cylinder 1-category associated to an

   126 (2) want answer independent of handle decomp (i.e. don't

   125 $n{-}1$-manifold $Y$ is semisimple for all $Y$).

   127 just go from coend to derived coend (e.g. Hochschild homology));

   126 \item For non-semiemple TQFTs, this approach is less satisfactory.

   128 (3) ...

   127 Our main motivating example (though we will not develop it in this paper)

   129 )

   128 is the $4{+}1$-dimensional TQFT associated to Khovanov homology.

   130 

   129 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together

   131 

   130 with a link $L \subset \bd W$.

   131 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$.

   132 \item How would we go about computing $A_{Kh}(W^4, L)$?

   133 For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence)

   134 \nn{... $L_1, L_2, L_3$}.

   135 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt

   136 to compute $A_{Kh}(S^1\times B^3, L)$.

   137 According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$

   138 corresponds to taking a coend (self tensor product) over the cylinder category

   139 associated to $B^3$ (with appropriate boundary conditions).

   140 The coend is not an exact functor, so the exactness of the triangle breaks.

   141 \item The obvious solution to this problem is to replace the coend with its derived counterpart.

   142 This presumably works fine for $S^1\times B^3$ (the answer being to Hochschild homology

   143 of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired.

   144 If we build our manifold up via a handle decomposition, the computation

   145 would be a sequence of derived coends.

   146 A different handle decomposition of the same manifold would yield a different

   147 sequence of derived coends.

   148 To show that our definition in terms of derived coends is well-defined, we

   149 would need to show that the above two sequences of derived coends yield the same answer.

   150 This is probably not easy to do.

   151 \item Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$

   152 which is manifestly invariant.

   153 (That is, a definition that does not

   154 involve choosing a decomposition of $W$.

   155 After all, one of the virtues of our starting point --- TQFTs via field and local relations ---

   156 is that it has just this sort of manifest invariance.)

   157 \item The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient

   158 \[

   159  \text{linear combinations of fields} \;\big/\; \text{local relations} ,

   160 \]

   161 with an appropriately free resolution (the ``blob complex")

   162 \[

   163 	\cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) .

   164 \]

   165 Here $\bc_0$ is linear combinations of fields on $W$,

   166 $\bc_1$ is linear combinations of local relations on $W$,

   167 $\bc_1$ is linear combinations of relations amongst relations on $W$,

   168 and so on.

   169 \item None of the above ideas depend on the details of the Khovanov homology example,

   170 so we develop the general theory in the paper and postpone specific applications

   171 to later papers.

   172 \item The blob complex enjoys the following nice properties \nn{...}

   173 \end{itemize}

   174 

   175 \bigskip

   176 \hrule

   177 \bigskip