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189 Our main motivating example (though we will not develop it in this paper) |
189 Our main motivating example (though we will not develop it in this paper) |
190 is the $(4{+}\varepsilon)$-dimensional TQFT associated to Khovanov homology. |
190 is the $(4{+}\varepsilon)$-dimensional TQFT associated to Khovanov homology. |
191 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
191 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
192 with a link $L \subset \bd W$. |
192 with a link $L \subset \bd W$. |
193 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
193 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
194 %\todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S} |
194 %\todo{I'm tempted to replace $A_{Kh}$ with $\colimit{Kh}$ throughout this page -S} |
195 |
195 |
196 How would we go about computing $A_{Kh}(W^4, L)$? |
196 How would we go about computing $A_{Kh}(W^4, L)$? |
197 For the Khovanov homology of a link in $S^3$ the main tool is the exact triangle (long exact sequence) |
197 For the Khovanov homology of a link in $S^3$ the main tool is the exact triangle (long exact sequence) |
198 relating resolutions of a crossing. |
198 relating resolutions of a crossing. |
199 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
199 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
413 We think of this disk-like $A_\infty$ $n$-category as a free resolution of the ordinary $n$-category. |
413 We think of this disk-like $A_\infty$ $n$-category as a free resolution of the ordinary $n$-category. |
414 \end{rem} |
414 \end{rem} |
415 |
415 |
416 There is a version of the blob complex for $\cC$ a disk-like $A_\infty$ $n$-category |
416 There is a version of the blob complex for $\cC$ a disk-like $A_\infty$ $n$-category |
417 instead of an ordinary $n$-category; this is described in \S \ref{sec:ainfblob}. |
417 instead of an ordinary $n$-category; this is described in \S \ref{sec:ainfblob}. |
418 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
418 The definition is in fact simpler, almost tautological, and we use a different notation, $\colimit{\cC}(M)$. |
419 The next theorem describes the blob complex for product manifolds |
419 The next theorem describes the blob complex for product manifolds |
420 in terms of the $A_\infty$ blob complex of the disk-like $A_\infty$ $n$-categories constructed as in the previous example. |
420 in terms of the $A_\infty$ blob complex of the disk-like $A_\infty$ $n$-categories constructed as in the previous example. |
421 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
421 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
422 |
422 |
423 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
423 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
427 Let $\cC$ be an $n$-category. |
427 Let $\cC$ be an $n$-category. |
428 Let $\bc_*(Y;\cC)$ be the disk-like $A_\infty$ $k$-category associated to $Y$ via blob homology |
428 Let $\bc_*(Y;\cC)$ be the disk-like $A_\infty$ $k$-category associated to $Y$ via blob homology |
429 (see Example \ref{ex:blob-complexes-of-balls}). |
429 (see Example \ref{ex:blob-complexes-of-balls}). |
430 Then |
430 Then |
431 \[ |
431 \[ |
432 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
432 \bc_*(Y\times W; \cC) \simeq \colimit{\bc_*(Y;\cC)}(W). |
433 \] |
433 \] |
434 \end{thm:product} |
434 \end{thm:product} |
435 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
435 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
436 (see \S \ref{ss:product-formula}). |
436 (see \S \ref{ss:product-formula}). |
437 |
437 |