411 For example, the gluing rule for 3-manifolds in Ozsv\'ath-Szab\'o/Seiberg-Witten theory |
411 For example, the gluing rule for 3-manifolds in Ozsv\'ath-Szab\'o/Seiberg-Witten theory |
412 involves a tensor product over an $A_\infty$ 1-category associated to 2-manifolds \cite{1003.0598,1005.1248}. |
412 involves a tensor product over an $A_\infty$ 1-category associated to 2-manifolds \cite{1003.0598,1005.1248}. |
413 Long exact sequences are important computational tools in these theories, |
413 Long exact sequences are important computational tools in these theories, |
414 and also in Khovanov homology, but the colimit construction breaks exactness. |
414 and also in Khovanov homology, but the colimit construction breaks exactness. |
415 For these reasons and others, it is desirable to |
415 For these reasons and others, it is desirable to |
416 extend to above framework to incorporate ideas from derived categories. |
416 extend the above framework to incorporate ideas from derived categories. |
417 |
417 |
418 One approach to such a generalization might be to simply define a |
418 One approach to such a generalization might be to simply define a |
419 TQFT via its gluing formulas, replacing tensor products with |
419 TQFT via its gluing formulas, replacing tensor products with |
420 derived tensor products (c.f. \cite{1011.1958}). |
420 derived tensor products (c.f. \cite{1011.1958}). |
421 However, it is probably difficult to prove |
421 However, it is probably difficult to prove |