192 Turaev-Viro theories, and Dijkgraaf-Witten theories. |
192 Turaev-Viro theories, and Dijkgraaf-Witten theories. |
193 These can all be given satisfactory accounts in the framework outlined above. |
193 These can all be given satisfactory accounts in the framework outlined above. |
194 (The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories with only a weak dependence on interiors in order to be |
194 (The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories with only a weak dependence on interiors in order to be |
195 extended all the way down to dimension 0.) |
195 extended all the way down to dimension 0.) |
196 |
196 |
197 For other TQFT-like invariants, however, the above framework seems to be inadequate. |
197 For other non-semisimple TQFT-like invariants, however, the above framework seems to be inadequate. |
198 |
198 For example, the gluing rule for 3-manifolds in Ozsv\'{a}th-Szab\'{o}/Seiberg-Witten theory |
199 \nn{kevin's rewrite stops here} |
199 involves a tensor product over an $A_\infty$ 1-category associated to 2-manifolds \cite{1003.0598,1005.1248}. |
200 |
200 Long exact sequences are important computational tools in these theories, |
201 However new invariants on manifolds, particularly those coming from |
201 and also in Khovanov homology, but the colimit construction breaks exactness. |
202 Seiberg-Witten theory and Ozsv\'{a}th-Szab\'{o} theory, do not fit the framework well. |
202 For these reasons and others, it is desirable to |
203 In particular, they have more complicated gluing formulas, involving derived or |
203 extend to above framework to incorporate ideas from derived categories. |
204 $A_\infty$ tensor products \cite{1003.0598,1005.1248}. |
204 |
205 It seems worthwhile to find a more general notion of TQFT that explain these. |
205 One approach to such a generalization might be to simply define a |
206 While we don't claim to fulfill that goal here, our notions of $n$-category and |
206 TQFT via its gluing formulas, replacing tensor products with |
207 of the blob complex are hopefully a step in the right direction, |
207 derived tensor products. |
208 and provide similar gluing formulas. |
208 \nn{maybe cite Kh's paper on links in $S^1\times S^2$} |
209 |
209 However, it is probably difficult to prove |
210 One approach to such a generalization might be simply to define a |
|
211 TQFT invariant via its gluing formulas, replacing tensor products with |
|
212 derived tensor products. However, it is probably difficult to prove |
|
213 the invariance of such a definition, as the object associated to a manifold |
210 the invariance of such a definition, as the object associated to a manifold |
214 will a priori depend on the explicit presentation used to apply the gluing formulas. |
211 will a priori depend on the explicit presentation used to apply the gluing formulas. |
215 We instead give a manifestly invariant construction, and |
212 We instead give a manifestly invariant construction, and |
216 deduce gluing formulas based on $A_\infty$ tensor products. |
213 deduce from it the gluing formulas based on $A_\infty$ tensor products. |
217 |
214 |
218 \nn{Triangulated categories are important; often calculations are via exact sequences, |
215 This paper is organized as follows. |
219 and the standard TQFT constructions are quotients, which destroy exactness.} |
216 We first give an account of our version of $n$-categories. |
|
217 According to our definition, $n$-categories are, among other things, |
|
218 functorial invariants of $k$-balls, $0\le k \le n$, which behave well with respect to gluing. |
|
219 We then describe how to use [homotopy] colimits to extend $n$-categories |
|
220 from balls to arbitrary $k$-manifolds. |
|
221 This extension is the desired derived version of a TQFT, which we call the blob complex. |
|
222 (The name comes from the ``blobs" which feature prominently |
|
223 in a concrete version of the homotopy colimit.) |
220 |
224 |
221 \nn{In many places we omit details; they can be found in MW. |
225 \nn{In many places we omit details; they can be found in MW. |
222 (Blanket statement in order to avoid too many citations to MW.)} |
226 (Blanket statement in order to avoid too many citations to MW.)} |
223 |
227 |
224 \nn{perhaps say something explicit about the relationship of this paper to big blob paper. |
228 \nn{perhaps say something explicit about the relationship of this paper to big blob paper. |