191 and $A(Y)$ is a finite-dimensional vector space for all $n$-manifolds $Y$. |
191 and $A(Y)$ is a finite-dimensional vector space for all $n$-manifolds $Y$. |
192 Examples of semisimple TQFTS include Witten-Reshetikhin-Turaev theories, |
192 Examples of semisimple TQFTS include Witten-Reshetikhin-Turaev theories, |
193 Turaev-Viro theories, and Dijkgraaf-Witten theories. |
193 Turaev-Viro theories, and Dijkgraaf-Witten theories. |
194 These can all be given satisfactory accounts in the framework outlined above. |
194 These can all be given satisfactory accounts in the framework outlined above. |
195 (The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories in order to be |
195 (The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories in order to be |
196 extended all the way down to 0 dimensions.) |
196 extended all the way down to 0-manifolds.) |
197 |
197 |
198 For other TQFT-like invariants, however, the above framework seems to be inadequate. |
198 For other non-semisimple TQFT-like invariants, however, the above framework seems to be inadequate. |
199 |
199 |
200 However new invariants on manifolds, particularly those coming from |
200 \nn{temp} |
201 Seiberg-Witten theory and Ozsv\'{a}th-Szab\'{o} theory, do not fit the framework well. |
201 |
202 In particular, they have more complicated gluing formulas, involving derived or |
202 For example, the gluing rule for 3-manifolds in Ozsv\'{a}th-Szab\'{o}/Seiberg-Witten theory |
203 $A_\infty$ tensor products \cite{1003.0598,1005.1248}. |
203 involves a tensor product over an $A_\infty$ 1-category associated to 2-manifolds \cite{1003.0598,1005.1248}. |
204 It seems worthwhile to find a more general notion of TQFT that explain these. |
204 Long exact sequences are important computational tools in these theories, |
205 While we don't claim to fulfill that goal here, our notions of $n$-category and |
205 and also in Khovanov homology, but the colimit construction breaks exactness. |
206 of the blob complex are hopefully a step in the right direction, |
206 For these reasons and others, it is desirable to |
207 and provide similar gluing formulas. |
207 extend to above framework to incorporate ideas from derived categories. |
208 |
208 |
209 One approach to such a generalization might be simply to define a |
209 One approach to such a generalization might be to simply define a |
210 TQFT invariant via its gluing formulas, replacing tensor products with |
210 TQFT via its gluing formulas, replacing tensor products with |
211 derived tensor products. However, it is probably difficult to prove |
211 derived tensor products. |
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212 \nn{maybe cite Kh's paper on links in $S^1\times S^2$} |
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213 However, it is probably difficult to prove |
212 the invariance of such a definition, as the object associated to a manifold |
214 the invariance of such a definition, as the object associated to a manifold |
213 will a priori depend on the explicit presentation used to apply the gluing formulas. |
215 will a priori depend on the explicit presentation used to apply the gluing formulas. |
214 We instead give a manifestly invariant construction, and |
216 We instead give a manifestly invariant construction, and |
215 deduce gluing formulas based on $A_\infty$ tensor products. |
217 deduce from it the gluing formulas based on $A_\infty$ tensor products. |
216 |
218 |
217 \nn{Triangulated categories are important; often calculations are via exact sequences, |
219 This paper is organized as follows. |
218 and the standard TQFT constructions are quotients, which destroy exactness.} |
220 We first give an account of our version of $n$-categories. |
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221 According to our definition, $n$-categories are, among other things, |
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222 functorial invariants of $k$-balls, $0\le k \le n$, which behave well with respect to gluing. |
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223 We then describe how to use [homotopy] colimits to extend $n$-categories |
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224 from balls to arbitrary $k$-manifolds. |
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225 This extension is the desired derived version of a TQFT, which we call the blob complex. |
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226 (The name comes from the ``blobs" which feature prominently |
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227 in a concrete version of the homotopy colimit.) |
219 |
228 |
220 \nn{In many places we omit details; they can be found in MW. |
229 \nn{In many places we omit details; they can be found in MW. |
221 (Blanket statement in order to avoid too many citations to MW.)} |
230 (Blanket statement in order to avoid too many citations to MW.)} |
222 |
231 |
223 \nn{perhaps say something explicit about the relationship of this paper to big blob paper. |
232 \nn{perhaps say something explicit about the relationship of this paper to big blob paper. |