equal
deleted
inserted
replaced
134 |
134 |
135 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
135 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
136 \begin{article} |
136 \begin{article} |
137 |
137 |
138 \begin{abstract} |
138 \begin{abstract} |
139 \nn{needs revision} |
139 We summarize our axioms for higher categories, and describe the ``blob complex". |
140 We explain the need for new axioms for topological quantum field theories that include ideas from derived |
|
141 categories and homotopy theory. We summarize our axioms for higher categories, and describe the ``blob complex". |
|
142 Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$. |
140 Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$. |
143 The $0$-th homology of this chain complex recovers the usual TQFT invariants of $W$. |
141 The $0$-th homology of this chain complex recovers the usual topological quantum field theory invariants of $W$. |
144 The higher homology groups should be viewed as generalizations of Hochschild homology. |
142 The higher homology groups should be viewed as generalizations of Hochschild homology (indeed, when $W=S^1$ they coincide). |
145 The blob complex has a very natural definition in terms of homotopy colimits along decompositions of the manifold $W$. |
143 The blob complex has a very natural definition in terms of homotopy colimits along decompositions of the manifold $W$. |
146 We outline the important properties of the blob complex, and sketch the proof of a generalization of |
144 We outline the important properties of the blob complex, and sketch the proof of a generalization of |
147 Deligne's conjecture on Hochschild cohomology and the little discs operad to higher dimensions. |
145 Deligne's conjecture on Hochschild cohomology and the little discs operad to higher dimensions. |
148 \end{abstract} |
146 \end{abstract} |
149 |
147 |
652 \item a linear combination $s$ of string diagrams on $W$, |
650 \item a linear combination $s$ of string diagrams on $W$, |
653 \end{itemize} |
651 \end{itemize} |
654 such that |
652 such that |
655 \begin{itemize} |
653 \begin{itemize} |
656 \item there is a permissible decomposition of $W$, compatible with the $k$ blobs, such that |
654 \item there is a permissible decomposition of $W$, compatible with the $k$ blobs, such that |
657 $s$ is the product of linear combinations of fields $s_i$ on the initial pieces $X_i$ of the decomposition |
655 $s$ is the result of gluing together linear combinations of fields $s_i$ on the initial pieces $X_i$ of the decomposition |
658 (for fixed restrictions to the boundaries of the pieces), |
656 (for fixed restrictions to the boundaries of the pieces), |
659 \item the $s_i$'s corresponding to innermost blobs evaluate to zero in $\cC$, and |
657 \item the $s_i$'s corresponding to innermost blobs evaluate to zero in $\cC$, and |
660 \item the $s_i$'s corresponding to the other pieces are single fields (linear combinations with only one term). |
658 \item the $s_i$'s corresponding to the other pieces are single fields (linear combinations with only one term). |
661 \end{itemize} |
659 \end{itemize} |
662 %that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. |
660 %that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. |