442 We describe only the case $k=1$ below. |
442 We describe only the case $k=1$ below. |
443 |
443 |
444 The construction of the $n{+}1$-dimensional part of the theory (the path integral) |
444 The construction of the $n{+}1$-dimensional part of the theory (the path integral) |
445 requires that the starting data (fields and local relations) satisfy additional |
445 requires that the starting data (fields and local relations) satisfy additional |
446 conditions. |
446 conditions. |
447 We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT |
447 (Specifically, $A(X; c)$ is finite dimensional for all $n$-manifolds $X$ and the inner products |
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448 on $A(B^n; c)$ induced by the path integral of $B^{n+1}$ are positive definite for all $c$.) |
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449 We do not assume these conditions here, so when we say ``TQFT" we mean a ``decapitated" TQFT |
448 that lacks its $n{+}1$-dimensional part. |
450 that lacks its $n{+}1$-dimensional part. |
449 Such a ``decapitated'' TQFT is sometimes also called an $n+\epsilon$ or |
451 Such a decapitated TQFT is sometimes also called an $n{+}\epsilon$ or |
450 $n+\frac{1}{2}$ dimensional TQFT, referring to the fact that it assigns maps to $n{+}1$-dimensional |
452 $n{+}\frac{1}{2}$ dimensional TQFT, referring to the fact that it assigns linear maps to $n{+}1$-dimensional |
451 mapping cylinders between $n$-manifolds, but nothing to arbitrary $n{+}1$-manifolds. |
453 mapping cylinders between $n$-manifolds, but nothing to general $n{+}1$-manifolds. |
452 |
454 |
453 Let $Y$ be an $n{-}1$-manifold. |
455 Let $Y$ be an $n{-}1$-manifold. |
454 Define a linear 1-category $A(Y)$ as follows. |
456 Define a linear 1-category $A(Y)$ as follows. |
455 The set of objects of $A(Y)$ is $\cC(Y)$. |
457 The set of objects of $A(Y)$ is $\cC(Y)$. |
456 The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, |
458 The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, |