435 |
435 |
436 The above construction can be extended to higher codimensions, assigning |
436 The above construction can be extended to higher codimensions, assigning |
437 a $k$-category $A(Y)$ to an $n{-}k$-manifold $Y$, for $0 \le k \le n$. |
437 a $k$-category $A(Y)$ to an $n{-}k$-manifold $Y$, for $0 \le k \le n$. |
438 These invariants fit together via actions and gluing formulas. |
438 These invariants fit together via actions and gluing formulas. |
439 We describe only the case $k=1$ below. |
439 We describe only the case $k=1$ below. |
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440 |
440 The construction of the $n{+}1$-dimensional part of the theory (the path integral) |
441 The construction of the $n{+}1$-dimensional part of the theory (the path integral) |
441 requires that the starting data (fields and local relations) satisfy additional |
442 requires that the starting data (fields and local relations) satisfy additional |
442 conditions. |
443 conditions. |
443 We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT |
444 We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT |
444 that lacks its $n{+}1$-dimensional part. |
445 that lacks its $n{+}1$-dimensional part. |
445 Such a ``decapitated'' TQFT is sometimes also called an $n+\epsilon$ or |
446 Such a ``decapitated'' TQFT is sometimes also called an $n+\epsilon$ or |
446 $n+\frac{1}{2}$ dimensional TQFT, referring to the fact that it assigns maps to |
447 $n+\frac{1}{2}$ dimensional TQFT, referring to the fact that it assigns maps to $n{+}1$-dimensional |
447 mapping cylinders between $n$-manifolds, but nothing to arbitrary $n{+}1$-manifolds. |
448 mapping cylinders between $n$-manifolds, but nothing to arbitrary $n{+}1$-manifolds. |
448 |
449 |
449 Let $Y$ be an $n{-}1$-manifold. |
450 Let $Y$ be an $n{-}1$-manifold. |
450 Define a linear 1-category $A(Y)$ as follows. |
451 Define a linear 1-category $A(Y)$ as follows. |
451 The set of objects of $A(Y)$ is $\cC(Y)$. |
452 The set of objects of $A(Y)$ is $\cC(Y)$. |