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529 We will use the term `field on $W$' to refer to a point of this functor, |
529 We will use the term `field on $W$' to refer to a point of this functor, |
530 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
530 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
531 |
531 |
532 |
532 |
533 \subsubsection{Colimits} |
533 \subsubsection{Colimits} |
534 \nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?} |
534 Our definition of an $n$-category is essentially a collection of functors defined on $k$-balls (and homeomorphisms) for $k \leq n$ satisfying certain axioms. It is natural to consider extending such functors to the larger categories of all $k$-manifolds (again, with homeomorphisms). In fact, the axioms stated above explictly require such an extension to $k$-spheres for $k<n$. |
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535 |
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536 The natural construction achieving this is the colimit. |
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537 \nn{continue} |
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538 |
535 \nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?} |
539 \nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?} |
536 \nn{Explain codimension colimits here too} |
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537 |
540 |
538 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
541 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
539 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ |
542 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ |
540 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
543 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
541 |
544 |