408 |
408 |
409 |
409 |
410 |
410 |
411 |
411 |
412 \subsection{From $n$-categories to systems of fields} |
412 \subsection{From $n$-categories to systems of fields} |
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413 \label{ss:ncat_fields} |
413 |
414 |
414 We can extend the functors $\cC$ above from $k$-balls to arbitrary $k$-manifolds as follows. |
415 We can extend the functors $\cC$ above from $k$-balls to arbitrary $k$-manifolds as follows. |
415 |
416 |
416 Let $W$ be a $k$-manifold, $1\le k \le n$. |
417 Let $W$ be a $k$-manifold, $1\le k \le n$. |
417 We will define a set $\cC(W)$. |
418 We will define a set $\cC(W)$. |
464 $a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x) |
465 $a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x) |
465 \to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$. |
466 \to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$. |
466 |
467 |
467 In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit |
468 In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit |
468 is as follows. |
469 is as follows. |
469 Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_{m-1}$ of permissible decompositions. |
470 Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions. |
470 Such sequences (for all $m$) form a simplicial set. |
471 Such sequences (for all $m$) form a simplicial set. |
471 Let |
472 Let |
472 \[ |
473 \[ |
473 V = \bigoplus_{(x_i)} \psi_\cC(x_0) , |
474 V = \bigoplus_{(x_i)} \psi_\cC(x_0) , |
474 \] |
475 \] |
475 where the sum is over all $m$-sequences and all $m$. |
476 where the sum is over all $m$-sequences and all $m$, and each summand is degree shifted by $m$. |
476 We endow $V$ with a differential which is the sum of the differential of the $\psi_\cC(x_0)$ |
477 We endow $V$ with a differential which is the sum of the differential of the $\psi_\cC(x_0)$ |
477 summands plus another term using the differential of the simplicial set of $m$-sequences. |
478 summands plus another term using the differential of the simplicial set of $m$-sequences. |
478 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
479 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
479 summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
480 summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
480 \[ |
481 \[ |
484 is the usual map. |
485 is the usual map. |
485 \nn{need to say this better} |
486 \nn{need to say this better} |
486 \nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
487 \nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
487 combine only two balls at a time; for $n=1$ this version will lead to usual definition |
488 combine only two balls at a time; for $n=1$ this version will lead to usual definition |
488 of $A_\infty$ category} |
489 of $A_\infty$ category} |
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490 |
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491 We will call $m$ the filtration degree of the complex. |
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492 We can think of this construction as starting with a disjoint copy of a complex for each |
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493 permissible decomposition (filtration degree 0). |
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494 Then we glue these together with mapping cylinders coming from gluing maps |
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495 (filtration degree 1). |
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496 Then we kill the extra homology we just introduced with mapping cylinder between the mapping cylinders (filtration degree 2). |
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497 And so on. |
489 |
498 |
490 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
499 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
491 |
500 |
492 It is easy to see that |
501 It is easy to see that |
493 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
502 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |