591 |
591 |
592 |
592 |
593 |
593 |
594 \subsection{From $n$-categories to systems of fields} |
594 \subsection{From $n$-categories to systems of fields} |
595 \label{ss:ncat_fields} |
595 \label{ss:ncat_fields} |
596 In this section we describe how to extend an $n$-category as described above (of either the plain or $A_\infty$ variation) to a system of fields. That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension, from $k$-balls and $k$-spheres to arbitrary $k$-manifolds. |
596 In this section we describe how to extend an $n$-category as described above (of either the plain or $A_\infty$ variety) to a system of fields. That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension, from $k$-balls and $k$-spheres to arbitrary $k$-manifolds. |
597 |
597 |
598 We will first define the `cell-decomposition' poset $\cJ(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. An $n$-category $\cC$ provides a functor from this poset to the category of sets, and we will define $\cC(W)$ as a suitable colimit (or homotopy colimit) of this functor. We'll later give a more explicit description of this colimit. In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complex to $n$-manifolds with boundary data), then the resulting system of fields is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). |
598 We will first define the `cell-decomposition' poset $\cJ(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
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599 An $n$-category $\cC$ provides a functor from this poset to the category of sets, and we will define $\cC(W)$ as a suitable colimit (or homotopy colimit in the $A_\infty$ case) of this functor. |
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600 We'll later give a more explicit description of this colimit. In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), then the resulting system of fields is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). |
599 |
601 |
600 \begin{defn} |
602 \begin{defn} |
601 Say that a `permissible decomposition' of $W$ is a cell decomposition |
603 Say that a `permissible decomposition' of $W$ is a cell decomposition |
602 \[ |
604 \[ |
603 W = \bigcup_a X_a , |
605 W = \bigcup_a X_a , |
619 \label{partofJfig} |
621 \label{partofJfig} |
620 \end{figure} |
622 \end{figure} |
621 |
623 |
622 |
624 |
623 |
625 |
624 \nn{resume revising here} |
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625 |
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626 An $n$-category $\cC$ determines |
626 An $n$-category $\cC$ determines |
627 a functor $\psi_{\cC;W}$ from $\cJ(W)$ to the category of sets |
627 a functor $\psi_{\cC;W}$ from $\cJ(W)$ to the category of sets |
628 (possibly with additional structure if $k=n$). |
628 (possibly with additional structure if $k=n$). |
629 For a $k$-cell $X$ in a cell composition of $W$, we can consider the `splittable fields' $\cC(X)_{\bdy X}$, the subset of $\cC(X)$ consisting of fields which are splittable with respect to each boundary $k-1$-cell. |
629 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
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630 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
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631 are splittable along this decomposition. |
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632 %For a $k$-cell $X$ in a cell composition of $W$, we can consider the `splittable fields' $\cC(X)_{\bdy X}$, the subset of $\cC(X)$ consisting of fields which are splittable with respect to each boundary $k-1$-cell. |
630 |
633 |
631 \begin{defn} |
634 \begin{defn} |
632 Define the functor $\psi_{\cC;W} : \cJ(W) \to \Set$ as follows. |
635 Define the functor $\psi_{\cC;W} : \cJ(W) \to \Set$ as follows. |
633 For a decomposition $x = \bigcup_a X_a$ in $\cJ(W)$, $\psi_{\cC;W}(x)$ is the subset |
636 For a decomposition $x = \bigcup_a X_a$ in $\cJ(W)$, $\psi_{\cC;W}(x)$ is the subset |
634 \begin{equation} |
637 \begin{equation} |
635 \label{eq:psi-C} |
638 \label{eq:psi-C} |
636 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)_{\bdy X_a} |
639 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
637 \end{equation} |
640 \end{equation} |
638 where the restrictions to the various pieces of shared boundaries amongst the cells |
641 where the restrictions to the various pieces of shared boundaries amongst the cells |
639 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). |
642 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). |
640 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
643 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
641 \end{defn} |
644 \end{defn} |
642 |
645 |
643 When the $n$-category $\cC$ is enriched in some monoidal category $(A,\boxtimes)$, and $W$ is an $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ |
646 When the $n$-category $\cC$ is enriched in some monoidal category $(A,\boxtimes)$, and $W$ is a |
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647 closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and |
644 we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$. (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.) |
648 we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$. (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.) |
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649 Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we |
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650 fix a field on $\bd W$ |
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651 (i.e. fix an element of the colimit associated to $\bd W$). |
645 |
652 |
646 Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
653 Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
647 |
654 |
648 \begin{defn}[System of fields functor] |
655 \begin{defn}[System of fields functor] |
649 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cC(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
656 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cC(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
656 When $\cC$ is an $A_\infty$ $n$-category, $\cC(W)$ for $W$ a $k$-manifold with $k < n$ is defined as above, as the colimit of $\psi_{\cC;W}$. When $W$ is an $n$-manifold, the chain complex $\cC(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
663 When $\cC$ is an $A_\infty$ $n$-category, $\cC(W)$ for $W$ a $k$-manifold with $k < n$ is defined as above, as the colimit of $\psi_{\cC;W}$. When $W$ is an $n$-manifold, the chain complex $\cC(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
657 \end{defn} |
664 \end{defn} |
658 |
665 |
659 We can specify boundary data $c \in \cC(\bdy W)$, and define functors $\psi_{\cC;W,c}$ with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
666 We can specify boundary data $c \in \cC(\bdy W)$, and define functors $\psi_{\cC;W,c}$ with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
660 |
667 |
661 We can now give a more concrete description of the colimit in each case. If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold, we can take the vector space $\cC(W,c)$ to be the direct sum over all permissible decompositions of $W$ |
668 We now give a more concrete description of the colimit in each case. If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold, we can take the vector space $\cC(W,c)$ to be the direct sum over all permissible decompositions of $W$ |
662 \begin{equation*} |
669 \begin{equation*} |
663 \cC(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K |
670 \cC(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K |
664 \end{equation*} |
671 \end{equation*} |
665 where $K$ is the vector space spanned by elements $a - g(a)$, with |
672 where $K$ is the vector space spanned by elements $a - g(a)$, with |
666 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
673 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
667 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
674 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
668 |
675 |
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676 \nn{ ** resume revising here} |
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677 |
669 In the $A_\infty$ case enriched over chain complexes, the concrete description of the homotopy colimit |
678 In the $A_\infty$ case enriched over chain complexes, the concrete description of the homotopy colimit |
670 is slightly more involved. |
679 is more involved. |
671 %\nn{should probably rewrite this to be compatible with some standard reference} |
680 %\nn{should probably rewrite this to be compatible with some standard reference} |
672 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
681 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
673 Such sequences (for all $m$) form a simplicial set in $\cJ(W)$. |
682 Such sequences (for all $m$) form a simplicial set in $\cJ(W)$. |
674 Define $V$ as a vector space via |
683 Define $V$ as a vector space via |
675 \[ |
684 \[ |