445 |
450 |
446 \subsection{Examples of $n$-categories} |
451 \subsection{Examples of $n$-categories} |
447 |
452 |
448 \nn{these examples need to be fleshed out a bit more} |
453 \nn{these examples need to be fleshed out a bit more} |
449 |
454 |
450 We know describe several classes of examples of $n$-categories satisfying our axioms. |
455 We now describe several classes of examples of $n$-categories satisfying our axioms. |
451 |
456 |
452 \begin{example}{Maps to a space} |
457 \begin{example}[Maps to a space] |
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458 \rm |
453 \label{ex:maps-to-a-space}% |
459 \label{ex:maps-to-a-space}% |
454 Fix $F$ a closed $m$-manifold (keep in mind the case where $F$ is a point). Fix a `target space' $T$, any topological space. |
460 Fix a `target space' $T$, any topological space. We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
455 For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\cC(X)$ to be the set of |
461 For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
456 all maps from $X\times F$ to $T$. |
462 all continuous maps from $X$ to $T$. |
457 For $X$ an $n$-ball define $\cC(X)$ to be maps from $X\times F$ to $T$ modulo |
463 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
458 homotopies fixed on $\bd X \times F$. |
464 homotopies fixed on $\bd X \times F$. |
459 (Note that homotopy invariance implies isotopy invariance.) |
465 (Note that homotopy invariance implies isotopy invariance.) |
460 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
466 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
461 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
467 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
462 \end{example} |
468 \end{example} |
463 |
469 |
464 \begin{example}{Linearized, twisted, maps to a space} |
470 \begin{example}[Maps to a space, with a fiber] |
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471 \rm |
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472 \label{ex:maps-to-a-space-with-a-fiber}% |
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473 We can modify the example above, by fixing an $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. Taking $F$ to be a point recovers the previous case. |
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474 \end{example} |
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475 |
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476 \begin{example}[Linearized, twisted, maps to a space] |
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477 \rm |
465 \label{ex:linearized-maps-to-a-space}% |
478 \label{ex:linearized-maps-to-a-space}% |
466 We can linearize the above example as follows. |
479 We can linearize Examples \ref{ex:maps-to-a-space} and \ref{ex:maps-to-a-space-with-a-fiber} as follows. |
467 Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$ |
480 Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$ |
468 (e.g.\ the trivial cocycle). |
481 (have in mind the trivial cocycle). |
469 For $X$ of dimension less than $n$ define $\cC(X)$ as before. |
482 For $X$ of dimension less than $n$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X)$ as before, ignoring $\alpha$. |
470 For $X$ an $n$-ball and $c\in \cC(\bd X)$ define $\cC(X; c)$ to be |
483 For $X$ an $n$-ball and $c\in \Maps(\bdy X \times F \to T)$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X; c)$ to be |
471 the $R$-module of finite linear combinations of maps from $X\times F$ to $T$, |
484 the $R$-module of finite linear combinations of continuous maps from $X\times F$ to $T$, |
472 modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
485 modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
473 $h: X\times F\times I \to T$, then $a \sim \alpha(h)b$. |
486 $h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
474 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
487 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
475 \end{example} |
488 \end{example} |
476 |
489 |
477 \begin{itemize} |
490 The next example is only intended to be illustrative, as we don't specify which definition of a `traditional $n$-category' we intend. Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here. |
478 |
491 \begin{example}[Traditional $n$-categories] |
479 \item \nn{Continue converting these into examples} |
492 \rm |
480 |
493 \label{ex:traditional-n-categories} |
481 \item Given a traditional $n$-category $C$ (with strong duality etc.), |
494 Given a `traditional $n$-category with strong duality' $C$ |
482 define $\cC(X)$ (with $\dim(X) < n$) |
495 define $\cC(X)$, for $X$ a $k$-ball or $k$-sphere with $k < n$, |
483 to be the set of all $C$-labeled sub cell complexes of $X$. |
496 to be the set of all $C$-labeled sub cell complexes of $X$. |
484 (See Subsection \ref{sec:fields}.) |
497 (See Subsection \ref{sec:fields}.) |
485 For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
498 For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
486 combinations of $C$-labeled sub cell complexes of $X$ |
499 combinations of $C$-labeled sub cell complexes of $X$ |
487 modulo the kernel of the evaluation map. |
500 modulo the kernel of the evaluation map. |
488 Define a product morphism $a\times D$ to be the product of the cell complex of $a$ with $D$, |
501 Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
489 and with the same labeling as $a$. |
502 with each cell labelled by the $m$-th iterated identity morphism of the corresponding cell for $a$. |
490 More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
503 More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
491 Define $\cC(X)$, for $\dim(X) < n$, |
504 Define $\cC(X)$, for $\dim(X) < n$, |
492 to be the set of all $C$-labeled sub cell complexes of $X\times F$. |
505 to be the set of all $C$-labeled sub cell complexes of $X\times F$. |
493 Define $\cC(X; c)$, for $X$ an $n$-ball, |
506 Define $\cC(X; c)$, for $X$ an $n$-ball, |
494 to be the dual Hilbert space $A(X\times F; c)$. |
507 to be the dual Hilbert space $A(X\times F; c)$. |
495 \nn{refer elsewhere for details?} |
508 \nn{refer elsewhere for details?} |
496 |
509 \end{example} |
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510 |
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511 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
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512 \newcommand{\Bord}{\operatorname{Bord}} |
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513 \begin{example}[The bordism $n$-category] |
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514 \rm |
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515 \label{ex:bordism-category} |
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516 For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
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517 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
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518 to $\bd X$. \nn{spheres} |
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519 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such submanifolds; |
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520 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
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521 $W \to W'$ which restricts to the identity on the boundary |
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522 \end{example} |
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523 |
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524 \begin{itemize} |
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525 |
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526 \item \nn{Continue converting these into examples} |
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527 |
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528 \item |
497 \item Variation on the above examples: |
529 \item Variation on the above examples: |
498 We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
530 We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
499 for example product boundary conditions or take the union over all boundary conditions. |
531 for example product boundary conditions or take the union over all boundary conditions. |
500 %\nn{maybe should not emphasize this case, since it's ``better" in some sense |
532 %\nn{maybe should not emphasize this case, since it's ``better" in some sense |
501 %to think of these guys as affording a representation |
533 %to think of these guys as affording a representation |
502 %of the $n{+}1$-category associated to $\bd F$.} |
534 %of the $n{+}1$-category associated to $\bd F$.} |
503 |
535 |
504 \item Here's our version of the bordism $n$-category. |
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505 For a $k$-ball $X$, $k<n$, define $\cC(X)$ to be the set of all $k$-dimensional |
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506 submanifolds $W$ of $X\times \r^\infty$ such that the projection $W \to X$ is transverse |
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507 to $\bd X$. |
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508 For $k=n$ define $\cC(X)$ to be homeomorphism classes (rel boundary) of such submanifolds; |
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509 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
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510 $W\to W'$ which restricts to the identity on the boundary. |
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511 |
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512 \item \nn{sphere modules; ref to below} |
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513 |
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514 \end{itemize} |
536 \end{itemize} |
515 |
537 |
516 |
538 |
517 We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
539 We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
518 |
540 |
519 \begin{example}{Chains of maps to a space} |
541 \begin{example}[Chains of maps to a space] |
520 We can modify Example \ref{ex:maps-to-a-space} above by defining $\cC(X; c)$ for an $n$-ball $X$ to be the chain complex |
542 \rm |
521 $C_*(\Maps_c(X\times F \to T))$, where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
543 \label{ex:chains-of-maps-to-a-space} |
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544 We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$. |
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545 For $k$-balls and $k$-spheres $X$, with $k < n$, the sets $\pi^\infty_{\leq n}(T)(X)$ are just $\Maps{X \to T}$. |
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546 Define $\pi^\infty_{\leq n}(T)(X; c)$ for an $n$-ball $X$ and $c \in \pi^\infty_{\leq n}(T)(\bdy X)$ to be the chain complex |
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547 $$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
522 and $C_*$ denotes singular chains. |
548 and $C_*$ denotes singular chains. |
523 \end{example} |
549 \end{example} |
524 |
550 |
525 \begin{example}{Blob complexes of balls (with a fiber)} |
551 See ??? below, recovering $C_*(\Maps{M \to T})$ as (up to homotopy) the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
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552 |
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553 \begin{example}[Blob complexes of balls (with a fiber)] |
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554 \rm |
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555 \label{ex:blob-complexes-of-balls} |
526 Fix an $m$-dimensional manifold $F$. |
556 Fix an $m$-dimensional manifold $F$. |
527 Given a plain $n$-category $C$, |
557 Given a plain $n$-category $C$, |
528 when $X$ is a $k$-ball or $k$-sphere, with $k<n-m$, define $\cC(X) = C(X)$. When $X$ is an $(n-m)$-ball, |
558 when $X$ is a $k$-ball or $k$-sphere, with $k<n-m$, define $\cC(X) = C(X)$. When $X$ is an $(n-m)$-ball, |
529 define $\cC(X; c) = \bc^C_*(X\times F; c)$ |
559 define $\cC(X; c) = \bc^C_*(X\times F; c)$ |
530 where $\bc^C_*$ denotes the blob complex based on $C$. |
560 where $\bc^C_*$ denotes the blob complex based on $C$. |
531 \end{example} |
561 \end{example} |
532 |
562 |
533 \begin{defn} |
563 This example will be essential for ???, which relates ... |
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564 |
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565 \begin{example} |
534 \nn{should add $\infty$ version of bordism $n$-cat} |
566 \nn{should add $\infty$ version of bordism $n$-cat} |
535 \end{defn} |
567 \end{example} |
536 |
568 |
537 |
569 |
538 |
570 |
539 |
571 |
540 |
572 |
541 |
573 |
542 \subsection{From $n$-categories to systems of fields} |
574 \subsection{From $n$-categories to systems of fields} |
543 \label{ss:ncat_fields} |
575 \label{ss:ncat_fields} |
544 |
576 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. |
545 We can extend the functors $\cC$ above from $k$-balls to arbitrary $k$-manifolds as follows. |
577 |
546 |
578 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). |
547 Let $W$ be a $k$-manifold, $1\le k \le n$. |
579 |
548 We will define a set $\cC(W)$. |
580 \begin{defn} |
549 (If $k = n$ and our $k$-categories are enriched, then |
581 Say that a `permissible decomposition' of $W$ is a cell decomposition |
550 $\cC(W)$ will have additional structure; see below.) |
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551 $\cC(W)$ will be the colimit of a functor defined on a category $\cJ(W)$, |
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552 which we define next. |
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553 |
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554 Define a permissible decomposition of $W$ to be a cell decomposition |
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555 \[ |
582 \[ |
556 W = \bigcup_a X_a , |
583 W = \bigcup_a X_a , |
557 \] |
584 \] |
558 where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
585 where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
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586 |
559 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
587 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
560 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
588 of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$. |
561 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
589 |
562 (The objects of $\cJ(W)$ are permissible decompositions of $W$, and there is a unique |
590 The category $\cJ(W)$ has objects the permissible decompositions of $W$, and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
563 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
591 See Figure \ref{partofJfig} for an example. |
564 See Figure \ref{partofJfig}.) |
592 \end{defn} |
565 |
593 |
566 \begin{figure}[!ht] |
594 \begin{figure}[!ht] |
567 \begin{equation*} |
595 \begin{equation*} |
568 \mathfig{.63}{tempkw/zz2} |
596 \mathfig{.63}{tempkw/zz2} |
569 \end{equation*} |
597 \end{equation*} |
570 \caption{A small part of $\cJ(W)$} |
598 \caption{A small part of $\cJ(W)$} |
571 \label{partofJfig} |
599 \label{partofJfig} |
572 \end{figure} |
600 \end{figure} |
573 |
601 |
574 |
602 |
575 $\cC$ determines |
603 |
576 a functor $\psi_\cC$ from $\cJ(W)$ to the category of sets |
604 |
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605 An $n$-category $\cC$ determines |
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606 a functor $\psi_{\cC;W}$ from $\cJ(W)$ to the category of sets |
577 (possibly with additional structure if $k=n$). |
607 (possibly with additional structure if $k=n$). |
578 For a decomposition $x = (X_a)$ in $\cJ(W)$, define $\psi_\cC(x)$ to be the subset |
608 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. |
579 \[ |
609 |
580 \psi_\cC(x) \sub \prod_a \cC(X_a) |
610 \begin{defn} |
581 \] |
611 Define the functor $\psi_{\cC;W} : \cJ(W) \to \Set$ as follows. |
582 such that the restrictions to the various pieces of shared boundaries amongst the |
612 For a decomposition $x = \bigcup_a X_a$ in $\cJ(W)$, $\psi_{\cC;W}(x)$ is the subset |
583 $X_a$ all agree. |
613 \begin{equation} |
584 (Think fibered product.) |
614 \label{eq:psi-C} |
585 If $x$ is a refinement of $y$, define a map $\psi_\cC(x)\to\psi_\cC(y)$ |
615 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)_{\bdy X_a} |
586 via the composition maps of $\cC$. |
616 \end{equation} |
587 (If $\dim(W) = n$ then we need to also make use of the monoidal |
617 where the restrictions to the various pieces of shared boundaries amongst the cells |
588 product in the enriching category. |
618 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). |
589 \nn{should probably be more explicit here}) |
619 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 $\cC$. |
590 |
620 \end{defn} |
591 Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$. |
621 |
592 When $k<n$ or $k=n$ and we are in the plain (non-$A_\infty$) case, this means that |
622 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 $A$ |
593 for each decomposition $x$ there is a map |
623 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.) |
594 $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps |
624 |
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625 Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
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626 |
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627 \begin{defn}[System of fields functor] |
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628 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cC(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
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629 That is, for each decomposition $x$ there is a map |
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630 $\psi_{\cC;W}(x)\to \cC(W)$, these maps are compatible with the refinement maps |
595 above, and $\cC(W)$ is universal with respect to these properties. |
631 above, and $\cC(W)$ is universal with respect to these properties. |
596 When $k=n$ and we are in the $A_\infty$ case, it means |
632 \end{defn} |
597 homotopy colimit. |
633 |
598 |
634 \begin{defn}[System of fields functor, $A_\infty$ case] |
599 More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take |
635 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 $\psi_{\cC;W}$. |
600 \[ |
636 \end{defn} |
601 \cC(W) = \left( \oplus_x \psi_\cC(x)\right) \big/ K |
637 |
602 \] |
638 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 $W$. |
603 where $K$ is generated by all things of the form $a - g(a)$, where |
639 |
604 $a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x) |
640 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 $W$ |
605 \to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$. |
641 \begin{equation*} |
606 |
642 \cC(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K |
607 In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit |
643 \end{equation*} |
608 is as follows. |
644 where $K$ is the vector space spanned by elements $a - g(a)$, with |
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645 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
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646 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
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647 |
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648 In the $A_\infty$ case enriched over chain complexes, the concrete description of the homotopy colimit |
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649 is slightly more involved. |
609 %\nn{should probably rewrite this to be compatible with some standard reference} |
650 %\nn{should probably rewrite this to be compatible with some standard reference} |
610 Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions. |
651 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
611 Such sequences (for all $m$) form a simplicial set. |
652 Such sequences (for all $m$) form a simplicial set in $\cJ(W)$. |
612 Let |
653 Define $V$ as a vector space via |
613 \[ |
654 \[ |
614 V = \bigoplus_{(x_i)} \psi_\cC(x_0) , |
655 V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
615 \] |
656 \] |
616 where the sum is over all $m$-sequences and all $m$, and each summand is degree shifted by $m$. |
657 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. (Our homological conventions are obtuse: if a complex $U$ is concentrated in degree $0$, the complex $U[m]$ is concentrated in degree $m$.) |
617 We endow $V$ with a differential which is the sum of the differential of the $\psi_\cC(x_0)$ |
658 We endow $V$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
618 summands plus another term using the differential of the simplicial set of $m$-sequences. |
659 summands plus another term using the differential of the simplicial set of $m$-sequences. |
619 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
660 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
620 summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
661 summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
621 \[ |
662 \[ |
622 \bd (a, \bar{x}) = (\bd a, \bar{x}) \pm (g(a), d_0(\bar{x})) + \sum_{j=1}^k \pm (a, d_j(\bar{x})) , |
663 \bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
623 \] |
664 \] |
624 where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
665 where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
625 is the usual map. |
666 is the usual gluing map coming from the antirefinement $x_0 < x_1$. |
626 \nn{need to say this better} |
667 \nn{need to say this better} |
627 \nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
668 \nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
628 combine only two balls at a time; for $n=1$ this version will lead to usual definition |
669 combine only two balls at a time; for $n=1$ this version will lead to usual definition |
629 of $A_\infty$ category} |
670 of $A_\infty$ category} |
630 |
671 |