482 \label{ex:maps-to-a-space}% |
482 \label{ex:maps-to-a-space}% |
483 Fix a `target space' $T$, any topological space. We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
483 Fix a `target space' $T$, any topological space. We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
484 For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
484 For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
485 all continuous maps from $X$ to $T$. |
485 all continuous maps from $X$ to $T$. |
486 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
486 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
487 homotopies fixed on $\bd X \times F$. |
487 homotopies fixed on $\bd X$. |
488 (Note that homotopy invariance implies isotopy invariance.) |
488 (Note that homotopy invariance implies isotopy invariance.) |
489 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
489 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
490 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
490 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
491 \end{example} |
491 \end{example} |
492 |
492 |
493 \begin{example}[Maps to a space, with a fiber] |
493 \begin{example}[Maps to a space, with a fiber] |
494 \rm |
494 \rm |
495 \label{ex:maps-to-a-space-with-a-fiber}% |
495 \label{ex:maps-to-a-space-with-a-fiber}% |
496 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. |
496 We can modify the example above, by fixing a |
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497 closed $m$-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. |
497 \end{example} |
498 \end{example} |
498 |
499 |
499 \begin{example}[Linearized, twisted, maps to a space] |
500 \begin{example}[Linearized, twisted, maps to a space] |
500 \rm |
501 \rm |
501 \label{ex:linearized-maps-to-a-space}% |
502 \label{ex:linearized-maps-to-a-space}% |
534 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
535 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
535 \newcommand{\Bord}{\operatorname{Bord}} |
536 \newcommand{\Bord}{\operatorname{Bord}} |
536 \begin{example}[The bordism $n$-category] |
537 \begin{example}[The bordism $n$-category] |
537 \rm |
538 \rm |
538 \label{ex:bordism-category} |
539 \label{ex:bordism-category} |
539 For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
540 For a $k$-ball or $k$-sphere $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
540 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
541 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
541 to $\bd X$. \nn{spheres} |
542 to $\bd X$. |
542 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such submanifolds; |
543 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such submanifolds; |
543 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
544 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
544 $W \to W'$ which restricts to the identity on the boundary |
545 $W \to W'$ which restricts to the identity on the boundary. |
545 \end{example} |
546 \end{example} |
546 |
547 |
547 \begin{itemize} |
548 %\nn{the next example might be an unnecessary distraction. consider deleting it.} |
548 |
549 |
549 \item \nn{Continue converting these into examples} |
550 %\begin{example}[Variation on the above examples] |
550 |
551 %We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
551 \item |
552 %for example product boundary conditions or take the union over all boundary conditions. |
552 \item Variation on the above examples: |
553 %%\nn{maybe should not emphasize this case, since it's ``better" in some sense |
553 We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
554 %%to think of these guys as affording a representation |
554 for example product boundary conditions or take the union over all boundary conditions. |
555 %%of the $n{+}1$-category associated to $\bd F$.} |
555 %\nn{maybe should not emphasize this case, since it's ``better" in some sense |
556 %\end{example} |
556 %to think of these guys as affording a representation |
|
557 %of the $n{+}1$-category associated to $\bd F$.} |
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558 |
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559 \end{itemize} |
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560 |
557 |
561 |
558 |
562 We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
559 We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
563 |
560 |
564 \begin{example}[Chains of maps to a space] |
561 \begin{example}[Chains of maps to a space] |
622 \label{partofJfig} |
619 \label{partofJfig} |
623 \end{figure} |
620 \end{figure} |
624 |
621 |
625 |
622 |
626 |
623 |
|
624 \nn{resume revising here} |
627 |
625 |
628 An $n$-category $\cC$ determines |
626 An $n$-category $\cC$ determines |
629 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 |
630 (possibly with additional structure if $k=n$). |
628 (possibly with additional structure if $k=n$). |
631 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 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. |