669 (restriction maps, gluing, product morphisms, action of homeomorphisms) is usually obvious. |
669 (restriction maps, gluing, product morphisms, action of homeomorphisms) is usually obvious. |
670 |
670 |
671 \begin{example}[Maps to a space] |
671 \begin{example}[Maps to a space] |
672 \rm |
672 \rm |
673 \label{ex:maps-to-a-space}% |
673 \label{ex:maps-to-a-space}% |
674 Let $T$be a topological space. |
674 Let $T$ be a topological space. |
675 We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
675 We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
676 For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
676 For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
677 all continuous maps from $X$ to $T$. |
677 all continuous maps from $X$ to $T$. |
678 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
678 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
679 homotopies fixed on $\bd X$. |
679 homotopies fixed on $\bd X$. |
711 $h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
711 $h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
712 (In order for this to be well-defined we must choose $\alpha$ to be zero on degenerate simplices. |
712 (In order for this to be well-defined we must choose $\alpha$ to be zero on degenerate simplices. |
713 Alternatively, we could equip the balls with fundamental classes.) |
713 Alternatively, we could equip the balls with fundamental classes.) |
714 \end{example} |
714 \end{example} |
715 |
715 |
716 The next example is only intended to be illustrative, as we don't specify which definition of a ``traditional $n$-category" we intend. |
716 \begin{example}[$n$-categories from TQFTs] |
717 Further, most of these definitions don't even have an agreed-upon notion of ``strong duality", which we assume here. |
717 \rm |
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718 \label{ex:ncats-from-tqfts}% |
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719 Let $\cF$ be a TQFT in the sense of \S\ref{sec:fields}: an $n$-dimensional |
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720 system of fields (also denoted $\cF$) and local relations. |
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721 Let $W$ be an $n{-}j$-manifold. |
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722 Define the $j$-category $\cF(W)$ as follows. |
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723 If $X$ is a $k$-ball with $k<j$, let $\cF(W)(X) \deq \cF(W\times X)$. |
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724 If $X$ is a $j$-ball and $c\in \cl{\cF(W)}(\bd X)$, |
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725 let $\cF(W)(X; c) \deq A_\cF(W\times X; c)$. |
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726 \end{example} |
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727 |
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728 The next example is only intended to be illustrative, as we don't specify |
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729 which definition of a ``traditional $n$-category" we intend. |
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730 Further, most of these definitions don't even have an agreed-upon notion of |
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731 ``strong duality", which we assume here. |
718 \begin{example}[Traditional $n$-categories] |
732 \begin{example}[Traditional $n$-categories] |
719 \rm |
733 \rm |
720 \label{ex:traditional-n-categories} |
734 \label{ex:traditional-n-categories} |
721 Given a ``traditional $n$-category with strong duality" $C$ |
735 Given a ``traditional $n$-category with strong duality" $C$ |
722 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
736 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
1386 \medskip |
1400 \medskip |
1387 |
1401 |
1388 We now give some examples of modules over topological and $A_\infty$ $n$-categories. |
1402 We now give some examples of modules over topological and $A_\infty$ $n$-categories. |
1389 |
1403 |
1390 \begin{example}[Examples from TQFTs] |
1404 \begin{example}[Examples from TQFTs] |
1391 \nn{need to add corresponding ncat example} |
1405 \rm |
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1406 Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold, |
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1407 and $\cF(W)$ the $j$-category associated to $W$. |
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1408 Let $Y$ be an $(n{-}j{+}1)$-manifold with $\bd Y = W$. |
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1409 Define a $\cF(W)$ module $\cF(Y)$ as follows. |
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1410 If $M = (B, N)$ is a marked $k$-ball with $k<j$ let |
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1411 $\cF(Y)(M)\deq \cF((B\times W) \cup (N\times Y))$. |
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1412 If $M = (B, N)$ is a marked $j$-ball and $c\in \cl{\cF(Y)}(\bd M)$ let |
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1413 $\cF(Y)(M)\deq A_\cF((B\times W) \cup (N\times Y); c)$. |
1392 \end{example} |
1414 \end{example} |
1393 |
1415 |
1394 \begin{example} |
1416 \begin{example} |
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1417 \rm |
1395 Suppose $S$ is a topological space, with a subspace $T$. |
1418 Suppose $S$ is a topological space, with a subspace $T$. |
1396 We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ |
1419 We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ |
1397 for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs |
1420 for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs |
1398 $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all |
1421 $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all |
1399 such maps modulo homotopies fixed on $\bdy B \setminus N$. |
1422 such maps modulo homotopies fixed on $\bdy B \setminus N$. |