579 \nn{maybe should also mention version where we enrich over spaces rather than chain complexes} |
579 \nn{maybe should also mention version where we enrich over spaces rather than chain complexes} |
580 \end{example} |
580 \end{example} |
581 |
581 |
582 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
582 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
583 |
583 |
584 \todo{Yikes! We can't do this yet, because we haven't explained how to produce a system of fields and local relations from a topological $n$-category.} |
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585 |
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586 \begin{example}[Blob complexes of balls (with a fiber)] |
584 \begin{example}[Blob complexes of balls (with a fiber)] |
587 \rm |
585 \rm |
588 \label{ex:blob-complexes-of-balls} |
586 \label{ex:blob-complexes-of-balls} |
589 Fix an $m$-dimensional manifold $F$. We will define an $A_\infty$ $(n-m)$-category $\cC$. |
587 Fix an $m$-dimensional manifold $F$ and system of fields $\cE$. |
590 Given a plain $n$-category $C$, |
588 We will define an $A_\infty$ $(n-m)$-category $\cC$. |
591 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, |
589 When $X$ is a $k$-ball or $k$-sphere, with $k<n-m$, define $\cC(X) = \cE(X\times F)$. |
592 define $\cC(X; c) = \bc^C_*(X\times F; c)$ |
590 When $X$ is an $(n-m)$-ball, |
593 where $\bc^C_*$ denotes the blob complex based on $C$. |
591 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
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592 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
594 \end{example} |
593 \end{example} |
595 |
594 |
596 This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. Notice that with $F$ a point, the above example is a construction turning a topological $n$-category into an $A_\infty$ $n$-category. We think of this as providing a `free resolution' of the topological $n$-category. \todo{Say more here!} In fact, there is also a trivial way to do this: we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, and take $\CD{B}$ to act trivially. |
595 This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. Notice that with $F$ a point, the above example is a construction turning a topological $n$-category into an $A_\infty$ $n$-category. We think of this as providing a `free resolution' of the topological $n$-category. \todo{Say more here!} In fact, there is also a trivial way to do this: we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, and take $\CD{B}$ to act trivially. |
597 |
596 |
598 Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. It's easy to see that with $n=0$, the corresponding system of fields is just linear combinations of connected components of $T$, and the local relations are trivial. There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
597 Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. It's easy to see that with $n=0$, the corresponding system of fields is just linear combinations of connected components of $T$, and the local relations are trivial. There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |