570 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 $T$. |
570 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 $T$. |
571 For $k$-balls and $k$-spheres $X$, with $k < n$, the sets $\pi^\infty_{\leq n}(T)(X)$ are just $\Maps{X \to T}$. |
571 For $k$-balls and $k$-spheres $X$, with $k < n$, the sets $\pi^\infty_{\leq n}(T)(X)$ are just $\Maps{X \to T}$. |
572 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 |
572 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 |
573 $$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
573 $$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
574 and $C_*$ denotes singular chains. |
574 and $C_*$ denotes singular chains. |
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575 \nn{maybe should also mention version where we enrich over spaces rather than chain complexes} |
575 \end{example} |
576 \end{example} |
576 |
577 |
577 See ??? below, recovering $C_*(\Maps{M \to T})$ as (up to homotopy) the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
578 See \ref{thm:map-recon} below, recovering $C_*(\Maps{M \to T})$ as (up to homotopy) the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
578 |
579 |
579 \begin{example}[Blob complexes of balls (with a fiber)] |
580 \begin{example}[Blob complexes of balls (with a fiber)] |
580 \rm |
581 \rm |
581 \label{ex:blob-complexes-of-balls} |
582 \label{ex:blob-complexes-of-balls} |
582 Fix an $m$-dimensional manifold $F$. |
583 Fix an $m$-dimensional manifold $F$. |