942 |
942 |
943 Note that if we think of an ordinary 1-category as an $A_\infty$ 1-category where $k$-morphisms are identities for $k>1$, |
943 Note that if we think of an ordinary 1-category as an $A_\infty$ 1-category where $k$-morphisms are identities for $k>1$, |
944 then Axiom \ref{axiom:families} implies Axiom \ref{axiom:extended-isotopies}. |
944 then Axiom \ref{axiom:families} implies Axiom \ref{axiom:extended-isotopies}. |
945 |
945 |
946 Another variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
946 Another variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
947 In fact, the alternative construction of the blob complex described in \S \ref{ss:alt-def} |
947 In fact, the alternative construction $\btc_*(X)$ of the blob complex described in \S \ref{ss:alt-def} |
948 gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom; |
948 gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom; |
949 since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across. |
949 since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across. |
950 |
950 |
951 \noop{ |
951 \noop{ |
952 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
952 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
1141 \label{ex:chains-of-maps-to-a-space} |
1141 \label{ex:chains-of-maps-to-a-space} |
1142 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 $T$. |
1142 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 $T$. |
1143 For a $k$-ball $X$, with $k < n$, the set $\pi^\infty_{\leq n}(T)(X)$ is just $\Maps(X \to T)$. |
1143 For a $k$-ball $X$, with $k < n$, the set $\pi^\infty_{\leq n}(T)(X)$ is just $\Maps(X \to T)$. |
1144 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 |
1144 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 |
1145 \[ |
1145 \[ |
1146 C_*(\Maps_c(X\times F \to T)), |
1146 C_*(\Maps_c(X \to T)), |
1147 \] |
1147 \] |
1148 where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
1148 where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
1149 and $C_*$ denotes singular chains. |
1149 and $C_*$ denotes singular chains. |
1150 Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X\times F \to T)$, |
1150 Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X \to T)$, |
1151 we get an $A_\infty$ $n$-category enriched over spaces. |
1151 we get an $A_\infty$ $n$-category enriched over spaces. |
1152 \end{example} |
1152 \end{example} |
1153 |
1153 |
1154 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
1154 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
1155 homotopy as the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
1155 homotopy as the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |