equal
deleted
inserted
replaced
1840 The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. |
1840 The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. |
1841 We use the same type of colimit construction. |
1841 We use the same type of colimit construction. |
1842 |
1842 |
1843 In our example, $\cl\cM(H) = \cD(H\times\bd W \cup \bd H\times W)$. |
1843 In our example, $\cl\cM(H) = \cD(H\times\bd W \cup \bd H\times W)$. |
1844 |
1844 |
1845 \begin{module-axiom}[Module boundaries (maps)] |
1845 \begin{module-axiom}[Module boundaries] |
1846 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cl\cM(\bd M)$. |
1846 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cl\cM(\bd M)$. |
1847 These maps, for various $M$, comprise a natural transformation of functors.} |
1847 These maps, for various $M$, comprise a natural transformation of functors.} |
1848 \end{module-axiom} |
1848 \end{module-axiom} |
1849 |
1849 |
1850 Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
1850 Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
1948 \[ |
1948 \[ |
1949 \gl_Y : \cM(M_1) \trans E \times_{\cM(Y)} \cM(M_2) \trans E \to \cM(M) \trans E |
1949 \gl_Y : \cM(M_1) \trans E \times_{\cM(Y)} \cM(M_2) \trans E \to \cM(M) \trans E |
1950 \] |
1950 \] |
1951 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
1951 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
1952 to the intersection of the boundaries of $M$ and $M_i$. |
1952 to the intersection of the boundaries of $M$ and $M_i$. |
1953 If $k < n$, |
1953 If $k < n$ we require that $\gl_Y$ is injective.} |
1954 or if $k=n$ and we are in the $A_\infty$ case, |
|
1955 we require that $\gl_Y$ is injective. |
|
1956 (For $k=n$ in the ordinary (non-$A_\infty$) case, see below.)} |
|
1957 \end{module-axiom} |
1954 \end{module-axiom} |
1958 |
1955 |
1959 |
1956 |
1960 Second, we can compose an $n$-category morphism with a module morphism to get another |
1957 Second, we can compose an $n$-category morphism with a module morphism to get another |
1961 module morphism. |
1958 module morphism. |
1972 \[ |
1969 \[ |
1973 \gl_Y :\cC(X)\trans E \times_{\cC(Y)} \cM(M')\trans E \to \cM(M) \trans E |
1970 \gl_Y :\cC(X)\trans E \times_{\cC(Y)} \cM(M')\trans E \to \cM(M) \trans E |
1974 \] |
1971 \] |
1975 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
1972 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
1976 to the intersection of the boundaries of $X$ and $M'$. |
1973 to the intersection of the boundaries of $X$ and $M'$. |
1977 If $k < n$, |
1974 If $k < n$ we require that $\gl_Y$ is injective.} |
1978 or if $k=n$ and we are in the $A_\infty$ case, |
|
1979 we require that $\gl_Y$ is injective. |
|
1980 (For $k=n$ in the ordinary (non-$A_\infty$) case, see below.)} |
|
1981 \end{module-axiom} |
1975 \end{module-axiom} |
1982 |
1976 |
1983 \begin{module-axiom}[Strict associativity] |
1977 \begin{module-axiom}[Strict associativity] |
1984 The composition and action maps above are strictly associative. |
1978 The composition and action maps above are strictly associative. |
1985 Given any decomposition of a large marked ball into smaller marked and unmarked balls |
1979 Given any decomposition of a large marked ball into smaller marked and unmarked balls |