43 Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
43 Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
44 to the standard $k$-ball. |
44 to the standard $k$-ball. |
45 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
45 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
46 standard $k$-ball. |
46 standard $k$-ball. |
47 We {\it do not} assume that it is equipped with a |
47 We {\it do not} assume that it is equipped with a |
48 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. \nn{List the axiom numbers here, mentioning alternate versions, and also the same in the module section.} |
48 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
|
49 |
|
50 The axioms for an $n$-category are spread throughout this section. |
|
51 Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product} and \ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. |
|
52 |
49 |
53 |
50 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
54 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
51 the boundary), we want a corresponding |
55 the boundary), we want a corresponding |
52 bijection of sets $f:\cC(X)\to \cC(Y)$. |
56 bijection of sets $f:\cC(X)\to \cC(Y)$. |
53 (This will imply ``strong duality", among other things.) Putting these together, we have |
57 (This will imply ``strong duality", among other things.) Putting these together, we have |
216 (For example, vertical and horizontal composition of 2-morphisms.) |
220 (For example, vertical and horizontal composition of 2-morphisms.) |
217 In the presence of strong duality, these $k$ distinct compositions are subsumed into |
221 In the presence of strong duality, these $k$ distinct compositions are subsumed into |
218 one general type of composition which can be in any ``direction". |
222 one general type of composition which can be in any ``direction". |
219 |
223 |
220 \begin{axiom}[Composition] |
224 \begin{axiom}[Composition] |
|
225 \label{axiom:composition} |
221 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
226 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
222 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
227 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
223 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
228 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
224 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
229 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
225 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
230 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
465 same (traditional) $i$-morphism as the corresponding codimension $i$ cell $c$. |
470 same (traditional) $i$-morphism as the corresponding codimension $i$ cell $c$. |
466 |
471 |
467 |
472 |
468 %\addtocounter{axiom}{-1} |
473 %\addtocounter{axiom}{-1} |
469 \begin{axiom}[Product (identity) morphisms] |
474 \begin{axiom}[Product (identity) morphisms] |
|
475 \label{axiom:product} |
470 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
476 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
471 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
477 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
472 These maps must satisfy the following conditions. |
478 These maps must satisfy the following conditions. |
473 \begin{enumerate} |
479 \begin{enumerate} |
474 \item |
480 \item |
610 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
616 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
611 |
617 |
612 |
618 |
613 %\addtocounter{axiom}{-1} |
619 %\addtocounter{axiom}{-1} |
614 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
620 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
|
621 \label{axiom:families} |
615 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
622 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
616 \[ |
623 \[ |
617 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
624 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
618 \] |
625 \] |
619 These action maps are required to be associative up to homotopy, |
626 These action maps are required to be associative up to homotopy, |
1829 %For the time being, let's say they are.} |
1836 %For the time being, let's say they are.} |
1830 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, |
1837 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, |
1831 where $B^j$ is the standard $j$-ball. |
1838 where $B^j$ is the standard $j$-ball. |
1832 A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either |
1839 A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either |
1833 (a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls. |
1840 (a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls. |
1834 (See Figure \nn{need figure, and improve caption on other figure}.) |
1841 (See Figure \ref{subdividing1marked}.) |
1835 We now proceed as in the above module definitions. |
1842 We now proceed as in the above module definitions. |
1836 |
1843 |
1837 \begin{figure}[t] \centering |
1844 \begin{figure}[t] \centering |
1838 \begin{tikzpicture}[baseline,line width = 2pt] |
1845 \begin{tikzpicture}[baseline,line width = 2pt] |
1839 \draw[blue][fill=blue!15!white] (0,0) circle (2); |
1846 \draw[blue][fill=blue!15!white] (0,0) circle (2); |
1845 \draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3); |
1852 \draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3); |
1846 } |
1853 } |
1847 \end{tikzpicture} |
1854 \end{tikzpicture} |
1848 \caption{Cone on a marked circle, the prototypical 1-marked ball} |
1855 \caption{Cone on a marked circle, the prototypical 1-marked ball} |
1849 \label{feb21d} |
1856 \label{feb21d} |
|
1857 \end{figure} |
|
1858 |
|
1859 \begin{figure}[t] \centering |
|
1860 \begin{tikzpicture}[baseline,line width = 2pt] |
|
1861 \draw[blue][fill=blue!15!white] (0,0) circle (2); |
|
1862 \fill[red] (0,0) circle (0.1); |
|
1863 \foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} { |
|
1864 \draw[red] (0,0) -- (\qm:2); |
|
1865 % \path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
|
1866 % \path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |
|
1867 % \draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3); |
|
1868 } |
|
1869 |
|
1870 |
|
1871 \begin{scope}[black, thin] |
|
1872 \clip (0,0) circle (2); |
|
1873 \draw (0:1) -- (90:1) -- (180:1) -- (270:1) -- cycle; |
|
1874 \draw (90:1) -- (90:2.1); |
|
1875 \draw (180:1) -- (180:2.1); |
|
1876 \draw (270:1) -- (270:2.1); |
|
1877 \draw (0:1) -- (15:2.1); |
|
1878 \draw (0:1) -- (315:1.5) -- (270:1); |
|
1879 \draw (315:1.5) -- (315:2.1); |
|
1880 \end{scope} |
|
1881 |
|
1882 \node(0marked) at (2.5,2.25) {$0$-marked ball}; |
|
1883 \node(1marked) at (3.5,1) {$1$-marked ball}; |
|
1884 \node(plain) at (3,-1) {plain ball}; |
|
1885 \draw[line width=1pt, green!50!brown, ->] (0marked.270) to[out=270,in=45] (50:1.1); |
|
1886 \draw[line width=1pt, green!50!brown, ->] (1marked.225) to[out=270,in=45] (0.4,0.1); |
|
1887 \draw[line width=1pt, green!50!brown, ->] (plain.90) to[out=135,in=45] (-45:1); |
|
1888 |
|
1889 \end{tikzpicture} |
|
1890 \caption{Subdividing a $1$-marked ball into plain, $0$-marked and $1$-marked balls.} |
|
1891 \label{subdividing1marked} |
1850 \end{figure} |
1892 \end{figure} |
1851 |
1893 |
1852 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
1894 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
1853 \[ |
1895 \[ |
1854 \cD(X) \deq \cM(X\times C(S)) . |
1896 \cD(X) \deq \cM(X\times C(S)) . |
2211 \end{proof} |
2253 \end{proof} |
2212 |
2254 |
2213 For $n=1$ we have to check an additional ``global" relations corresponding to |
2255 For $n=1$ we have to check an additional ``global" relations corresponding to |
2214 rotating the 0-sphere $E$ around the 1-sphere $\bd X$. |
2256 rotating the 0-sphere $E$ around the 1-sphere $\bd X$. |
2215 But if $n=1$, then we are in the case of ordinary algebroids and bimodules, |
2257 But if $n=1$, then we are in the case of ordinary algebroids and bimodules, |
2216 and this is just the well-known ``Frobenius reciprocity" result for bimodules. |
2258 and this is just the well-known ``Frobenius reciprocity" result for bimodules \cite{MR1424954}. |
2217 \nn{find citation for this. Evans and Kawahigashi? Bisch!} |
|
2218 |
2259 |
2219 \medskip |
2260 \medskip |
2220 |
2261 |
2221 We have now defined $\cS(X; c)$ for any $n{+}1$-ball $X$ with boundary decoration $c$. |
2262 We have now defined $\cS(X; c)$ for any $n{+}1$-ball $X$ with boundary decoration $c$. |
2222 We must also define, for any homeomorphism $X\to X'$, an action $f: \cS(X; c) \to \cS(X', f(c))$. |
2263 We must also define, for any homeomorphism $X\to X'$, an action $f: \cS(X; c) \to \cS(X', f(c))$. |