diff -r 748cd16881bf -r 4067c74547bb text/ncat.tex --- a/text/ncat.tex Tue Dec 08 01:08:53 2009 +0000 +++ b/text/ncat.tex Fri Dec 11 22:44:25 2009 +0000 @@ -18,8 +18,8 @@ Before proceeding, we need more appropriate definitions of $n$-categories, $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. -(As is the case throughout this paper, by ``$n$-category" we mean -a weak $n$-category with strong duality.) +(As is the case throughout this paper, by ``$n$-category" we implicitly intend some notion of +a `weak' $n$-category with `strong duality'.) The definitions presented below tie the categories more closely to the topology and avoid combinatorial questions about, for example, the minimal sufficient @@ -46,10 +46,11 @@ the standard $k$-ball. In other words, -\xxpar{Morphisms (preliminary version):} -{For any $k$-manifold $X$ homeomorphic +\begin{preliminary-axiom}{\ref{axiom:morphisms}}{Morphisms} +For any $k$-manifold $X$ homeomorphic to the standard $k$-ball, we have a set of $k$-morphisms -$\cC_k(X)$.} +$\cC_k(X)$. +\end{preliminary-axiom} Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the standard $k$-ball. @@ -64,15 +65,17 @@ (This will imply ``strong duality", among other things.) So we replace the above with -\xxpar{Morphisms:} -%\xxpar{Axiom 1 -- Morphisms:} -{For each $0 \le k \le n$, we have a functor $\cC_k$ from +\begin{axiom}[Morphisms] +\label{axiom:morphisms} +For each $0 \le k \le n$, we have a functor $\cC_k$ from the category of $k$-balls and -homeomorphisms to the category of sets and bijections.} +homeomorphisms to the category of sets and bijections. +\end{axiom} + (Note: We usually omit the subscript $k$.) -We are being deliberately vague about what flavor of manifolds we are considering. +We are so far being deliberately vague about what flavor of manifolds we are considering. They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. They could be topological or PL or smooth. \nn{need to check whether this makes much difference --- see pseudo-isotopy below} @@ -93,16 +96,18 @@ boundary of a morphism. Morphisms are modeled on balls, so their boundaries are modeled on spheres: -\xxpar{Boundaries (domain and range), part 1:} -{For each $0 \le k \le n-1$, we have a functor $\cC_k$ from +\begin{axiom}[Boundaries (spheres)] +For each $0 \le k \le n-1$, we have a functor $\cC_k$ from the category of $k$-spheres and -homeomorphisms to the category of sets and bijections.} +homeomorphisms to the category of sets and bijections. +\end{axiom} (In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.) -\xxpar{Boundaries, part 2:} -{For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. -These maps, for various $X$, comprise a natural transformation of functors.} +\begin{axiom}[Boundaries (maps)] +For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. +These maps, for various $X$, comprise a natural transformation of functors. +\end{axiom} (Note that the first ``$\bd$" above is part of the data for the category, while the second is the ordinary boundary of manifolds.) @@ -141,16 +146,17 @@ That is, given compatible domain and range, we should be able to combine them into the full boundary of a morphism: -\xxpar{Domain $+$ range $\to$ boundary:} -{Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere ($0\le k\le n-1$), +\begin{axiom}[Boundary from domain and range] +Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere $(0\le k\le n-1)$, $B_i$ is a $k$-ball, and $E = B_1\cap B_2$ is a $k{-}1$-sphere (Figure \ref{blah3}). Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the two maps $\bd: \cC(B_i)\to \cC(E)$. -Then (axiom) we have an injective map +Then we have an injective map \[ \gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S) \] -which is natural with respect to the actions of homeomorphisms.} +which is natural with respect to the actions of homeomorphisms. +\end{axiom} \begin{figure}[!ht] $$ @@ -187,8 +193,8 @@ In the presence of strong duality, these $k$ distinct compositions are subsumed into one general type of composition which can be in any ``direction". -\xxpar{Composition:} -{Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) +\begin{axiom}[Composition] +Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). Let $E = \bd Y$, which is a $k{-}2$-sphere. Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. @@ -201,14 +207,16 @@ which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions to the intersection of the boundaries of $B$ and $B_i$. If $k < n$ we require that $\gl_Y$ is injective. -(For $k=n$, see below.)} +(For $k=n$, see below.) +\end{axiom} \begin{figure}[!ht] $$\mathfig{.4}{tempkw/blah5}$$ \caption{From two balls to one ball}\label{blah5}\end{figure} -\xxpar{Strict associativity:} -{The composition (gluing) maps above are strictly associative.} +\begin{axiom}[Strict associativity] +The composition (gluing) maps above are strictly associative. +\end{axiom} \begin{figure}[!ht] $$\mathfig{.65}{tempkw/blah6}$$ @@ -242,8 +250,8 @@ The next axiom is related to identity morphisms, though that might not be immediately obvious. -\xxpar{Product (identity) morphisms:} -{Let $X$ be a $k$-ball and $D$ be an $m$-ball, with $k+m \le n$. +\begin{axiom}[Product (identity) morphisms] +Let $X$ be a $k$-ball and $D$ be an $m$-ball, with $k+m \le n$. Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram \[ \xymatrix{ @@ -274,7 +282,7 @@ \res_{X\times E}(a\times D) = a\times E \] for $E\sub \bd D$ and $a\in \cC(X)$. -} +\end{axiom} \nn{need even more subaxioms for product morphisms?} @@ -301,10 +309,11 @@ We start with the plain $n$-category case. -\xxpar{Isotopy invariance in dimension $n$ (preliminary version):} -{Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts +\begin{preliminary-axiom}{\ref{axiom:extended-isotopies}}{Isotopy invariance in dimension $n$} +Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. -Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.} +Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. +\end{preliminary-axiom} This axiom needs to be strengthened to force product morphisms to act as the identity. Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. @@ -333,10 +342,12 @@ The revised axiom is -\xxpar{Extended isotopy invariance in dimension $n$:} -{Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts +\begin{axiom}[Extended isotopy invariance in dimension $n$] +\label{axiom:extended-isotopies} +Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. -Then $f$ acts trivially on $\cC(X)$.} +Then $f$ acts trivially on $\cC(X)$. +\end{axiom} \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} @@ -346,8 +357,8 @@ isotopy invariance with the requirement that families of homeomorphisms act. For the moment, assume that our $n$-morphisms are enriched over chain complexes. -\xxpar{Families of homeomorphisms act in dimension $n$.} -{For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes +\begin{axiom}[Families of homeomorphisms act in dimension $n$] +For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes \[ C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . \] @@ -357,7 +368,8 @@ \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that a diagram like the one in Proposition \ref{CDprop} commutes. \nn{repeat diagram here?} -\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?}} +\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} +\end{axiom} We should strengthen the above axiom to apply to families of extended homeomorphisms. To do this we need to explain how extended homeomorphisms form a topological space.