# HG changeset patch # User kevin@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1248461550 0 # Node ID a5f6a2ef9c9e93c4e57f42b2e831c99fc28a82da # Parent 9e5716a79abef67cfe82bfd4e89adde4b2ee9b15 ... diff -r 9e5716a79abe -r a5f6a2ef9c9e text/ncat.tex --- a/text/ncat.tex Thu Jul 23 22:13:48 2009 +0000 +++ b/text/ncat.tex Fri Jul 24 18:52:30 2009 +0000 @@ -29,16 +29,22 @@ \xxpar{Morphisms (preliminary version):} {For any $k$-manifold $X$ homeomorphic -to a $k$-ball, we have a set of $k$-morphisms +to the standard $k$-ball, we have a set of $k$-morphisms $\cC(X)$.} -Given a homeomorphism $f:X\to Y$ between such $k$-manifolds, we want a corresponding +Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the +standard $k$-ball. +We {\it do not} assume that it is equipped with a +preferred homeomorphism to the standard $k$-ball. +The same goes for ``a $k$-sphere" below. + +Given a homeomorphism $f:X\to Y$ between $k$-balls, we want a corresponding bijection of sets $f:\cC(X)\to \cC(Y)$. So we replace the above with \xxpar{Morphisms:} {For each $0 \le k \le n$, we have a functor $\cC_k$ from -the category of manifolds homeomorphic to the $k$-ball and +the category of $k$-balls and homeomorphisms to the category of sets and bijections.} (Note: We usually omit the subscript $k$.) @@ -55,6 +61,7 @@ of morphisms). The 0-sphere is unusual among spheres in that it is disconnected. Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. +(Actually, this is only true in the oriented case.) For $k>1$ and in the presence of strong duality the domain/range division makes less sense. \nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} We prefer to combine the domain and range into a single entity which we call the @@ -63,13 +70,13 @@ \xxpar{Boundaries (domain and range), part 1:} {For each $0 \le k \le n-1$, we have a functor $\cC_k$ from -the category of manifolds homeomorphic to the $k$-sphere and +the category of $k$-spheres and homeomorphisms to the category of sets and bijections.} (In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.) \xxpar{Boundaries, part 2:} -{For each $X$ homeomorphic to a $k$-ball, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. +{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.} (Note that the first ``$\bd$" above is part of the data for the category, @@ -78,7 +85,7 @@ Given $c\in\cC(\bd(X))$, let $\cC(X; c) = \bd^{-1}(c)$. Most of the examples of $n$-categories we are interested in are enriched in the following sense. -The various sets of $n$-morphisms $\cC(X; c)$, for all $X$ homeomorphic to an $n$-ball and +The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category (e.g.\ vector spaces, or modules over some ring, or chain complexes), and all the structure maps of the $n$-category should be compatible with the auxiliary @@ -104,8 +111,8 @@ the full boundary of a morphism: \xxpar{Domain $+$ range $\to$ boundary:} -{Let $S = B_1 \cup_E B_2$, where $S$ is homeomorphic to a $k$-sphere ($0\le k\le n-1$), -$B_i$ is homeomorphic to a $k$-ball, and $E = B_1\cap B_2$ is homeomorphic to a $k{-}1$-sphere. +{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. 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 @@ -119,7 +126,7 @@ We have ``restriction" maps $\cC(S)_E \to \cC(B_i)$, which can be thought of as domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$. -If $B$ is homeomorphic to a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls +If $B$ is a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$. Next we consider composition of morphisms. @@ -130,9 +137,9 @@ 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 homeomorphic to $k$-balls ($0\le k\le n$) -and $Y = B_1\cap B_2$ is homeomorphic to a $k{-}1$-ball. -Let $E = \bd Y$, which is homeomorphic to a $k{-}2$-sphere. +{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. +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 $E$. We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. @@ -162,7 +169,7 @@ The next axiom is related to identity morphisms, though that might not be immediately obvious. \xxpar{Product (identity) morphisms:} -{Let $X$ be homeomorphic to a $k$-ball and $D$ be homeomorphic to an $m$-ball, with $k+m \le n$. +{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{ @@ -180,7 +187,7 @@ We start with the plain $n$-category case. \xxpar{Isotopy invariance in dimension $n$ (preliminary version):} -{Let $X$ be homeomorphic to the $n$-ball and $f: X\to X$ be a homeomorphism which restricts +{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)$.} @@ -196,7 +203,7 @@ \nn{is there a non-TQFT reason to require this?} Second, we require that product (a.k.a.\ identity) $n$-morphisms act as the identity. -Let $X$ be an $n$-ball and $Y\sub\bd X$ be at $n{-}1$-ball. +Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. Let $J$ be a 1-ball (interval). We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. We define a map @@ -218,7 +225,7 @@ The revised axiom is \xxpar{Pseudo and extended isotopy invariance in dimension $n$:} -{Let $X$ be homeomorphic to the $n$-ball and $f: X\to X$ be a homeomorphism which restricts +{Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts to the identity on $\bd X$ and is pseudo-isotopic or extended isotopic (rel boundary) to the identity. Then $f$ acts trivially on $\cC(X)$.} @@ -250,8 +257,8 @@ \nn{need to also say something about collaring homeomorphisms.} \nn{this paragraph needs work.} -Note that if take homology of chain complexes, we turn an $A_\infty$ $n$-category -into a plain $n$-category. +Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category +into a plain $n$-category (enriched over graded groups). \nn{say more here?} In the other direction, if we enrich over topological spaces instead of chain complexes, we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting @@ -289,7 +296,7 @@ For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\cC(X)$ to be the set of all maps from $X\times F$ to $T$. For $X$ an $n$-ball define $\cC(X)$ to be maps from $X\times F$ to $T$ modulo -homotopies fixed on $\bd X$. +homotopies fixed on $\bd X \times F$. (Note that homotopy invariance implies isotopy invariance.) For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. @@ -320,7 +327,7 @@ \nn{refer elsewhere for details?} \item Variation on the above examples: -We could allow $F$ to have boundary and specify boundary conditions on $(\bd X)\times F$, +We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, for example product boundary conditions or take the union over all boundary conditions. \nn{maybe should not emphasize this case, since it's ``better" in some sense to think of these guys as affording a representation @@ -373,8 +380,8 @@ In our example, let $\cM(B, N) = \cD((B\times \bd W)\cup_{N\times \bd W} (N\times W))$, where $\cD$ is the fields functor for the TQFT. -Define the boundary of a marked ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. -Call such a thing a {marked hemisphere}. +Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. +Call such a thing a {marked $k{-}1$-hemisphere}. \xxpar{Module boundaries, part 1:} {For each $0 \le k \le n-1$, we have a functor $\cM_k$ from @@ -402,8 +409,130 @@ \] which is natural with respect to the actions of homeomorphisms.} +\xxpar{Axiom yet to be named:} +{For each marked $k$-hemisphere $H$ there is a restriction map +$\cM(H)\to \cC(H)$. +($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) +These maps comprise a natural transformation of functors.} +Note that combining the various boundary and restriction maps above +we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ +a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. +This fact will be used below. +\nn{need to say more about splitableness/transversality in various places above} +We stipulate two sorts of composition (gluing) for modules, corresponding to two ways +of splitting a marked $k$-ball into two (marked or plain) $k$-balls. +First, we can compose two module morphisms to get another module morphism. + +\nn{need figures for next two axioms} + +\xxpar{Module composition:} +{Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls ($0\le k\le n$) +and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. +Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. +Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. +We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. +Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps. +Then (axiom) we have a map +\[ + \gl_Y : \cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E \to \cM(M)_E +\] +which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions +to the intersection of the boundaries of $M$ and $M_i$. +If $k < n$ we require that $\gl_Y$ is injective. +(For $k=n$, see below.)} + +Second, we can compose an $n$-category morphism with a module morphism to get another +module morphism. +We'll call this the action map to distinguish it from the other kind of composition. + +\xxpar{$n$-category action:} +{Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), +$X$ is a plain $k$-ball, +and $Y = X\cap M'$ is a $k{-}1$-ball. +Let $E = \bd Y$, which is a $k{-}2$-sphere. +We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. +Let $\cC(X)_E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. +Then (axiom) we have a map +\[ + \gl_Y :\cC(X)_E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E +\] +which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions +to the intersection of the boundaries of $X$ and $M'$. +If $k < n$ we require that $\gl_Y$ is injective. +(For $k=n$, see below.)} + +\xxpar{Module strict associativity:} +{The composition and action maps above are strictly associative.} + +The above two axioms are equivalent to the following axiom, +which we state in slightly vague form. +\nn{need figure for this} + +\xxpar{Module multi-composition:} +{Given any decomposition +\[ + M = X_1 \cup\cdots\cup X_p \cup M_1\cup\cdots\cup M_q +\] +of a marked $k$-ball $M$ +into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a +map from an appropriate subset (like a fibered product) +of +\[ + \cC(X_1)\times\cdots\times\cC(X_p) \times \cM(M_1)\times\cdots\times\cM(M_q) +\] +to $\cM(M)$, +and these various multifold composition maps satisfy an +operad-type strict associativity condition.} + +(The above operad-like structure is analogous to the swiss cheese operad +\nn{need citation}.) +\nn{need to double-check that this is true.} + +\xxpar{Module product (identity) morphisms:} +{Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$. +Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$. +If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram +\[ \xymatrix{ + M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\ + M \ar[r]^{f} & M' +} \] +commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} + +\nn{Need to say something about compatibility with gluing (of both $M$ and $D$) above.} + +There are two alternatives for the next axiom, according whether we are defining +modules for plain $n$-categories or $A_\infty$ $n$-categories. +In the plain case we require + +\xxpar{Pseudo and extended isotopy invariance in dimension $n$:} +{Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts +to the identity on $\bd M$ and is pseudo-isotopic or extended isotopic (rel boundary) to the identity. +Then $f$ acts trivially on $\cM(M)$.} + +\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} + +We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. +In other words, if $M = (B, N)$ then we require only that isotopies are fixed +on $\bd B \setmin N$. + +For $A_\infty$ modules we require + +\xxpar{Families of homeomorphisms act.} +{For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes +\[ + C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . +\] +Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ +which fix $\bd M$. +These action maps are required to be associative up to homotopy +\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 $\Homeo(M\to M')$? what about boundary fixing property?}} + +\medskip