blob: comparison text/ncat.tex

equal deleted inserted replaced

-:9bf409eb5040
+:7a5a73ec8961
 At first it might seem that we need another axiom for this, but in fact once we have
 all the axioms in the subsection for $0$ through $k-1$ we can use a colimit
 construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$
 to spheres (and any other manifolds):
-\begin{prop}
+\begin{lem}
-\label{axiom:spheres}
+\label{lem:spheres}
 For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from
 the category of $k{-}1$-spheres and
 homeomorphisms to the category of sets and bijections.
-\end{prop}
+\end{lem}
 We postpone the proof \todo{} of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in the other Axioms at lower levels.
 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point.
 We have just argued that the boundary of a morphism has no preferred splitting into
 domain and range, but the converse meets with our approval.
 That is, given compatible domain and range, we should be able to combine them into
 the full boundary of a morphism.
-The following proposition follows from the colimit construction used to define $\cl{\cC}_{k-1}$
+The following lemma follows from the colimit construction used to define $\cl{\cC}_{k-1}$
 on spheres.
-\begin{prop}[Boundary from domain and range]
+\begin{lem}[Boundary from domain and range]
+\label{lem:domain-and-range}
 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$,
 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}).
 Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the
 two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$.
 Then we have an injective map
 	\gl_E : \cC(B_1) \times_{\\cl{cC}(E)} \cC(B_2) \into \cl{\cC}(S)
 \]
 which is natural with respect to the actions of homeomorphisms.
 (When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product
 becomes a normal product.)
-\end{prop}
+\end{lem}
 \begin{figure}[!ht]
 $$
 \begin{tikzpicture}[%every label/.style={green}
 ]
 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$
 We call $B$ the ball and $N$ the marking.
 A homeomorphism between marked $k$-balls is a homeomorphism of balls which
 restricts to a homeomorphism of markings.
-\mmpar{Module axiom 1}{Module morphisms}
+\begin{module-axiom}[Module morphisms]
 {For each $0 \le k \le n$, we have a functor $\cM_k$ from
 the category of marked $k$-balls and
 homeomorphisms to the category of sets and bijections.}
+\end{module-axiom}
 (As with $n$-categories, we will usually omit the subscript $k$.)
 For example, let $\cD$ be the $m{+}1$-dimensional TQFT which assigns to a $k$-manifold $N$ the set
 of maps from $N$ to $T$, modulo homotopy (and possibly linearized) if $k=m$.
 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure}
 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}.
-\mmpar{Module axiom 2}{Module boundaries (hemispheres)}
+\begin{lem}
+\label{lem:hemispheres}
 {For each $0 \le k \le n-1$, we have a functor $\cM_k$ from
 the category of marked $k$-hemispheres and
 homeomorphisms to the category of sets and bijections.}
+\end{lem}
+The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. We use the same type of colimit construction.
 In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$.
-\mmpar{Module axiom 3}{Module boundaries (maps)}
+\begin{module-axiom}[Module boundaries (maps)]
 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$.
 These maps, for various $M$, comprise a natural transformation of functors.}
+\end{module-axiom}
 Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$.
 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces),
 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$
 and $c\in \cC(\bd M)$.
-\mmpar{Module axiom 4}{Boundary from domain and range}
+\begin{lem}[Boundary from domain and range]
 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$),
 $M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere.
 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the
 two maps $\bd: \cM(M_i)\to \cM(E)$.
 Then (axiom) we have an injective map
 \[
 	\gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \hookrightarrow \cM(H)
 \]
 which is natural with respect to the actions of homeomorphisms.}
+\end{lem}
+Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}.
 Let $\cM(H)_E$ denote the image of $\gl_E$.
 We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$".
-\mmpar{Module axiom 5}{Module to category restrictions}
+\begin{module-axiom}[Module to category restrictions]
 {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.}
+\end{module-axiom}
 Note that combining the various boundary and restriction maps above
 (for both modules and $n$-categories)
 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 $\cC(Y)$.
 \label{zzz3}
 \end{figure}
 First, we can compose two module morphisms to get another module morphism.
-\mmpar{Module axiom 6}{Module composition}
+\begin{module-axiom}[Module composition]
 {Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls (with $0\le k\le n$)
 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 $\cM(M_i)_E \to \cM(Y)$.
 \]
 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.)}
+\end{module-axiom}
 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.
-\mmpar{Module axiom 7}{$n$-category action}
+\begin{module-axiom}[$n$-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 $\cC(X)_E\to \cC(Y)$.
 \]
 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.)}
+\end{module-axiom}
-\mmpar{Module axiom 8}{Strict associativity}
+\begin{module-axiom}[Strict associativity]
 {The composition and action maps above are strictly associative.}
+\end{module-axiom}
 Note that the above associativity axiom applies to mixtures of module composition,
 action maps and $n$-category composition.
 See Figure \ref{zzz1b}.
 (The above operad-like structure is analogous to the swiss cheese operad
 \cite{MR1718089}.)
 %\nn{need to double-check that this is true.}
-\mmpar{Module axiom 9}{Product/identity morphisms}
+\begin{module-axiom}[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 $\tilde{f}(a\times D) = f(a)\times D'$.}
+\end{module-axiom}
 \nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.}
 \nn{postpone finalizing the above axiom until the n-cat version is finalized}
 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
-\mmpar{Module axiom 10a}{Extended isotopy invariance in dimension $n$}
+\begin{module-axiom}[\textup{\textbf{[topological version]}} 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 extended isotopic (rel boundary) to the identity.
 Then $f$ acts trivially on $\cM(M)$.}
+\end{module-axiom}
 \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
-\mmpar{Module axiom 10b}{Families of homeomorphisms act}
+\addtocounter{module-axiom}{-1}
+\begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} 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 $M$
 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{CHprop} commutes.
 \nn{repeat diagram here?}
 \nn{restate this with $\Homeo(M\to M')$?  what about boundary fixing property?}}
+\end{module-axiom}
 \medskip
 Note that the above axioms imply that an $n$-category module has the structure
 of an $n{-}1$-category.

changeset 336	7a5a73ec8961
parent 335	9bf409eb5040
child 339	9698f584e732