blob: comparison text/ncat.tex

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 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 $k < n$,
 or if $k=n$ and we are in the $A_\infty$ case,
 we require that $\gl_Y$ is injective.
-(For $k=n$ in the plain (non-$A_\infty$) case, see below.)
+(For $k=n$ in the ordinary (non-$A_\infty$) case, see below.)
 \end{axiom}
 \begin{figure}[!ht] \centering
 \begin{tikzpicture}[%every label/.style={green},
 				x=1.5cm,y=1.5cm]
 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories.
 The last axiom (below), concerning actions of
 homeomorphisms in the top dimension $n$, distinguishes the two cases.
-We start with the plain $n$-category case.
+We start with the ordinary $n$-category case.
 \begin{axiom}[\textup{\textbf{[preliminary]}} 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 $\cC(X)$; that is $f(a) = a$ for all $a\in \cC(X)$.
 isotopic (rel boundary) to the identity {\it extended isotopy}.
 The revised axiom is
 %\addtocounter{axiom}{-1}
-\begin{axiom}[\textup{\textbf{[plain  version]}} Extended isotopy invariance in dimension $n$.]
+\begin{axiom}[\textup{\textbf{[ordinary  version]}} 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 isotopic (rel boundary) to the identity.
 Then $f$ acts trivially on $\cC(X)$.
 In addition, collar maps act trivially on $\cC(X)$.
 We will not pursue this in detail here.
 A variant on the above axiom would be to drop the ``up to homotopy" and require a strictly associative action.
 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).
+into a ordinary $n$-category (enriched over graded groups).
 In a different 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
 instead of  $C_*(\Homeo_\bd(X))$.
 Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex
 type $A_\infty$ $n$-category.
 \medskip
-The alert reader will have already noticed that our definition of a (plain) $n$-category
+The alert reader will have already noticed that our definition of a (ordinary) $n$-category
 is extremely similar to our definition of a system of fields.
 There are two differences.
 First, for the $n$-category definition we restrict our attention to balls
 (and their boundaries), while for fields we consider all manifolds.
 Second,  in category definition we directly impose isotopy
 but (string diagrams)/(relations) is isomorphic to
 (pasting diagrams composed of smaller string diagrams)/(relations)}
 }
-\begin{example}[The bordism $n$-category, plain version]
+\begin{example}[The bordism $n$-category, ordinary version]
 \label{ex:bord-cat}
 \rm
 \label{ex:bordism-category}
 For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional
 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse
 \subsection{From balls to manifolds}
 \label{ss:ncat_fields} \label{ss:ncat-coend}
 In this section we show how to extend an $n$-category $\cC$ as described above
-(of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$.
+(of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$.
 This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this.
-In the case of plain $n$-categories, this construction factors into a construction of a
+In the case of ordinary $n$-categories, this construction factors into a construction of a
 system of fields and local relations, followed by the usual TQFT definition of a
 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}.
 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead.
-Recall that we can take a plain $n$-category $\cC$ and pass to the ``free resolution",
+Recall that we can take a ordinary $n$-category $\cC$ and pass to the ``free resolution",
 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls
 (recall Example \ref{ex:blob-complexes-of-balls} above).
 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant
 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the
 same as the original blob complex for $M$ with coefficients in $\cC$.
 %\nn{need to finish explaining why we have a system of fields;
 %define $k$-cat $\cC(\cdot\times W)$}
 \subsection{Modules}
-Next we define plain and $A_\infty$ $n$-category modules.
+Next we define ordinary and $A_\infty$ $n$-category modules.
 The definition will be very similar to that of $n$-categories,
 but with $k$-balls replaced by {\it marked $k$-balls,} defined below.
 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary
 in the context of an $m{+}1$-dimensional TQFT.
 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 $k < n$,
 or if $k=n$ and we are in the $A_\infty$ case,
 we require that $\gl_Y$ is injective.
-(For $k=n$ in the plain (non-$A_\infty$) case, see below.)}
+(For $k=n$ in the ordinary (non-$A_\infty$) case, see below.)}
 \end{module-axiom}
 Second, we can compose an $n$-category morphism with a module morphism to get another
 module morphism.
 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 $k < n$,
 or if $k=n$ and we are in the $A_\infty$ case,
 we require that $\gl_Y$ is injective.
-(For $k=n$ in the plain (non-$A_\infty$) case, see below.)}
+(For $k=n$ in the ordinary (non-$A_\infty$) case, see below.)}
 \end{module-axiom}
 \begin{module-axiom}[Strict associativity]
 The composition and action maps above are strictly associative.
 Given any decomposition of a large marked ball into smaller marked and unmarked balls
 $a$ along a map associated to $\pi$.
 \medskip
 There are two alternatives for the next axiom, according whether we are defining
-modules for plain $n$-categories or $A_\infty$ $n$-categories.
+modules for ordinary $n$-categories or $A_\infty$ $n$-categories.
-In the plain case we require
+In the ordinary case we require
-\begin{module-axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$]
+\begin{module-axiom}[\textup{\textbf{[ordinary 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 isotopic (rel boundary) to the identity.
 Then $f$ acts trivially on $\cM(M)$.}
 In addition, collar maps act trivially on $\cM(M)$.
 \end{module-axiom}

changeset 680	0591d017e698
parent 679	72a1d5014abc
child 682	5f22b4501e5f