blob: comparison pnas/pnas.tex

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 They are less general in that we impose stronger duality requirements in dimensions 0 through $n$.
 Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional {\it unoriented} or {\it oriented} TQFTs, while
 Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional {\it framed} TQFTs.
 We will define two variations simultaneously,  as all but one of the axioms are identical in the two cases.
-These variations are ``plain $n$-categories", where homeomorphisms fixing the boundary
+These variations are ``ordinary $n$-categories", where homeomorphisms fixing the boundary
 act trivially on the sets associated to $n$-balls
 (and these sets are usually vector spaces or more generally modules over a commutative ring)
 and ``$A_\infty$ $n$-categories",  where there is a homotopy action of
 $k$-parameter families of homeomorphisms on these sets
 (which are usually chain complexes or topological spaces).
 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 $n$-category case, see Axiom \ref{axiom:extended-isotopies}.)
+(For $k=n$ in the ordinary $n$-category case, see Axiom \ref{axiom:extended-isotopies}.)
 \end{axiom}
 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity}
 The gluing maps above are strictly associative.
 Given any decomposition of a ball $B$ into smaller balls
 where the first arrow is gluing with a product morphism on $Y\times_p J$ and
 the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts
 to the identity on the boundary.
-\begin{axiom}[\textup{\textbf{[for plain  $n$-categories]}} Extended isotopy invariance in dimension $n$.]
+\begin{axiom}[\textup{\textbf{[for ordinary  $n$-categories]}} 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 $\cC(X)$.
 It is natural to hope to extend such functors to the
 larger categories of all $k$-manifolds (again, with homeomorphisms).
 In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$.
 The natural construction achieving this is a colimit along the poset of permissible decompositions.
-Given a plain $n$-category $\cC$,
+Given an ordinary $n$-category $\cC$,
 we will denote its extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$,
 this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$.
 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity}
 imply that $\cl{\cC}(X)  \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$.
 Suppose that $\cC$ is enriched in vector spaces: this means that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball,
 %When $\cC$ is a topological $n$-category,
 %the flexibility available in the construction of a homotopy colimit allows
 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$.
 %\todo{either need to explain why this is the same, or significantly rewrite this section}
-When $\cC$ is the plain $n$-category based on string diagrams for a traditional
+When $\cC$ is the ordinary $n$-category based on string diagrams for a traditional
 $n$-category $C$,
 one can show \cite{1009.5025} that the above two constructions of the homotopy colimit
 are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$.
 Roughly speaking, the generators of $\bc_k(W; \cC)$ are string diagrams on $W$ together with
 a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested.
 The blob complex has several important special cases.
 \begin{thm}[Skein modules]
 \label{thm:skein-modules}
-Suppose $\cC$ is a plain $n$-category.
+Suppose $\cC$ is an ordinary $n$-category.
 The $0$-th blob homology of $X$ is the usual
 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
 by $\cC$.
 \begin{equation*}
 H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X)
 blob complexes for the $A_\infty$ $n$-categories constructed as above.
 \begin{thm}[Product formula]
 \label{thm:product}
 Let $W$ be a $k$-manifold and $Y$ be an $n{-}k$ manifold.
-Let $\cC$ be a plain $n$-category.
+Let $\cC$ be an ordinary $n$-category.
 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above.
 Then
 \[
 	\bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W).
 \]

changeset 671	6088d0b8611b
parent 670	7a4fc5a873ac
child 672	3f0f4f4ad048