diff -r 6ef67f13b69c -r bc22926d4fb0 text/ncat.tex --- a/text/ncat.tex Wed Jun 02 22:46:31 2010 -0700 +++ b/text/ncat.tex Thu Jun 03 09:47:18 2010 -0700 @@ -1343,17 +1343,17 @@ \label{ssec:spherecat} In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" -whose objects correspond to $n$-categories. +whose objects are $n$-categories. When $n=2$ -this is a version of the familiar algebras-bimodules-intertwiners 2-category. -(Terminology: It is clearly appropriate to call an $S^0$ module a bimodule, +this is a version of the familiar algebras-bimodules-intertwiners $2$-category. +While it is clearly appropriate to call an $S^0$ module a bimodule, but this is much less true for higher dimensional spheres, -so we prefer the term ``sphere module" for the general case.) +so we prefer the term ``sphere module" for the general case. The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe these first. The $n{+}1$-dimensional part of $\cS$ consists of intertwiners -(of garden-variety $1$-category modules associated to decorated $n$-balls). +of (garden-variety) $1$-category modules associated to decorated $n$-balls. We will see below that in order for these $n{+}1$-morphisms to satisfy all of the duality requirements of an $n{+}1$-category, we will have to assume that our $n$-categories and modules have non-degenerate inner products. @@ -1367,10 +1367,10 @@ (This, in turn, is very similar to our definition of $n$-category.) Because of this similarity, we only sketch the definitions below. -We start with 0-sphere modules, which also could reasonably be called (categorified) bimodules. +We start with $0$-sphere modules, which also could reasonably be called (categorified) bimodules. (For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) -Define a 0-marked $k$-ball $(X, M)$, $1\le k \le n$, to be a pair homeomorphic to the standard -$(B^k, B^{k-1})$, where $B^{k-1}$ is properly embedded in $B^k$. +Define a $0$-marked $k$-ball $(X, M)$, $1\le k \le n$, to be a pair homeomorphic to the standard +$(B^k, B^{k-1})$. See Figure \ref{feb21a}. Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. @@ -1382,20 +1382,20 @@ \label{feb21a} \end{figure} -0-marked balls can be cut into smaller balls in various ways. -These smaller balls could be 0-marked or plain. -We can also take the boundary of a 0-marked ball, which is 0-marked sphere. +The $0$-marked balls can be cut into smaller balls in various ways. We only consider those decompositions in which the smaller balls are either + $0$-marked (i.e. intersect the $0$-marking of the large ball in a disc) or plain (don't intersect the $0$-marking of the large ball). +We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere. Fix $n$-categories $\cA$ and $\cB$. -These will label the two halves of a 0-marked $k$-ball. -The 0-sphere module we define next will depend on $\cA$ and $\cB$ +These will label the two halves of a $0$-marked $k$-ball. +The $0$-sphere module we define next will depend on $\cA$ and $\cB$ (it's an $\cA$-$\cB$ bimodule), but we will suppress that from the notation. -An $n$-category 0-sphere module $\cM$ is a collection of functors $\cM_k$ from the category -of 0-marked $k$-balls, $1\le k \le n$, +An $n$-category $0$-sphere module $\cM$ is a collection of functors $\cM_k$ from the category +of $0$-marked $k$-balls, $1\le k \le n$, (with the two halves labeled by $\cA$ and $\cB$) to the category of sets. If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. -Given a decomposition of a 0-marked $k$-ball $X$ into smaller balls $X_i$, we have +Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). @@ -1404,7 +1404,7 @@ \medskip -Part of the structure of an $n$-category 0-sphere module is captured by saying it is +Part of the structure of an $n$-category 0-sphere module $\cM$ is captured by saying it is a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) of $\cA$ and $\cB$. Let $J$ be some standard 0-marked 1-ball (i.e.\ an interval with a marked point in its interior). @@ -1413,7 +1413,7 @@ \cD(X) \deq \cM(X\times J) . \] The product is pinched over the boundary of $J$. -$\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ +The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ (see Figure \ref{feb21b}). These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. @@ -1429,9 +1429,9 @@ of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled by $\cA_i$-$\cA_{i+1}$ bimodules $\cM_i$. (See Figure \ref{feb21c}.) -To this data we can apply to coend construction as in Subsection \ref{moddecss} above -to obtain an $\cA_0$-$\cA_l$ bimodule and, forgetfully, an $n{-}1$-category. -This amounts to a definition of taking tensor products of bimodules over $n$-categories. +To this data we can apply the coend construction as in Subsection \ref{moddecss} above +to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. +This amounts to a definition of taking tensor products of $0$-sphere module over $n$-categories. \begin{figure}[!ht] \begin{equation*} @@ -1445,7 +1445,7 @@ associated to the marked and labeled circle. (See Figure \ref{feb21c}.) If the circle is divided into two intervals, we can think of this $n{-}1$-category -as the 2-ended tensor product of the two bimodules associated to the two intervals. +as the 2-sided tensor product of the two bimodules associated to the two intervals. \medskip @@ -1558,18 +1558,4 @@ \item functors \end{itemize} -\bigskip -\hrule -\nn{Some salvaged paragraphs that we might want to work back in:} -\bigskip - -Appendix \ref{sec:comparing-A-infty} explains the translation between this definition of an $A_\infty$ $1$-category and the usual one expressed in terms of `associativity up to higher homotopy', as in \cite{MR1854636}. (In this version of the paper, that appendix is incomplete, however.) - -The motivating example is `chains of maps to $M$' for some fixed target space $M$. This is a topological $A_\infty$ category $\Xi_M$ with $\Xi_M(J) = C_*(\Maps(J \to M))$. The gluing maps $\Xi_M(J) \tensor \Xi_M(J') \to \Xi_M(J \cup J')$ takes the product of singular chains, then glues maps to $M$ together; the associativity condition is automatically satisfied. The evaluation map $\ev_{J,J'} : \CD{J \to J'} \tensor \Xi_M(J) \to \Xi_M(J')$ is the composition -\begin{align*} -\CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)), -\end{align*} -where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism. - -\hrule