357 or if $k=n$ and we are in the $A_\infty$ case, |
357 or if $k=n$ and we are in the $A_\infty$ case, |
358 we require that $\gl_Y$ is injective. |
358 we require that $\gl_Y$ is injective. |
359 (For $k=n$ in the plain (non-$A_\infty$) case, see below.) |
359 (For $k=n$ in the plain (non-$A_\infty$) case, see below.) |
360 \end{axiom} |
360 \end{axiom} |
361 |
361 |
362 \begin{axiom}[Strict associativity] \label{nca-assoc} |
362 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity} |
363 The gluing maps above are strictly associative. |
363 The gluing maps above are strictly associative. |
364 Given any decomposition of a ball $B$ into smaller balls |
364 Given any decomposition of a ball $B$ into smaller balls |
365 $$\bigsqcup B_i \to B,$$ |
365 $$\bigsqcup B_i \to B,$$ |
366 any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result. |
366 any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result. |
367 \end{axiom} |
367 \end{axiom} |
495 There is an $A_\infty$ analogue enriched in topological spaces, where at the top level we take all such submanifolds, rather than homeomorphism classes. For each fixed $\bdy W \subset \bdy X \times \bbR^\infty$, we can topologize the set of submanifolds by ambient isotopy rel boundary. |
495 There is an $A_\infty$ analogue enriched in topological spaces, where at the top level we take all such submanifolds, rather than homeomorphism classes. For each fixed $\bdy W \subset \bdy X \times \bbR^\infty$, we can topologize the set of submanifolds by ambient isotopy rel boundary. |
496 |
496 |
497 \subsection{The blob complex} |
497 \subsection{The blob complex} |
498 \subsubsection{Decompositions of manifolds} |
498 \subsubsection{Decompositions of manifolds} |
499 |
499 |
500 A \emph{ball decomposition} of $W$ is a |
500 A \emph{ball decomposition} of a $k$-manifold $W$ is a |
501 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
501 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
502 $\du_a X_a$ and each $M_i$ is a manifold. |
502 $\du_a X_a$ and each $M_i$ is a manifold. |
503 If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself. |
503 If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself. |
504 A {\it permissible decomposition} of $W$ is a map |
504 A {\it permissible decomposition} of $W$ is a map |
505 \[ |
505 \[ |
534 \begin{equation*} |
534 \begin{equation*} |
535 %\label{eq:psi-C} |
535 %\label{eq:psi-C} |
536 \psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl |
536 \psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl |
537 \end{equation*} |
537 \end{equation*} |
538 where the restrictions to the various pieces of shared boundaries amongst the cells |
538 where the restrictions to the various pieces of shared boundaries amongst the cells |
539 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). When $k=n$, the `subset' and `product' in the above formula should be interpreted in the appropriate enriching category. |
539 $X_a$ all agree (this is a fibered product of all the labels of $k$-cells over the labels of $k-1$-cells). When $k=n$, the `subset' and `product' in the above formula should be interpreted in the appropriate enriching category. |
540 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
540 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
541 \end{defn} |
541 \end{defn} |
542 |
542 |
543 We will use the term `field on $W$' to refer to a point of this functor, |
543 We will use the term `field on $W$' to refer to a point of this functor, |
544 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
544 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
545 |
545 |
546 |
546 |
547 \subsubsection{Colimits} |
547 \subsubsection{Colimits} |
548 Our definition of an $n$-category is essentially a collection of functors defined on $k$-balls (and homeomorphisms) for $k \leq n$ satisfying certain axioms. It is natural to consider extending such functors to the larger categories of all $k$-manifolds (again, with homeomorphisms). In fact, the axioms stated above explictly require such an extension to $k$-spheres for $k<n$. |
548 Our definition of an $n$-category is essentially a collection of functors defined on $k$-balls (and homeomorphisms) for $k \leq n$ satisfying certain axioms. It is natural to consider extending such functors to the larger categories of all $k$-manifolds (again, with homeomorphisms). In fact, the axioms stated above explicitly require such an extension to $k$-spheres for $k<n$. |
549 |
549 |
550 The natural construction achieving this is the colimit. |
550 The natural construction achieving this is the colimit. For an $n$-category $\cC$, we denote the extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, this is defined to be the colimit of the function $\psi_{\cC;W}$. 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 $k<n$. Recall that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, the set $\cC(X;c)$ |
551 \nn{continue} |
551 \nn{continue} |
552 |
552 |
553 \nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?} |
553 \nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?} |
554 |
554 |
555 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
555 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |