# HG changeset patch # User Kevin Walker # Date 1300232989 25200 # Node ID 1b49432f3aef62df34a8540fcea0c3460620556b # Parent c5256040e58fb60b58ff96787d68249915eb5eb0 tried to clarify the spirally nature of the axioms in the intro to the ncat section; other word-smithing in that intro; added remark about C(0-sphere) diff -r c5256040e58f -r 1b49432f3aef text/ncat.tex --- a/text/ncat.tex Tue Mar 15 08:11:27 2011 -0700 +++ b/text/ncat.tex Tue Mar 15 16:49:49 2011 -0700 @@ -14,13 +14,15 @@ (As is the case throughout this paper, by ``$n$-category" we mean some notion of a ``weak" $n$-category with ``strong duality".) -The definitions presented below tie the categories more closely to the topology -and avoid combinatorial questions about, for example, the minimal sufficient -collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. +Compared to other definitions in the literature, +the definitions presented below tie the categories more closely to the topology +and avoid combinatorial questions about, for example, finding a minimal sufficient +collection of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. It is easy to show that examples of topological origin -(e.g.\ categories whose morphisms are maps into spaces or decorated balls), +(e.g.\ categories whose morphisms are maps into spaces or decorated balls, or bordism categories), satisfy our axioms. -For examples of a more purely algebraic origin, one would typically need the combinatorial +To show that examples of a more purely algebraic origin satisfy our axioms, +one would typically need the combinatorial results that we have avoided here. See \S\ref{n-cat-names} for a discussion of $n$-category terminology. @@ -30,6 +32,15 @@ \medskip +The axioms for an $n$-category are spread throughout this section. +Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product} and \ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. + +Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms +for $k{-}1$-morphisms. +So readers who prefer things to be presented in a strictly logical order should read this subsection $n$ times, first imagining that $k=0$, then that $k=1$, and so on until they reach $k=n$. + +\medskip + There are many existing definitions of $n$-categories, with various intended uses. In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. Generally, these sets are indexed by instances of a certain typical shape. @@ -49,13 +60,9 @@ We {\it do not} assume that it is equipped with a preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. -The axioms for an $n$-category are spread throughout this section. -Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product} and \ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. - - Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on the boundary), we want a corresponding -bijection of sets $f:\cC(X)\to \cC(Y)$. +bijection of sets $f:\cC_k(X)\to \cC_k(Y)$. (This will imply ``strong duality", among other things.) Putting these together, we have \begin{axiom}[Morphisms] @@ -103,7 +110,8 @@ Morphisms are modeled on balls, so their boundaries are modeled on spheres. In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for $1\le k \le n$. -At first it might seem that we need another axiom for this, but in fact once we have +At first it might seem that we need another axiom +(more specifically, additional data) for this, but in fact once we have all the axioms in this subsection for $0$ through $k-1$ we can use a colimit construction, as described in \S\ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ to spheres (and any other manifolds): @@ -197,6 +205,10 @@ The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. %\nn{we might want a more official looking proof...} +If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union +of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified +with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$. + Let $\cl{\cC}(S)\trans E$ denote the image of $\gl_E$. We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$".