# HG changeset patch # User Kevin Walker # Date 1304723062 25200 # Node ID 4c9e16fbe09b7d682f7223ba0f11672f416f27b3 # Parent 2c9f09286beb40a5a95a0fd07cb07d87dba024af remark about apparent vacuity of lemma 6.1.2 (spheres) diff -r 2c9f09286beb -r 4c9e16fbe09b text/ncat.tex --- a/text/ncat.tex Fri May 06 15:23:26 2011 -0700 +++ b/text/ncat.tex Fri May 06 16:04:22 2011 -0700 @@ -124,10 +124,13 @@ \end{lem} We postpone the proof of this result until after we've actually given all the axioms. -Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, -along with the data described in the other axioms at lower levels. +Note that defining this functor for fixed $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, +along with the data described in the other axioms for smaller values of $k$. -%In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. +Of course, Lemma \ref{lem:spheres}, as stated, is satisfied by the trivial functor. +What we really mean is that there is exists a functor which interacts with other data of $\cC$ as specified +in the other axioms below. + \begin{axiom}[Boundaries]\label{nca-boundary} For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$.