--- a/blob to-do	Fri Jun 17 20:56:27 2011 -0600
+++ b/blob to-do	Sun Jun 19 15:31:28 2011 -0600
@@ -30,6 +30,9 @@
 * leftover: we used to require that composition of A-infinity n-morphisms was injective (just like lower morphisms).  Should we stick this back in?  I don't think we use it anywhere.
 
 * should we require, for A-inf n-cats, that families which preserve product morphisms act trivially?  as now defined, this is only true up to homotopy for the blob complex, so maybe best not to open that can of worms
+(but since the strict version of this is true for BT_*, maybe we're OK)
+
+* probably should go through and refer to new splitting axiom when we need to choose refinements etc.
 
 
 

--- a/blob_changes_v3	Fri Jun 17 20:56:27 2011 -0600
+++ b/blob_changes_v3	Sun Jun 19 15:31:28 2011 -0600
@@ -27,7 +27,7 @@
 - added details to the construction of traditional 1-categories from disklike 1-categories (Appendix C.1)
 - extended the lemmas of Appendix B (about adapting families of homeomorphisms to open covers) to the topological category
 - modified families-of-homeomorphisms-action axiom for A-infinity n-categories, and added discussion of alternatives
-
+- added n-cat axiom for existence of splittings
 
 
 

--- a/text/kw_macros.tex	Fri Jun 17 20:56:27 2011 -0600
+++ b/text/kw_macros.tex	Sun Jun 19 15:31:28 2011 -0600
@@ -31,6 +31,7 @@
 \def\ol{\overline}
 \def\BD{BD}
 \def\bbc{{\mathcal{BBC}}}
+\def\vcone{\text{V-Cone}}
 
 \def\spl{_\pitchfork}
 \def\trans#1{_{\pitchfork #1}}
@@ -63,7 +64,7 @@
 
 % \DeclareMathOperator{\pr}{pr} etc.
 \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}}
-\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}{Coll};
+\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}{Coll}{Cone};
 
 \DeclareMathOperator*{\colim}{colim}
 \DeclareMathOperator*{\hocolim}{hocolim}

--- a/text/ncat.tex	Fri Jun 17 20:56:27 2011 -0600
+++ b/text/ncat.tex	Sun Jun 19 15:31:28 2011 -0600
@@ -568,7 +568,7 @@
 %We start with the ordinary $n$-category case.
 
 The next axiom says, roughly, that we have strict associativity in dimension $n$, 
-even we we reparameterize our $n$-balls.
+even when we reparametrize our $n$-balls.
 
 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$]
 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which 
@@ -649,7 +649,7 @@
 The revised axiom is
 
 %\addtocounter{axiom}{-1}
-\begin{axiom}[\textup{\textbf{[ordinary  version]}} Extended isotopy invariance in dimension $n$.]
+\begin{axiom}[\textup{\textbf{[ordinary  version]}} Extended isotopy invariance in dimension $n$]
 \label{axiom:extended-isotopies}
 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which 
 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$.
@@ -661,6 +661,45 @@
 
 \medskip
 
+We need one additional axiom, in order to constrain the poset of decompositions of a given morphism.
+We will soon want to take colimits (and homotopy colimits) indexed by such posets, and we want to require
+that these colimits are in some sense locally acyclic.
+Before stating the axiom we need a few preliminary definitions.
+If $P$ is a poset let $P\times I$ denote the product poset, where $I = \{0, 1\}$ with ordering $0\le 1$.
+Let $\Cone(P)$ denote $P$ adjoined an additional object $v$ (the vertex of the cone) with $p\le v$ for all objects $p$ of $P$.
+Finally, let $\vcone(P)$ denote $P\times I \cup \Cone(P)$, where we identify $P\times \{0\}$ with the base of the cone.
+We call $P\times \{1\}$ the base of $\vcone(P)$.
+(See Figure \nn{need figure}.)
+
+\nn{maybe call this ``splittings" instead of ``V-cones"?}
+
+\begin{axiom}[V-cones]
+\label{axiom:vcones}
+Let $c\in \cC_k(X)$ and
+let $P$ be a finite poset of splittings of $c$.
+Then we can embed $\vcone(P)$ into the splittings of $c$, with $P$ corresponding to the base of $\vcone(P)$.
+Furthermore, if $q$ is any decomposition of $X$, then we can take the vertex of $\vcone(P)$ to be $q$ up to a small perturbation.
+\end{axiom}
+
+It is easy to see that this axiom holds in our two motivating examples, 
+using standard facts about transversality and general position.
+One starts with $q$, perturbs it so that it is in general position with respect to $c$ (in the case of string diagrams)
+and also with respect to each of the decompositions of $P$, then chooses common refinements of each decomposition of $P$
+and the perturbed $q$.
+These common refinements form the middle ($P\times \{0\}$ above) part of $\vcone(P)$.
+
+We note two simple special cases of axiom \ref{axiom:vcones}.
+If $P$ is the empty poset, then $\vcone(P)$ consists of only the vertex, and the axiom says that any morphism $c$
+can be split along any decomposition of $X$, after a small perturbation.
+If $P$ is the disjoint union of two points, then $\vcone(P)$ looks like a letter W, and the axiom implies that the
+poset of splittings of $c$ is connected.
+Note that we do not require that any two splittings of $c$ have a common refinement (i.e.\ replace the letter W with the letter V).
+Two decompositions of $X$ might intersect in a very messy way, but one can always find a third
+decomposition which has common refinements with each of the original two decompositions.
+
+
+\medskip
+
 This completes the definition of an $n$-category.
 Next we define enriched $n$-categories.