661 

   662 \medskip

   663 

   664 We need one additional axiom, in order to constrain the poset of decompositions of a given morphism.

   665 We will soon want to take colimits (and homotopy colimits) indexed by such posets, and we want to require

   666 that these colimits are in some sense locally acyclic.

   667 Before stating the axiom we need a few preliminary definitions.

   668 If $P$ is a poset let $P\times I$ denote the product poset, where $I = \{0, 1\}$ with ordering $0\le 1$.

   669 Let $\Cone(P)$ denote $P$ adjoined an additional object $v$ (the vertex of the cone) with $p\le v$ for all objects $p$ of $P$.

   670 Finally, let $\vcone(P)$ denote $P\times I \cup \Cone(P)$, where we identify $P\times \{0\}$ with the base of the cone.

   671 We call $P\times \{1\}$ the base of $\vcone(P)$.

   672 (See Figure \nn{need figure}.)

   673 

   674 \nn{maybe call this ``splittings" instead of ``V-cones"?}

   675 

   676 \begin{axiom}[V-cones]

   677 \label{axiom:vcones}

   678 Let $c\in \cC_k(X)$ and

   679 let $P$ be a finite poset of splittings of $c$.

   680 Then we can embed $\vcone(P)$ into the splittings of $c$, with $P$ corresponding to the base of $\vcone(P)$.

   681 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.

   682 \end{axiom}

   683 

   684 It is easy to see that this axiom holds in our two motivating examples, 

   685 using standard facts about transversality and general position.

   686 One starts with $q$, perturbs it so that it is in general position with respect to $c$ (in the case of string diagrams)

   687 and also with respect to each of the decompositions of $P$, then chooses common refinements of each decomposition of $P$

   688 and the perturbed $q$.

   689 These common refinements form the middle ($P\times \{0\}$ above) part of $\vcone(P)$.

   690 

   691 We note two simple special cases of axiom \ref{axiom:vcones}.

   692 If $P$ is the empty poset, then $\vcone(P)$ consists of only the vertex, and the axiom says that any morphism $c$

   693 can be split along any decomposition of $X$, after a small perturbation.

   694 If $P$ is the disjoint union of two points, then $\vcone(P)$ looks like a letter W, and the axiom implies that the

   695 poset of splittings of $c$ is connected.

   696 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).

   697 Two decompositions of $X$ might intersect in a very messy way, but one can always find a third

   698 decomposition which has common refinements with each of the original two decompositions.

   699