v(K)$ define $e_v \colon V(K) \to \mathbb{R}$ by the formula $e_v(w) = 0$ for $w \neq v$ and $e_v(v) = \kappa$. We embed each vertex $v$ of $K$ as $e_v \in \ell_1(V(K))$ and extend this embedding to $K$ to be affine on each simplex of $K$. We consider $K$ to be a subspace of $\ell_1(V(K))$. We call the induced metric on $K$ the [[**metric of scale $\kappa$ on $K$**]{}]{}. For $\kappa=1$ it is the standard metric, as defined in [@hu1965 p. 100]. The topology induced by this metric is called [[**the metric topology**]{}]{}.

A [[**polyhedron**]{}]{} is a simplicial complex endowed with the metric topology.

Let $K$ and $L$ be polyhedra. We say that $K$ is a [[**full subpolyhedron**]{}]{} of $L$ if whenever the vertices $v_0,\ldots, v_n$ span a simplex in $K$ and each $v_i$ is a vertex in $L$, then the $v_i$ span a simplex in $L$.

\[lem:complex is ane\] A locally finite-dimensional polyhedron is a complete metric $ANE(\infty)$-space.

It is complete by [@hu1965 Lemma 11.5]. It is an $ANE(\infty)$-space by [@hu1965 Theorem 11.3].

Let $K$ and $L$ be simplicial complexes and let $p \colon K \to L$. We say that $p$ is [[**quasi-simplicial**]{}]{} if it is a simplicial map 