\[lem:simplicial regular\], hence it is a weak $n$-homotopy equivalence.

Let $\mathcal{A} \subset {{\mathcal{B}_{L}}}$ such that $\bigcap_{A \in \mathcal{A}} A \neq \emptyset$. By Lemma \[lem:stars\], $\bigcap_{A \in \mathcal{A}} A$ is an absolute extensor in dimension $n$. We have $B = \bigcap_{A \in \mathcal{A}} p^{-1}(A) = p^{-1}(\bigcap_{A \in \mathcal{A}} A)$. The map $p_{|B} \colon B \to \bigcap_{A \in \mathcal{A}} A$ satisfies the conditions of Lemma \[lem:simplicial regular\], hence it is a weak $n$-homotopy equivalence. Therefore $B$ has vanishing homotopy groups in dimensions less than $n$ and by Theorem \[thm:dugundji\], it is an absolute extensor in dimension $n$. Hence $p^{-1}({{\mathcal{B}_{L}}})$ is an $AE(n)$-cover.

A lifting condition
===================

\[lem:single lift\] Let $p \colon Y \to Z$ be a surjective map of metric spaces. Let ${{\mathcal{F}_{}}}$ be a cover of $Z$. Assume that either ${{\mathcal{F}_{}}}$ is an open cover or ${{\mathcal{F}_{}}}$ is a closed, locally finite, locally finite-dimensional cover. Assume that $p^{-1}({{\mathcal{F}_{}}})$ is an $AE(n)$-cover.

Let $X$ be an at most $n$-dimensional metric space. Let $f \colon X \to Z$. Let $A$ be a closed subset of $X$. Let $g_0 \colon A \to Y$ be a map such that $p \circ g_0 = f_{|A}$ (a lift of $f$ on $A$).

Then there exists a map 