Z$ be a map. Let ${{\mathcal{F}_{}}} = \{ F_i \}_{i \in I}$ be a cover of $Z$. A [[**pull-back of ${{\mathcal{F}_{}}}$**]{}]{} is a cover $p^{-1}({{\mathcal{F}_{}}})$ of $Y$ defined by the formula $$p^{-1}({{\mathcal{F}_{}}}) = \{ p^{-1}(F_i) \}_{i \in I}.$$ The pull-back $p^{-1}({{\mathcal{F}_{}}})$ retains the indexing set $I$ of ${{\mathcal{F}_{}}}$.

\[lem:pullback properties\] Let $f \colon X \to Y$ be a map, let ${{\mathcal{G}_{}}}$ be a cover of $Y$ and let $f^{-1}({{\mathcal{G}_{}}})$ be a pull-back of ${{\mathcal{G}_{}}}$.

1.  If $f$ is surjective, then the covers $f^{-1}({{\mathcal{G}_{}}})$ and ${{\mathcal{G}_{}}}$ are isomorphic.

2.  If ${{\mathcal{G}_{}}}$ is open / closed / locally finite / locally finite-dimensional, then so is $f^{-1}({{\mathcal{G}_{}}})$.

Nerve Theorem
=============

A Nerve Theorem in its abstract form states that if two spaces admit isomorphic $AE(n)$-covers, then they are weak $n$-homotopy equivalent. For general spaces, some local finiteness and dimension restrictions are placed on the covers [@nagorko2007]. In our case we need such a theorem without these restrictions, which we are able to prove for polyhedra covered by subcomplexes.

The goal of this section is to prove the following theorem.

\[thm:nregular pullback\] If a quasi-simplicial map $p \colon K \to L$ of polyhedra is surjective and $n$-regular, then it is a weak $n$-homotopy equivalence 