into $\beta L$.

\[lem:qs is lipschitz\] Assume that $K$ and $L$ are polyhedra endowed with metrics of scale $\kappa$ and $\lambda$ respectively. If $p \colon K \to L$ is quasi-simplicial, then it is $\frac{\lambda}{2\kappa}$-Lipschitz.

Weak $n$-homotopy
-----------------

We say that a map is a [[**weak $n$-homotopy equivalence**]{}]{} if it induces isomorphisms on homotopy groups of dimensions less than $n$, regardless of the choice of basepoint.

Let ${{\mathcal{F}_{}}}$ be a cover of a space $X$. We say that the two maps $f, g \colon Y \to X$ are [[**${{\mathcal{F}_{}}}$-close**]{}]{} if for each $y \in Y$ there exists $F \in {{\mathcal{F}_{}}}$ such that $f(y), g(y) \in F$.

Let ${{\mathcal{U}_{}}}$ be a cover of a space $X$. We say that two maps $f, g \colon Y \to X$ are [[**${{\mathcal{U}_{}}}$-homotopic**]{}]{} if there exists a homotopy $H \colon Y \times [0, 1] \to X$ whose paths refine ${{\mathcal{U}_{}}}$, i.e. for each $y \in Y$ there exists $U \in {{\mathcal{U}_{}}}$ such that $H(\{ y \} \times [0, 1])\subset U$.

Carrier Theorem {#sec:carrier theorem}
---------------

Let $\mathcal{C}$ be a class of topological spaces. We say that a cover ${{\mathcal{F}_{}}}$ of a topological space is a $\mathcal{C}$-cover, if for each $\mathcal{A} \subset \mathcal{F}$ the intersection $\bigcap \mathcal{A}$ 