$k$-connectedness of an inverse limit of polyhedra that is easy to verify. It is designed to aid detection of local $k$-connectedness in many examples of spaces arising in geometric group theory. We prove the following theorem.

Let $K$ and $L$ be simplicial complexes. We say that a map $p \colon K \to L$ is [[**$n$-regular**]{}]{} if it is quasi-simplicial (i.e. it is simplicial into the first barycentric subdivision $\beta L$ of $L$) and if for each simplex $\delta$ of $\beta L$ the inverse image $p^{-1}(\delta)$ has vanishing homotopy groups in dimensions less than $n$ (regardless of the choice of basepoint).

\[thm:main theorem\] Let $$X = \lim_{\longleftarrow}\big( K_1 \xleftarrow{p_1} K_2 \xleftarrow{p_2} \cdots \big).$$ Assume that for each $i$ the following conditions are satisfied.

1.  $K_i$ is a locally finite-dimensional simplicial complex endowed with the metric topology.

2.  $p_i$ is a quasi-simplicial map that is surjective and $n$-regular.

Then $X$ is locally $k$-connected for each $k < n$, and each short projection $\pi^m_i \colon K_m \to K_i$ and each long projection $\pi_i \colon X \to K_i$ is a weak $n$-homotopy equivalence (it induces homomorphisms on homotopy groups in dimensions less than $n$, regardless of the choice of basepoint).

Note that there is 