  Let $K$ be a simplicial complex on vertex set $V$. $K$ is called $d$-Leray if
the homology groups of any induced subcomplex of $K$ are trivial in dimensions
$d$ and higher. $K$ is called $d$-collapsible if it can be reduced to the void
complex by sequentially removing a simplex of size at most $d$ that is
contained in a unique maximal face.
  We define the $t$-tolerance complex of $K$, $\mathcal{T}_t(K)$, as the
simplicial complex on vertex set $V$ whose simplices are formed as the union of
a simplex in $K$ and a set of size at most $t$. We prove that for any $d$ and
$t$ there exists a positive integer $h(t,d)$ such that, for every
$d$-collapsible complex $K$, the $t$-tolerance complex $\mathcal{T}_t(K)$ is
$h(t,d)$-Leray.
  The definition of the complex $\mathcal{T}_t(K)$ is motivated by results of
Montejano and Oliveros on "tolerant" versions of Helly's theorem. As an
application, we present some new tolerant versions of the colorful Helly
theorem.
