  An immersion $f : {\mathcal D} \rightarrow \mathcal C$ between
$\Delta$-complexes is a $\Delta$-map that induces injections from star sets of
$\mathcal D$ to star sets of $\mathcal C$. We study immersions between
finite-dimensional connected $\Delta$-complexes by replacing the fundamental
group of the base space by an appropriate inverse monoid. We show how conjugacy
classes of the closed inverse submonoids of this inverse monoid may be used to
classify connected immersions into the complex. This extends earlier results of
Margolis and Meakin for immersions between graphs and of Meakin and Szak\'acs
on immersions into $2$-dimensional $CW$-complexes.
