  It is well known that under mild conditions on a connected topological space
$\mathcal X$, connected covers of $\mathcal X$ may be classified via conjugacy
classes of subgroups of the fundamental group of $\mathcal X$. In this paper,
we extend these results to the study of immersions into 2-dimensional
CW-complexes. An immersion $f : {\mathcal D} \rightarrow \mathcal C$ between
CW-complexes is a cellular map such that each point $y \in {\mathcal D}$ has a
neighborhood $U$ that is mapped homeomorphically onto $f(U)$ by $f$. In order
to classify immersions into a 2-dimensional CW-complex $\mathcal C$, we need to
replace the fundamental group of $\mathcal C$ 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.
