We introduce notions of homotopy and cohomology for ordered groupoids.
We show that abstract homotopy theory can be used to define
a suitable notion of homotopy equivalence for ordered groupoids
(and hence inverse semigroups).
As an application of our theory we prove a theorem which is an exact
counterpart of the well-known result in topology which states that
every continuous function can be factorised into a homotopy equivalence
followed by a fibration.
We show that this factorisation is isomorphic to the one constructed by
Steinberg in his Fibration Theorem,
originally proved using a generalisation of Tilson's derived category.
We show that the cohomology of an ordered groupoid can be defined as
the cohomology of a suitable small category, in doing so
we generalise the cohomology of inverse semigroups due to Lausch.
We define extensions of ordered groupoids and show that these provide
an interpretation of low-dimensional cohomology groups.
