  This manuscript attempts to present a way in which the classical construction
of the Dirac operator can be carried over to the setting of diffeology. A more
specific aim is to describe a procedure for gluing together two usual Dirac
operators and to explain in what sense the result is again a Dirac operator.
Since versions of cut-and-paste (surgery) operations have already appeared in
the context of Atiyah-Singer theory, we specify that our gluing procedure is
designed to lead to spaces that are not smooth manifolds in any ordinary sense,
and since much attention has been paid in recent years to Dirac operators on
spaces with singularities, we also specify that our approach is more of a
piecewise-linear nature (although, hopefully, singular spaces in a more
analytic sense will enter the picture sooner or later; but this work is not yet
about them). Most of it is devoted to the diffeological versions of the
components that go into the standard definition of a Dirac operator as the
composition of a Clifford connection with Clifford action by sections of the
cotangent bundle; a diffeological Dirac operator is then standardly defined.
