  We describe a bicategory $(\mathcal{R}ed\,\mathcal{O}rb)$ of reduced
orbifolds in the framework of classical differential geometry (i.e. without any
explicit reference to notions of Lie groupoids or differentiable stacks, but
only using orbifold atlases, local lifts and changes of charts). In order to
construct such a bicategory, we first define a $2$-category
$(\mathcal{R}ed\,\mathcal{A}tl)$ whose objects are reduced orbifold atlases (on
any paracompact, second countable, Hausdorff topological space). The definition
of morphisms is obtained as a slight modification of a definition by A. Pohl,
while the definitions of $2$-morphisms and compositions of them is new in this
setup. Using the bicalculus of fractions described by D. Pronk, we are able to
construct the bicategory $(\mathcal{R}ed\,\mathcal{O}rb)$ from the $2$-category
$(\mathcal{R}ed\,\mathcal{A}tl)$. We prove that
$(\mathcal{R}ed\,\mathcal{O}rb)$ is equivalent to the bicategory of reduced
orbifolds described in terms of proper, effective, \'etale Lie groupoids by D.
Pronk and I. Moerdijk and to the $2$-category of reduced orbifolds described by
several authors in the past in terms of a suitable class of differentiable
Deligne-Mumford stacks.
