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We describe a bicategory $(\mathcal{R}ed\,\mathcal{O}rb)$ of reducedorbifolds in the framework of classical differential geometry (i.e. without anyexplicit reference to notions of Lie groupoids or differentiable stacks, butonly using orbifold atlases, local lifts and changes of charts). In order toconstruct such a bicategory, we first define a $2$-category$(\mathcal{R}ed\,\mathcal{A}tl)$ whose objects are reduced orbifold atlases (onany paracompact, second countable, Hausdorff topological space). The definitionof 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 thissetup. Using the bicalculus of fractions described by D. Pronk, we are able toconstruct 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 reducedorbifolds described in terms of proper, effective, \'etale Lie groupoids by D.Pronk and I. Moerdijk and to the $2$-category of reduced orbifolds described byseveral authors in the past in terms of a suitable class of differentiableDeligne-Mumford stacks.

A bicategory of reduced orbifolds from the point of view of differential geometry - I