We define a $2$-category structure $(\textbf{Pre-Orb})$ on the category of reduced complex orbifold atlases. We construct a $2$-functor $F$ from $(\textbf{Pre-Orb})$ to the $2$-category $(\textbf{Grp})$ of proper étale effective groupoid objects over the complex manifolds.

Both on $(\textbf{Pre-Orb})$ and on $(\textbf{Grp})$ there are natural equivalence relations on objects: (a natural extension of) equivalence of orbifold atlases in $(\textbf{Pre-Orb})$ and Morita equivalences in $(\textbf{Grp})$. We prove that $F$ induces a bijection between the equivalence classes of its source and target.

In this paper we investigate the construction of bicategories of fractions originally described by D. Pronk: given any bicategory $\mathbf{\mathscr{C}}$ together with a suitable class of morphisms $\mathbf{W},$ one can construct a bicategory $\mathbf{\mathscr{C}}\left[\mathbf{W}^{-1}\right],$ where all the morphisms of $\mathbf{W}$ are turned into internal equivalences$,$ and that is universal with respect to this property.

Most of the descriptions leading to this construction were long and heavily based on the axiom of choice. In this paper we considerably simplify the description of the equivalence relation on $2$-morphisms and the constructions of associators, vertical and horizontal compositions in $\mathbf{\mathscr{C}}\left[\mathbf{W}^{-1}\right],$ thus proving that the axiom of choice is not needed under certain conditions. The simplified description of associators and compositions will also play a crucial role in two forthcoming papers about pseudofunctors and equivalences between bicategories of fractions.

We describe a bicategory $(\mathcal{R}\mathbf{ed}\,\mathcal{O}\mathbf{rb})$ of reduced orbifolds in the framework of differential geometry (i.e. without any explicit reference to the 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 firstly define a $2$-category $(\mathcal{R}\mathbf{ed}\,\mathcal{A}\mathbf{tl})$ 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 definition 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 $(\mathcal{R}\mathbf{ed}\,\mathcal{O}\mathbf{rb})$ from the $2$-category $(\mathcal{R}\mathbf{ed}\,\mathcal{A}\mathbf{tl})$.

We prove that $(\mathcal{R}\mathbf{ed}\,\mathcal{O}\mathbf{rb})$ is equivalent to the bicategory of reduced orbifolds described in terms of proper$,$ effective$,$ étale Lie groupoids by D. Pronk and I. Moerdijk$,$ and to the well-known $2$-category of reduced orbifolds constructed from a suitable class of differentiable Deligne-Mumford stacks.

We fix any bicategory $\mathbf{\mathscr{A}}$ together with a class of morphisms $\mathbf{W}_{\mathbf{\mathscr{A}}},$ such that there is a bicategory of fractions $\mathbf{\mathscr{A}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{A}}}\right]$ (as described by D. Pronk).

Given another such pair $(\mathbf{\mathscr{B}},\mathbf{W}_{\mathbf{\mathscr{B}}})$ and any pseudofunctor $\mathcal{F}:\mathbf{\mathscr{A}}\rightarrow\mathbf{\mathscr{B}},$ we find necessary and sufficient conditions in order to have an induced pseudofunctor $\mathcal{G}:\mathbf{\mathscr{A}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{A}}}\right]\rightarrow\mathbf{\mathscr{B}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{B}}}\right]$.

Moreover$,$ we give a simple description of $\mathcal{G}$ in the case when the class $\mathbf{W}_{\mathbf{\mathscr{B}}}$ is "right saturated".

We prove a result of cohomology and base change for families of coherent systems over a curve. We use that in order to prove the existence of (non-split$,$ non-degenerate) universal families of extensions of coherent systems (in the spirit of the paper Universal families of extensions by H. Lange). Such results will be applied in subsequent papers in order to describe the wallcrossing for some moduli spaces of coherent systems.

We fix any bicategory $\mathbf{\mathscr{A}}$ together with a class of morphisms $\mathbf{W}_{\mathbf{\mathscr{A}}},$ such that there is a bicategory of fractions $\mathbf{\mathscr{A}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{A}}}\right]$. Given another such pair $(\mathbf{\mathscr{B}},\mathbf{W}_{\mathbf{\mathscr{B}}})$ and any pseudofunctor $\mathcal{F}:\mathbf{\mathscr{A}}\rightarrow\mathbf{\mathscr{B}},$ we find necessary and sufficient conditions in order to have an induced equivalence of bicategories from $\mathbf{\mathscr{A}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{A}}}\right]$ to $\mathbf{\mathscr{B}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{B}}}\right]$.

In the special case when $\mathbf{W}_{\mathbf{\mathscr{B}}}$ consists only of internal equivalences of $\mathbf{\mathscr{B}},$ the bicategories $\mathbf{\mathscr{B}}$ and $\mathbf{\mathscr{B}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{B}}}\right]$ are equivalent. So the previous result gives necessary and sufficient conditions in order to have an equivalence from any bicategory of fractions $\mathbf{\mathscr{A}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{A}}}\right]$ to any given bicategory $\mathbf{\mathscr{B}}$.

We fix any pair $(\mathbf{\mathscr{C}},\mathbf{W})$ consisting of a bicategory and a class of morphisms in it, admitting a bicalculus of fractions$,$ i.e. a "localization" of $\mathbf{\mathscr{C}}$ with respect to the class $\mathbf{W}.$ In the resulting bicategory of fractions$,$ we identify necessary and sufficient conditions for the existence of weak fiber products.

We give a definition of atlases for ineffective orbifolds$,$ and prove that this definition leads to the same notion of orbifold as that defined via topological groupoids.