Orbifold Atlas Groupoids

Abstract

We study orbifolds and strong maps of orbifolds. We begin with introducing a representation for orbifolds that consists of internal categories in the category of topological spaces. These categories are built from atlas charts and chart embeddings without equivalence relation. They represent orbifolds and atlas maps, but do not work well for general strong maps. We generalize the notion of category of fractions to internal categories in the category of topological spaces. We find its universal property for an internal category in the category of topological spaces. We apply this to the atlas category to obtain an atlas groupoid. We give a description of strong maps of orbifolds and the equivalence relation on them in terms of atlas groupoids. We define paths in orbifolds as strong maps. We use our construction to give an explicit description of the equivalence classes on such paths in terms of charts and chart embeddings.