Orbispaces and their mapping spaces via groupoids: A categorical approach

This paper gives an accessible introduction to how orbispaces can be represented by orbigroupoids, i.e., proper étale groupoids, and how the category of orbispaces can be created as a bicategory of fractions from these orbigroupoids. We also explore how these groupoid representations can be used to create mapping groupoids of maps between orbispaces. Numerous examples are worked out, including how to describe the orbispace maps from the one point orbispace with isotropy group G to an arbitrary orbispace, and relate this to the inertia groupoid.