In today's research, intrinsically noncommutative spaces are considered from the perspective of several branches of modern physics, including quantum gravity, string theory, and statistical physics. From this point of view, it seems necessary to have a concept of space and its geometry that is fundamentally noncommutative. Virtual Topology and Functor Geometry presents new ideas for the development of an intrinsically noncommutative geometry. Written in an easily accessible, colloquial style, the book provides numerous suggestions for possible projects, ranging from exercise level to research level. Topics covered include Grothendieck categorical representations and structure sheaves.