Abstract: In differential geometry, we know that an irrational torus is a pathological space rather than a smooth manifold. However, we can still do some kind of differential geometry by using a new categorical viewpoint towards geometry. In this talk, basic geometry, homotopy groups, smooth bundles, and some new work on tangent spaces for irrational torii will be introduced. If time permits, we will talk about differential forms, de Rham cohomology and foliation theory on irrational torii as well.