Homotopy in small categories

In this work we explore topological approximations to category theory. More precisely, we will study the classifying space and the homotopic distance between functors as an adaptation of the topological concept of homotopic distance between continuous maps to the context of small categories. This notion can be viewed as a generalization of the recently studied LS-category and categorical complexity, two important invariants by homotopic equivalence between small categories. Moreover we will see how homotopic distance can be generalized to the higher homotopic distance and how it relates to fibrations.