In "Higher algebraic K-theory I" Quillen defines a morphism inverting functor to be a functor from a category C to the category Sets which maps "arrows" in C to isomorphisms in Sets.

Proposition 1: The category of covering spaces of BC is canonically isomorphic to the category of morphism-inverting functors $F: C\rightarrow Sets$.

[For $C$ a small category, its classifying space $BC$ is the geometric realization of its nerve, $NC$]

This proposition plays an essential role in Quillen's Theorem 1 showing that his Q-construction agrees with Grothendieck's construction for $K_0$.

Theorem 1: $\pi_1(B(QC))$ is canonically isomorphic to the Grothendieck group $K_0(M)$

Questions: Have morphism-inverting functors played an important role in other contexts? Is there a more modern incarnation of morphism-inverting functors related to the fundamental groupoid of an infinity-category?

wasteof time, but there are certainly more modern treatments of the material on which your time would be better spent. – Harry Gindi Jan 28 '10 at 19:37