Categories of Components and Loop-free Categories

Given a groupoid G one has, in addition to the equivalence of categories E from G to its skeleton, a fibration F (Definition 1.11) from G to its set of connected components (seen as a discrete category). From the observation that E and F differ unless G[x, x] = {idx} for every object x of G, we prove there is a fibered equivalence (Definition 1.12) from C[Σ-1] (Proposition 1.1) to C/Σ (Proposition 1.8) when Σ is a Yoneda-system (Definition 2.5) of a loop-free category C (Definition 3.2). In fact, all the equivalences from C[Σ-1] to C/Σ are fibered (Corollary 4.5). Furthermore, since the quotient C/Σ shrinks as Σ grows, we define the component category of a loop-free category as C/Σ where Σ is the greatest Yoneda-system of C (Proposition 3.7).