# Categories of components and loop-free categories

## Emmanuel Haucourt

Given a groupoid G one has, in addition to the equivalence of categories E from G to its skeleton, a fibration F 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]=id_x for every object x of G, we prove there is a fibered equivalence from C[\Sigma^{-1}] to C/\Sigma when \Sigma is a Yoneda-system of a loop-free category C. In fact, all the equivalences from C[\Sigma^{-1}]\$ to C/\Sigma are fibered. Furthermore, since the quotient C/\Sigma shrinks as \Sigma grows, we define the component category of a loop-free category as C/{\overline{\Sigma}} where \overline{\Sigma} is the greatest Yoneda-system of C.

Keywords: category of fractions, generalized congruence, quotient category, scwol, small category without loop, Yoneda-morphism, Yoneda-system, concurrency

2000 MSC: 18A20, 18A22, 18A32, 18B35, 18D30, 18E35

Theory and Applications of Categories, Vol. 16, 2006, No. 27, pp 736-770.

http://www.tac.mta.ca/tac/volumes/16/27/16-27.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/27/16-27.pdf

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