On 18 Jan 2006, at 09:36, V. Schmitt wrote:
Marco I am not very awake this morning but i think that this construction of formally inverted some arrow is well known for long (cf for instance Borceux's handbooks on localizations). Am i wrong? Cheers, Vincent
Categories of fractions are indeed very well-known, but satisfy a different universal property: to make *invertible* the assigned arrows (instead of making them *identities*). But you can view categories of fractions at the light of what I was saying. Take in Cat the (closed) ideal of functors which send every map to an isomorphism, or equivalently of those functors which factor through a groupoid. With respect to this ideal, the kernel of a functor f: X -> Y is the (wide and replete) subcategory of maps which f turns into isomorphisms, while the cokernel is the category of fractions of Y which inverts all arrows reached by f. Best regards Marco G. PS. And - thinking of Jean Pradine's message - yes, of course, quotient of groupoids are important, but have special features of their own; as he is pointing out.