Re: Normal quotients of categories

[Marco Grandis <grandis@dima.unige.it>]

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.