Subject: categories: re: Normal quotients of categories

From: "Reinhard Boerger" <[email-masked]>
Date: Wed, 18 Jan 2006 09:36:09 +0200

Hello,

let me add some remarks to Marco Grandis' posting.

> 1. Generalised quotients of categories.
>
>      A very general notion of generalised congruence in a category -
> also involving objects - can be found in a paper by Bednarczyk,
> Borzyszkowski and Pawlowski [BBP].

I have not yet looked at that paper, but I think the "natural" thing is
to consider equivalence relations R on a category C, which are
subcategories of CxC (i.e. closed under composition, not
necessary full; identities are in R by relexivity of R). In my
diplomarbeit "Kongruenzrelationen auf Kategorien" from 1977, I
considered that, but I was not the first one. Some years earlier
there was a paper by Jacques Mersch from Liege (Belgium), which
unfortunately appeared only in an internal publication of the
university of Liege. Moreover, I think I remember that John Isbell did
something on this subject.

>      The quotient  p: X -> X/A  is determined by the obvious
> universal property:

The universal property is also obvious in the general situation. For
small categories, a functor with this property always exists, let's
call it a quotient functor. For large categories it my happen that the
hom-sets of the quotient become large, even if the hom-sets of the
original category are small. In general, a congruence as above is
not a kernel of a functor; the quotient functor may identify more
morphisms. In my diplomarbeit, I rediscovered an example, which
had already been found by Mersch. The quotient functors are
exactly the regular epis in CAT. But unfortunately, they are not
closed under compositon, so a quotient of a quotient of C need not
be a quotient of C.

>      It is interesting to note that  p  automatically satisfies a 2-
> dimensional universal property, as one can easily deduce from the fact
> that natural transformations can be viewed as functors  X -> Y^2,
> with values in the category of morphisms of  Y.

Of course, this argument also works in the general situation.

                                                             Greetings
                                                         Reinhard Boerger