categories: Re: colimits of categories

[Lutz Schroeder <lschrode@tzi.de>]

 > Is there a good reference for the construction of colimits of
 > categories?
 > Here, by 'categories', I mean 1-categories; and I'm considering
 > 1Cat as a weak 2-category.  I've been playing around with this for
 > the last week; I can construct limits without much trouble (at
 > least if the index category is a 1-category; I assume that changing
 > the index category to a 2-category wouldn't cause any substantial
 > problems), but constructing colimits seems noticeably messier.
 > It's not too bad if your index category is filtered, but in general
 > it seems like a pain.

The construction of quotients of 1-categories (which, together with 
coproducts, make up arbitrary colimits) is indeed somewhat messy: the 
naive quotient (equivalence classes of objects and morphisms) forms a
graph with a partial composition operation that obeys the identity law 
but no further axioms. Over this structure, one can construct a free 
category (the category of paths, modulo the smallest congruence that 
makes the quotient map a functor), which is, then, the actual quotient 
category; which equivalence classes of arrows are or are not identified 
in the quotient category is, in the general case, hard to predict. 
References include

M. Bednarczyk, M. Borzyszkowski, and W. Pawlowski: Generalized 
congruences --- epimorphisms in Cat. Theory and Applications of 
Categories 5 (1999), 266-280

L. Schroeder and H. Herrlich: Free adjunction of morphisms. Applied 
Categorical Structures 8 (2000), 595-606.


Greetings,

Lutz


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Lutz Schroeder                  Phone +49-421-218-4683
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