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 -- ----------------------------------------------------------------------------- Lutz Schroeder Phone +49-421-218-4683 Dept. of Computer Science Fax +49-421-218-3054 University of Bremen lschrode@informatik.uni-bremen.de P.O.Box 330440, D-28334 Bremen http://www.informatik.uni-bremen.de/~lschrode ----------------------------------------------------------------------------- 29-Jan-2002 20:56:44 -0400,3194;000000000000-00000000