Steve is right about the messiness. The very first paper I am aware of on the subject used 25 pages of detailed computations to show that the amalgamated sum of two categories gotten by identifying a single object of one category with an object of the other exists. That must have been sometime in the 60s. What a mess! A good mathematician, whom I won't embarrass by identifying. Michael On Tue, 29 Jan 2002 S.J.Vickers@open.ac.uk wrote:
David Carlton asks -
Is there a good reference for the construction of colimits of categories?
If I remember correctly, Philip Higgins's little book "Notes on categories and groupoids" (Van Nostrand Reinhold 1971) is good on that kind of thing.
You're right that the non-filtered colimits are distinctly messier than the limits. There are two reasons.
The first is that that is the way of algebra anyway - think of colimits of monoids or groups, for instance. Universal algebra says that colimits exist for every algebraic theory, but the construction is intricate. You first make an algebra of all possible terms (expressions) and then factor out a congruence to enforce the equational laws and the cocone commutativities.
The second reason is that categories are models not of an algebraic theory, but of an essentially algebraic theory (some operations - specifically here composition - are only partial, with domain of definition stipulated equationally). The techniques of universal algebra still work, by and large, but the proof is even more intricate than the 2-step process in algebra. This is because imposing equations can cause new terms to spring into existence.
Steve Vickers.
30-Jan-2002 09:03:32 -0400,3062;000000000001-00000000