categories: Re: colimits of categories

[S.J.Vickers@open.ac.uk]

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.