categories: Re: colimits of categories

[Michael Barr <barr@barrs.org>]

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.
> 
> 
>