categories: Re: Limits

[Ross Street <street@ics.mq.edu.au>]

I very much agree with James Dolan's response to the question of 
comparing categorical limits and adjoint functors with their abstract 
counterparts in analysis.

Other words that have been used for "objectification" and 
"categorification" are "laxification" and "identity breaking".

The original questions were a bit like asking: "Is the plus in an 
abelian group a categorical coproduct?"  Lots of abelian groups can 
arise by taking isomorphism classes and using a categorical 
coproduct: but then we lose the beautiful universal property.

Along the same lines, I enjoy bicategories, with coproducts in their 
homcategories (preserved by composition), much more than additive 
categories. Not only is every global coproduct in such a bicategory 
also a global product, but the projections from the global products 
are right adjoint to the coprojections into the coproduct.

Ross