Subject: categories: Re: Limits

From: [email-masked]
Date: Fri, 4 May 2001 14:04:36 -0700 (PDT)

i wrote:

|phenomenon of adjoint linear operators is, in yetter's terminology, a
|sort of decategorification of the phenomenon of adjoint functors.

now that i think about i guess it was crane rather than yetter who
started using the term "categorification".

is it correct that lawvere and schanuel use the term "objectification"
(or something like that) to mean pretty much the same thing as what
crane meant by "categorification"?  i think i might actually prefer
"objectification" here but i mostly hang out near sub-communities
where "categorification" has caught on to a certain extent.

---

From: Ross Street <[email-masked]>
Date: Sun, 6 May 2001 10:26:04 +1000

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