Vaughan Pratt wrote:
of what use are non-free cocompletions? Is there any reason not to define "cocompletion" to make it free?
Michael Shulman wrote:
To me the unadorned word "completion" connotes an idempotent operation
The presheaf category [C^op,Set] is a co-completion of C. Its full subcategory of limit-preserving functors is another co-completion of C. Both are "free", but with respect to a different criterion: the Yoneda embedding into the first one does not preserve existing colimits, whereas into the second one it does. In particular, the second operation is idempotent (up to equivalence of categories) if C is already co-complete (this also follows from the adjoint functor theorem). It would therefore qualify as a "completion" in the sense that Mike Shulman mentioned. Any co-complete category between these two extremes is presumably also a (non-free) co-completion. -- Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]