On 13 Jul 2018, at 15:10, Andrew Pitts <Andrew.Pitts@cl.cam.ac.uk> wrote:
Given a coverage on a small category, the condition for a Set-valued presheaf on the category to be a sheaf has an existence part and a uniqueness part. If only the uniqueness part holds, one says that the presheaf is separated. Is there a name out there for presheaves that only satisfy the existence part of the sheaf condition?
Many thanks to those who responded with suggestions. I won’t repeat replies that were also sent to categories@mta.ca, but there were a couple sent privately that I pass on: Peter Johnstone reminded me of the fact that category-theorists tend to use the prefix “weak” to remove the uniqueness part of a unique existence condition. So one could speak of "weak sheaves". (A slight problem with that is that when I say it, it sounds a lot like wheat sheaves :-).) Tom Hirschowitz pointed out that there is the concept of effective descent data in stack theory, so one might use the term “effective presheaf” for what I was referring to. I quite like that. Andy [For admin and other information see: http://www.mta.ca/~cat-dist/ ]