Sheaf terminology question
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? Andy Pitts [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Late Alexander L. Rosenberg used the term epipresheaf in that case; I am not sure if he learned it somewhere or he made it up. The reasoning behind the terminology is clear. Zoran Škoda On 16 Jul 2018 14:10, "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?
Andy Pitts
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
participants (2)
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Andrew Pitts -
Zoran Škoda