Sheaf terminology question

Sheaf terminology question

[Andrew Pitts]

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



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Re: Sheaf terminology question

[Andrew Pitts]

> 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




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Re: Sheaf terminology question

[Zoran Škoda]

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
>
>

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