Re: associated sheaf functor

Re: associated sheaf functor

[Krzysztof Worytkiewicz]

> I don't quite understand this question.
> I was interested since I am looking at the plus construction as part
> of my work at the moment.

Let P be a presheaf on the site (C,J) and consider the "classical" plus
construction

  P^+(c) = colim_{R \in J(c)}Match(R,P)

where Match(R,P) is the set of matching families for the cover R \in
J(c) and the colimit is taken over J(c) ordered by reverse inclusion
(cf. McLane & Moerdijk) . This is a nice filtered colimit so x \in
P^+(c) can be expressed as an equivalence class of matching families.

 Suppose now that J is given by a basis K. It is not immediately clear
(at least not for me) what happens in a variant of the above where
Match(R,P) is taken as the set of matching families for the K-cover R.
Indeed, the notion of "common refinement"  for K-covers is not as handy
as the one for J-covers for the task at hand since op-ordering K-covers
will not necessarily give a filtered category. The other obvious
candidate for a "category K(c)" where a factorisation witnessing a
refinement (of K-covers) is a morphism (in the opposite category) will
probably fail to be filtered as well, so I wondered if anybody allready
looked at such things.

I agree that a one-sentence prose might have been a bit messy...

Cheers

Krzysztof

associated sheaf functor

[Krzysztof Worytkiewicz]

Dear All,

Is anybody aware of a variant of the "plus" construction giving the
"associated separated  presheaf" wrt to a Grothendieck topology  which
works on a basis as only piece of data (ie without generating the whole
topology and then applying the classical plus functor)? Any hint welcome...

Cheers

Krzysztof