* Re: associated sheaf functor
[not found] <Law10-F41jkO22ZhCl8000005f4@hotmail.com>
@ 2003-09-17 11:41 ` Krzysztof Worytkiewicz
0 siblings, 0 replies; 2+ messages in thread
From: Krzysztof Worytkiewicz @ 2003-09-17 11:41 UTC (permalink / raw)
To: categories
> 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
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* associated sheaf functor
@ 2003-09-16 17:12 Krzysztof Worytkiewicz
0 siblings, 0 replies; 2+ messages in thread
From: Krzysztof Worytkiewicz @ 2003-09-16 17:12 UTC (permalink / raw)
To: categories
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
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2003-09-17 11:41 ` associated sheaf functor Krzysztof Worytkiewicz
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