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Sieves and sites


Bach
 Bach
(@bach)
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This was the first time that I learn the notion of sieves and I wonder if any sieve comes from a site, assuming that our category has all fiber products? Can you give me a counter example if this is wrong? 


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Bach
 Bach
(@bach)
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I am sorry for my stupid question. I learnt site before when I studied cohomology and always thought that site is a category with covering families. So my question should be if it is true that any Grothendieck topology is generated by some covering families (Grothendieck pretopology), assuming that our category has all fiber products, and give a counter example if this is false? 


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Joshua Wrigley
(@jwrigley)
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Joined: 12 months ago
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Hi Bach, every Grothendieck pretopology generates a Grothendieck topology.  In other terminology a pretopology is called a basis, which drives home the analogy that a basis generates a topology, but multiple different bases generate the same topology.

Specifically, if S \in K(c) is a covering family for c in a pretopology K then \{ f \colon d \to c \mid f = g\circ h, \, g \in S\} \in J(c) is the sieve generated by S in the topology J generated by K.


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Bach
 Bach
(@bach)
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@jwrigley I am sorry that it seems that my question is confusing. Of course I learnt that every Grothendieck pretopology generates a Grothendieck topology and you wrote the definition of such Grothendieck topology clearly in your reply. What I ask is some kind of inverse statement: starting with a Grothendieck topology in a category with all fiber products, can you show me a Grothendieck pretopology generating our Grothendieck topology (it seems that I repeat my words). I may say that this is similar to the trivial fact that every usual topology has a basis, but it is still something that one needs to prove, and different from the procedure of taking a basis and generating a topology from it.


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Joshua Wrigley
(@jwrigley)
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Joined: 12 months ago
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Yes, given a Grothendieck topology J, let K be the pretopology where K(c) = \{S \mid \langle S \rangle \in J(c)\} where \langle S \rangle = \{ f \mid f = g \circ h, g \in S \}.


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