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Edit again: this answer is wrong, see the comments. The new set of families (for each object $X$) is called the sieve generated by the existing covers of $X$. One term for a Grothendieck pretopology is a basis for a Grothendieck topology, and different bases can give rise to the same Grothendieck topology. All of them, and the topology they generate, have the same sheaves. See here for example. Edit: Actually it is proposition C.2.1.9 in Johnstone's Sketches of an Elephant (Google books ) |
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The new set of families (for each object $X$) is called the sieve generated by the existing covers of $X$. One term for a Grothendieck pretopology is a basis for a Grothendieck topology, and different bases can give rise to the same Grothendieck topology. All of them, and the topology they generate, have the same sheaves. See here for example. Edit: Actually it is proposition C.2.1.9 in Johnstone's Sketches of an Elephant (Google books ) |
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The new set of families (for each object $X$) is called the sieve generated by the existing covers of $X$. One term for a Grothendieck pretopology is a basis for a Grothendieck topology, and different bases can give rise to the same Grothendieck topology. All of them, and the topology they generate, have the same sheaves. See here for example. |
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