I am currently analyzing a category which is not small, but locally small. I have seen that any coverage on any small category $\mathcal{C}$ generates a unique Grothendieck Topology on $\mathcal{C}$ in the book Sketches of an Elephant. Is this however also the case for locally small categories, or maybe essentially small categories? If it is not, a counterexample would be really helpful. (Edit): Comments explain that this question is wrong.
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$\begingroup$ Are your covering families small? $\endgroup$– James E HansonCommented Nov 15, 2023 at 18:31
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$\begingroup$ I guess there's really two questions there: Are your covering families themselves small, and are the collections of covering families generated by small subcollections in the appropriate sense? $\endgroup$– James E HansonCommented Nov 15, 2023 at 18:35
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$\begingroup$ I think the question is confused. It's already the case for small categories that two different coverages can generate the same Grothendieck topology. "Any coverage generates a unique Grothendieck topology" means there is a function from coverages to topologies, but the function is not injective. $\endgroup$– Mike ShulmanCommented Nov 15, 2023 at 18:38
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$\begingroup$ @JamesHanson Yes, the covering families are small, should I maybe specify what category I am specifically working in? $\endgroup$– MaatCommented Nov 15, 2023 at 20:58
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2$\begingroup$ A topology is a particular kind of coverage, which generates itself. So any coverage that is not itself a topology provides an example: itself and the topology that it generates are two distinct coverages that generate the same topology. $\endgroup$– Mike ShulmanCommented Nov 16, 2023 at 5:48
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