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May 7, 2020 at 19:26 comment added D.-C. Cisinski Generalizations of these topologies are very alive nowadays in the work of Bhatt, Clausen, Mathiew, Morrow, Scholze on various flavours of $K$-theory and étale cohomology. Such descent result have their origin in Grothendieck's SGA 1 and SGA 4, where proper descent and invariance along universal homeomorphism (such as the Frobenius map) play a central role.
May 7, 2020 at 19:26 comment added D.-C. Cisinski Non-subcanonical topologies are very important: Voevodsky introduced a wealth of Grothendieck topologies on the category of schemes which are not subcanonical (h-topology and its variants) which express proper descent in various form and have many important applications in algebraic geometry (to define a reasonnable intersection theory for possibly singular schemes or in the recent proof of Weibel's conjecture).
May 7, 2020 at 13:52 history edited Praphulla Koushik CC BY-SA 4.0
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Sep 8, 2019 at 6:10 comment added Praphulla Koushik The two seemingly different functors in question are actually related. There is an equivalence of categories fibered in sets over $\mathcal{C}$ of the form $\underline{U}$ and functors of the form $h_U:\mathcal{C}^{op}\rightarrow \text{Set}$. This is special case of Proposition 3.26 (page $57$) of Angelo Vistoli's notes which says "(This is ) an equivalence of the category of categories fibered in sets over $\mathcal{C}$ and the category of functors $\mathcal{C}^{op}\rightarrow (\text{Set} )$ ."
Sep 8, 2019 at 5:39 history edited Praphulla Koushik CC BY-SA 4.0
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Sep 8, 2019 at 5:33 comment added Praphulla Koushik @GLe "A topology where all the representable functors are sheaves is called subcanonical. Since topologies on a small category form a locale, there exists a biggest topology such that all representable functors are sheaves, and is called the canonical one." is definitely a nice comment :) I have seen these subcanonical sites in Vistoli's notes sometime back, but did not gave attention.. I saw that now...
Sep 8, 2019 at 5:28 comment added display llvll no, it is not always true that the repr. functors are sheaves. Take for eg. a small category, where you declare all sieves covering. Then you get a topology where the only sheaf is the presheaf constant on the singleton. A topology where all the representable functors are sheaves is called subcanonical. Since topologies on a small category form a locale, there exists a biggest topology such that all representable functors are sheaves, and is called the canonical one.
Sep 8, 2019 at 2:59 history asked Praphulla Koushik CC BY-SA 4.0