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Let $\mathcal{C}$ be a category. Fixing an object $U$ of $\mathcal{C}$, there are some obvious functors we can associate to it, for example:

  1. $h_U:\mathcal{C}^{op}\rightarrow \text{Set}$ given by $V\mapsto \text{Hom}(V,U)$.
  2. $\Phi_U:\underline{U}\rightarrow \mathcal{C}$ given by $(V\rightarrow U)\mapsto V$ and $$(f:(V\rightarrow U)\rightarrow (V'\rightarrow U))\mapsto (f:V\rightarrow V').$$ This functor is actually a category fibered in groupoids over $\mathcal{C}$.

When defining Grothendieck topology on $\mathcal{C}$, did they think to make sure these functors behave the way we expect, in the sense $h_U:\mathcal{C}^{op}\rightarrow \text{Set}$ is a sheaf, the category fibered in groupoids $\Phi_U:\underline{U}\rightarrow \mathcal{C}$ is a stack? Was this one of the restrictions they had in mind or did it happen as coincidence? I do not know for all examples but, as mentioned in my other question, topologies on $\text{Sch}/S$ are such that $h_X:(\text{Sch}/S)^{op}\rightarrow \text{Set}$ are sheaves.

Question : Is it always true that representable functors $h_U:\mathcal{C}^{op}\rightarrow \text{Set}$ are sheaves? Is it always true that the CFG $\Phi_U:\underline{U}\rightarrow \mathcal{C}$ is a stack? Does making sure/expecting these things has some significance in defining the notion of Grothendieck topology?

Edit : User giuseppe mentions about subcanonical/canonical sites on a category. Angelo Vistoli's notes says (Definition 2.57, page $37$) "A topology $\mathcal{T}$ on a category $\mathcal{C}$ is called subcanonical if every representable functor on $\mathcal{C}$ is a sheaf with respect to $\mathcal{T}$. A subcanonical site is a category endowed with a subcanonical topology. There are examples of sites that are not subcanonical, but I have never had dealings with any of them." This seem to support my intuition that we are only interested in topologies that make representable functors to be sheaves.

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    $\begingroup$ 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. $\endgroup$
    – display llvll
    Commented Sep 8, 2019 at 5:28
  • $\begingroup$ @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... $\endgroup$
    – Praphulla Koushik
    Commented Sep 8, 2019 at 5:33
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    $\begingroup$ 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 $\mathcal{C}^{op}\rightarrow (\text{Set} )$ ." $\endgroup$
    – Praphulla Koushik
    Commented Sep 8, 2019 at 6:10
  • $\begingroup$ 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). $\endgroup$
    – D.-C. Cisinski
    Commented May 7, 2020 at 19:26
  • $\begingroup$ 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. $\endgroup$
    – D.-C. Cisinski
    Commented May 7, 2020 at 19:26

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