Let $C$ be a fixed category, and let $T_1$ and $T_2$ be two Grothendieck (pre)topologies on $C$. We say $T_1$ is subordinate to $T_2$ if every covering in $T_1$ has a refinement in …

Hi, The first axiom for a Grothendieck (pre)topology on a category $C$ says that for every object $X\in C$, the family consisting of just the identity $1_X : X\to X$ should be a c …

Let $A$ be a (commutative) ring. A family $(B_i)_{i\in I}$ of $A$-algebras is said to be an fppf cover if it satisfies three properties: (1) each $B_i$ is flat as an $A$-module, (2 …

To define a Grothendieck topology of a category, we usually require that the category is small. Question 1: Why do we need to require the category to be small? I thought that th …

Hi, The following result seems to be well known, but I can't come up with a proof. Suppose that $C$ is a topos and that $F\to G$ is an effective epimorphism in $C$. If $P$ is any …

For a presheaf $F$ on a category equipped with a pretopology, one has the sheafification $F^{\sharp}$ of $F$. I know well the plus-construction of sheafification, which is present …

Let $G$ be a topological group. Recall that its classifying space $BG$ is a CW-complex which is the base of a locally trivial principal bundle of group $G$, with contractible tota …

The wikipedia article(s) as well as the nlab article(s) about Grothendieck topologies and Grothendieck pretopologies are careful to differentiate the two very emphatically and to p …

I have a very quick question. Is there an easy example of a representable presheaf on a site that is not a sheaf? This certainly can't happen on a small FPPF site so I would expect …

(I find it easier to think in terms of Grothendieck pretopologies, instead of topologies. If this annoys any experts, please forgive me.) Suppose that $C$ is a (full) subcategory …

A coverage $J$ on a category $C$ assigns to an object $U$ of $C$ a set of covering families $J(U)$. The covering families are required to be stable under pullback, which amounts to …

The category of schemes has a large (and to me, slightly bewildering) number of what seem like different Grothendieck (pre)topologies. Zariski, ok, I get. Etale, that's alright, I …

I recently tried to explain the fppf site to a differential geometer. I started with the etale site, where I had two motivating claims: If X is a smooth projective variety over …

For each topos $\mathbb E$ let $\mathcal O(\mathbb E)$ be the locally presentable category of objects in $\mathbb E$. We can make $\mathcal O$ into a contravariant functor to the c …

I have the following Grothendieck pretopologies on the category of schemes. The first one, the covers are families of morphisms $\{ U_i \to X \}$ such that for every point $x \in …