Questions tagged [grothendieck-topologies]

For questions on the Zariski-open immersion, étale, flat, or other topologies defined via classes of morphisms in a category of schemes. Not to be confused with the usual Zariski topology in commutative algebra and algebraic geometry.

Filter by
Sorted by
Tagged with
0 votes
0 answers
39 views

Zariski opens of an affine space and (po)sites

There is a very neat presentation of the Zariski open sets in $\operatorname{Spec}(R)$: it is freely generated (under arbitrary unions and finite intersections, which distribute over each other) by ...
Trebor's user avatar
  • 4,610
1 vote
0 answers
42 views

Question regarding basis (for a Grothendieck topology)

I am studying a little bit of basis (for a Grothendieck topology), following MacLane's Sheaves in Geometry and Logic. Here they give an example as follows, Let $\mathbf{T}$ be a small category of ...
babu's user avatar
  • 169
3 votes
0 answers
28 views

Characterizing monos/epis of sheaves on a site as injective/surjective maps

Consider sheaves of sets on a site. In the stacks project injective maps of sheaves are defined to be maps that are injective along each component and surjective maps are ones in which each section ...
Jim's user avatar
  • 30.7k
1 vote
1 answer
49 views

Question about tag 03L7 in the stacks project (fpqc covers)

The stacks project omits most of the proof of tag 03L7. I think $(1) \Rightarrow (2) \Leftrightarrow (3)$ is pretty clear, but how do you prove $(2), (3) \Rightarrow (1)$? Say $(2)$ holds and $U \...
Jim's user avatar
  • 30.7k
0 votes
0 answers
41 views

Sheaves for a Grothendieck topology compatible with a pretopology

$ \newcommand{\cat}[1]{\mathcal{#1}} $ Let $ (\cat C,J) $ be a site. I'm trying to show that if the topology $ J $ comes from a pretopology $ K $ in the sense that for each $ X\in \cat C $ and for ...
GeometriaDifferenziale's user avatar
1 vote
0 answers
50 views

What is this Grothendieck's Theory

In his book Fields and Galois Theory, J.S. Milne remarks that (see p105): For Grothendieck, the classification of field extensions by Galois groups, and the classification of covering spaces by ...
JNF's user avatar
  • 87
2 votes
1 answer
60 views

Morphisms of frames induce morphisms of sites

One can associate a site with an arbitrary frame by defining coverages with suprema. According to Johnstone (Sketches of an Elephant, 2.3.20), "If $A$ and $B$ are frames, made into sites via ...
Daniel Rogozin's user avatar
4 votes
0 answers
83 views

Stack comparison lemma

Edit. I am realising that I was asking the wrong question. I was (without knowing it) interested in stack comparison lemmata similar to the lemma $$Sh(Sh(C,J),can) = Sh(C,J)$$ for sheaves. So my ...
Nico's user avatar
  • 4,220
2 votes
1 answer
69 views

vanishing cohomology for indiscrete Grothendieck topology

Let $\mathcal{C}$ be a category endowed with the indiscrete Grothendieck topology. Let $X$ be an object of $\mathcal{C}$ and $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$. Then I want to know how ...
Jean's user avatar
  • 277
3 votes
1 answer
100 views

What is the canonical topology on a complete heyting algebra?

I have troubles with catching the dividing line on Grothendieck topologies over posets between subcanonical and not subcanonical. In particular I have the following very specific basic question which ...
curious on mathematics's user avatar
2 votes
0 answers
125 views

Is there a universal sheaf?

I highly suspect the answer to the following question is no, but this has been bugging me for a while, so I figure I might as well ask. Throughout, "site" and "continuous functor" ...
Curious's user avatar
  • 593
0 votes
1 answer
304 views

Sheaves do not belong to algebraic geometry

In the n-cafe post https://golem.ph.utexas.edu/category/2010/02/sheaves_do_not_belong_to_algeb.html Tom Leinster points out that sheaves on a topological space may be viewed in light of the nerve ...
h3fr43nd's user avatar
  • 750
5 votes
1 answer
129 views

sheaves on a scheme

I work within the framework of Demazure & Gabriel`s book. (My schemes are all functors). Let $X$ be a scheme. I can define an underlying topological space $|X$| of $X$. Its points are equivalence ...
Nico's user avatar
  • 4,220
3 votes
2 answers
144 views

Can I recover the Zariski open subobjects from the Grothendieck topology they generate?

Let $\mathbf{cRing}$ be a category of commutative rings and let $\mathbf{Set}$ be a category of sets relative to which $\mathbf{cRing}$ is small (Grothendieck universes). The opposite $\mathbf{Aff}$ ...
Nico's user avatar
  • 4,220
0 votes
0 answers
113 views

Pullback of a Zariski Sheaf to the Etale SIte.

Suppose $F$ is a sheaf on the big etale site of a scheme $S$ (i..e, where the objects are all schemes over $S$). If we restrict $F$ to the small Zariski site of $S$ and then pull it back to the small ...
Jehu314's user avatar
  • 923
3 votes
0 answers
68 views

Does the Barr-embedding preserve coequalizers

Let $\mathbb C$ be a small regular category. Let $J$ be the Grothendieck topology generated by coverings $\{U'\twoheadrightarrow U\}$ consisting of precisely the regular epimorphisms in $\mathbb C$. ...
Nico's user avatar
  • 4,220
2 votes
0 answers
106 views

Universal property of the (Vistoli-)sheafification

Given a presheaf $F$, in Notes on Grothendieck topologies, fibered categories and descent theory there is a construction of the sheafification (Proof for theorem 2.64). The first part is the ...
Muster Maxfrau's user avatar
0 votes
0 answers
62 views

Confusion about nLab's definition of smooth $\infty$-groupoid

This nLab page defines (in Definition 2.6) a smooth $\infty$-groupoid as an $(\infty,1)$-sheaf on the $(\infty,1)$-category $C = \text{CartSp}_{\text{smooth}}$ whose objects are Cartesian spaces, ...
I.A.S. Tambe's user avatar
  • 2,431
0 votes
1 answer
55 views

What is the appropriate $(\infty,1)$-site structure on SmoothMfld, making the de Rham complex a $D(\mathbb{R})$-valued $(\infty,1)$-sheaf?

Let Mfld denote the category of smooth manifolds. This also has an $(\infty,1)$-category structure, with higher cells being smooth n-fold homotopies. We know Mfld has a 1-site structure / 1-...
I.A.S. Tambe's user avatar
  • 2,431
2 votes
0 answers
57 views

Does the standard Grothendieck topology on topological spaces, push down to the homotopy category?

Let $C$ be any class of topological spaces, such that for any $X \in C$, if $U$ is a topological space homeomorphic to an open subset of $X$, then $U \in C$. From now on we view $C$ as a full ...
I.A.S. Tambe's user avatar
  • 2,431
0 votes
1 answer
46 views

If $T$ is continuous and $F$ is a sheaf (on a general site) then is $TF$ always also a sheaf?

Let $C$ be a site, i.e. a category equipped with a Grothendieck topology (or coverage). Let $A,B$ be complete categories, and let $T : A \rightarrow B$ be a continuous functor. My question is, if $F$ ...
Indraneel Tambe 2's user avatar
1 vote
1 answer
69 views

What is the Grothendieck topology used for the site of open subsets of Euclidean spaces, in the definition of a diffeology?

In this nLab page on diffeological spaces, Definition 2.1 refers to a site $\mathcal{O}_\mathcal{P}$ whose objects are open subsets of Euclidean spaces, and whose morphisms are smooth maps between ...
Indraneel Tambe 2's user avatar
1 vote
0 answers
63 views

Attempting to prove a certain assignment of sieves on Top gives a Grothendieck topology

Let $C$ be the category of topological spaces. Given a topological space $X$, we say a collection $\{f_i : U_i \rightarrow X\}$ of morphisms in $C$ is a "covering family" of $X$ if each $f_i$...
Indraneel Tambe 2's user avatar
1 vote
2 answers
61 views

Is there a notion of "base" $\mathcal{B}$ for a site, and "$\mathcal{B}$-sheaf", such that a presheaf is a sheaf iff it is a "$\mathcal{B}$-sheaf"?

For a base $\mathcal{B}$ for the topology on a topological space $X$, there is the notion of a presheaf on $X$ being a "$\mathcal{B}$-sheaf", and a presheaf on $X$ is a sheaf iff it is a $\...
Indraneel Tambe 2's user avatar
1 vote
0 answers
53 views

Some questions on Grothendieck topologies

Let $\mathcal{C}$ be a small category and $J$ a Grothendieck topology on it, that is an assignment $J(C)$, for any $C \in \mathcal{C}$ assigning to any object a collection of sieves (think of sieves ...
TheWanderer's user avatar
  • 5,166
1 vote
0 answers
97 views

Compatibility of pullbacks with an equivalence relation

I'm currently working on the proof of the existence of the sheafification in Notes on Grothendieck topologies, fibered categories and descent theory , but i currently stuck on a statement in the proof ...
Muster Maxfrau's user avatar
0 votes
0 answers
114 views

How to generate Grothendieck Topologies from families of sieves

If $\mathcal{C}$ is a small category with pullbacks, a basis for a Grothendieck topology on $\mathcal{C}$ is a function which assigns to each object $C$ a collection $K(C)$ of families of morphisms ...
TheWanderer's user avatar
  • 5,166
1 vote
0 answers
158 views

global section of inverse image and etale cohomology

Let $f: X\to Y$ be a morphism of schemes. If it is needed, we can assume that $X,Y$ are quasi-compact and quasi-separated, and $f$ is affine. Consider the two small etale sites $X_{et}$ and $Y_{et}$. ...
DU CJ's user avatar
  • 81
3 votes
1 answer
161 views

meaning of sentence that a "presheaf/K-theory satisfies descent on a Grothendieck site"

I'm reading a post about Nisnevich topology and I would like to clarify what the author means in Definition 1.5: We define $\mathrm{Spc}_S = L_{\mathrm{Nis}}\mathcal{P}(\mathrm{Sm}_S)$ to be the ...
user267839's user avatar
  • 7,377
0 votes
1 answer
95 views

Definition of Grothendieck topology [closed]

I'm learning about Grothendieck topology in https://ncatlab.org/nlab/show/Grothendieck+topology There are two definitions that are Defintion2.1 and Definition 3.1 in the page. I think those are ...
神宮寺春姫's user avatar
2 votes
0 answers
108 views

Can any Grothendieck topos be presented by a singleton coverage?

If $\mathcal{E}$ is a Grothendieck topos, is it known whether one can always find a small site $(\mathbb{C}, \mathcal{J})$ with $\mathcal{E} \simeq \mathsf{Sh}(\mathbb{C}, \mathcal{J})$ and the ...
User7819's user avatar
  • 1,621
0 votes
1 answer
39 views

Uniqueness of amalgation when proving that presheaves are sheaves for the trivial topology

Let $\mathfrak{C}$ be a small category endowed with the trivial Grothendieck topology $J$. I must show that $Sh(\mathfrak{C},J)=[\mathfrak{C}^{op},{\bf Set}]$. I take an object $C$ and a matching ...
TheWanderer's user avatar
  • 5,166
2 votes
0 answers
257 views

fppf/ etale Cohomology calculate with Cech cohomology

Let $R$ be a commutative ring with one (so living in standard commutative algebra setting) and let $\phi: R \to S$ a faithfully flat. Then the so called Amitsur complex $R \to S^{\otimes \bullet +1}$ ...
user267839's user avatar
  • 7,377
1 vote
0 answers
44 views

When the pullback of etale sheaves commutes with the forgetful functor to the Zariski site?

Let $f:X\rightarrow Y$ be a morphism of schemes. Let $f^*$ be the pullback functor from the sheaves on the big etale site (or small etale/Zariski site depending on the context!) of $Y$ to $X$. Let $L$ ...
user127776's user avatar
  • 1,364
0 votes
0 answers
113 views

Do the pullback and restriction to the generic point coincide?

Assume we are given an etale sheaf $\mathcal{F}$ on the big etale site of schemes over $S$. Let $X$ be a $S$ scheme with the structure morphism $j$ and let $i:\eta_X\hookrightarrow X$ be its generic ...
user127776's user avatar
  • 1,364
2 votes
0 answers
89 views

Functorial characterization of the opposite category of schemes

Is there an approach to scheme theory in which schemes are viewed as algebraic objects instead of geometric ones? Here's what I mean: From the functorial perspective, a scheme is a certain kind of ...
Nick Mertes's user avatar
2 votes
0 answers
89 views

Sheaves on pushforward topologies?

Suppose $\mathcal{C}_i$ are sites, $\mathcal{C}$ is a category, and $\mathcal{F}_i:\mathcal{C}_i\rightarrow\mathcal{C}$ are functors. Assign to $\mathcal{C}$ the smallest topology such that all $\...
Curious's user avatar
  • 593
2 votes
1 answer
79 views

Meaning of certain cohomology classes and morphisms in Prop 3.3 of Deligne-Illusie's paper on mod $p^2$ liftings and decompos. of the de Rham complex

I am still struggling with Deligne and Illusie's paper (https://eudml.org/doc/143480). They say on page 261, in the course of the proof of theorem 3.3: The class $e(K)$ (which is associated to the ...
The Thin Whistler's user avatar
2 votes
0 answers
98 views

Reference for things "locally looking like" objects in a site?

Given a site $\mathcal{C}$ (with possibly extra structure), is there a good notion of a "completion" $\overline{\mathcal{C}}$ of $\mathcal{C}$ which results in a site of things that "...
Curious's user avatar
  • 593
1 vote
0 answers
115 views

Sheafification via compatible germs for Grothendieck topologies?

I currently been reading a bit about Grothendieck topology in the book "Introduction to Etalé Cohomology" by Tamme. I've just gotten to the part where the author define the sheafification ...
Najonathan's user avatar
0 votes
0 answers
60 views

Does sheafifying with respect to a new site destroy the already existent descent properties?

This is probably a rather obvious question (either obviously true or wrong). Given a presheaf on the category of schemes that satisfies descent with respect to some site. If we sheafify the presheaf ...
user127776's user avatar
  • 1,364
1 vote
0 answers
168 views

Why does there need to be more than one type of covering sieve?

Let $C$ be a category. If $X$ is an object of $C$, then write $\underline{X} = \text{Hom}(-, X): C^{\text{op}}\to \text{Sets}$ for the corresponding functor. Suppose that a "cover" of $X$ is ...
Nick Mertes's user avatar
2 votes
1 answer
164 views

$\operatorname{Hom}(Z/nZ, μ_n)$ is isomorphic to $μ_n$ as etale sheaf?

About etale sheaves, I saw $\operatorname{Hom}(Z/n, μ_n)\cong μ_n$ as isomorphism of etale sheaves here (https://mathoverflow.net/questions/52404/locally-constant-sheaves-for-the-%C3%A9tale-topology-...
aerile's user avatar
  • 1,447
2 votes
1 answer
103 views

Flabby representable sheaves

Let $S$ be a scheme. Consider some representable moduli functor $\mathcal{M}:(Sch/S)^{op}\rightarrow Set$ represented by some scheme $M$. Then for each $V\in (Sch/S)^{op}$, let define $$\mathcal{M}^{...
curious math guy's user avatar
2 votes
2 answers
197 views

Definition of comma category of a category.

I am reading Vistoli's notes: Notes on Grothendieck topologies, fibered categories and descent theory. In page 13, he gives a definition: For any category $\mathscr{C}$ and an object $X$ of $\mathscr{...
Mike's user avatar
  • 895
2 votes
0 answers
257 views

Obtain admissble cover from cover by admissble opens on rigid-analytic spaces

Suppose I have a rigid-analytic $K$-space $X$ and a cover $(U_{i})_{i \in I}$ of $X$ consisting of admissible open subsets, such that each $U_{i}$ satisfies a certain property $P$. Is there any ...
n7kvz's user avatar
  • 372
1 vote
0 answers
84 views

Universal closure operation examples

A universal closure operation on a topos $\mathcal{E}$ is a family of functions $c_{(-)}: Sub(-) \to Sub(-)$ such that for $A$ a subobject of $X$ we have $A \leq c_X(A)$ $c_X(c_X(A)) = c_X(A)$ For ...
user388557's user avatar
  • 2,544
1 vote
1 answer
66 views

Interesting Lawvere-Tierney topologies on Set

When doing topos theory I like working out hard theorems for Set first, and then translating back to general topoi. For stuff like subobjects and regular epi-mono factorization this works great, but I ...
user388557's user avatar
  • 2,544
2 votes
1 answer
183 views

A possible Grothendieck Topology/ Coverage of $\mathbf{Top}$?

For any topological space $X$ we have a canonical functor from its category (poset) of open subsets $\mathcal{O}(X)$ to category of topological spaces $\mathbf{Top}$ defined in the obvious way. This ...
Bumblebee's user avatar
  • 18.3k
1 vote
0 answers
87 views

Is the sheaf cohomology only a sheaf for flabby sheaves?

Let $X$ be a topological space and let $\mathcal{F}$ be an abelian sheaf on $X$. Choose an arbitary open cover $\{U_i\}_i$ of $X$. Then the Mayer-Vietoris long exact sequence is $$0\rightarrow \...
curious math guy's user avatar