Questions tagged [sites]

For questions related to the mathematical notion of site, which among others generalizes the notion of topological space. For questions on internet-sites, use online-resources, but note that these questions need to be very specific to research-level mathematics to be on-topic.

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Relationship between canonical topology on a topos and its site of definition

The canonical (Grothendieck) topology for a category $C$ is the largest (finest) topology such that every representable presheaf over $C$ is a sheaf. According to First Order Categorical Logic Lemma 1....
Joey Eremondi's user avatar
2 votes
1 answer
129 views

Is the slice of a subcanonical site also subcanonical?

A subcanonical site is one for which every representable functor is a sheaf. For a subcanonical site $C$, the fundamental theorem of topos theory says that there is an equivalence $Sh(C/c)\cong Sh(C)/...
Joey Eremondi's user avatar
6 votes
0 answers
228 views

Non-invertible map between principal bundles only locally trivial in a supercanonical Grothendieck topology?

It is a truth universally acknowledged that a map between principal bundles must be in want of an inverse. However, the construction of said inverse in the context of a more general site $(S,J)$ ...
David Roberts's user avatar
  • 33.8k
7 votes
1 answer
409 views

Tensor product of sites

Let $C, D$ two Grothendieck sites. Since the corresponding toposes $E, F$ are locally presentable categories, then (by Gabriel-Ulmer duality) they correspond to limit theories $X, Y$ (that is, small ...
Arshak Aivazian's user avatar
1 vote
0 answers
84 views

Why is the induced singleton pretopology closed under pullbacks?

Let $(C,\mathcal{T})$ be an arbitrary site with a pretopology $\mathcal{T}$. The category $C$ has coproducts. As a pretopology I mean the definition 2.24 of Grothendieck topology in Angelo Vistoli’s ...
Muster Maxfrau's user avatar
2 votes
1 answer
178 views

Necessary and sufficient conditions for all sheaves on a site to be continuous functors?

All representable functors are continuous. This makes it possible to associate additional natural operations with them, which are absent for arbitrary presheaves. What are the sufficient and what are ...
Arshak Aivazian's user avatar
4 votes
0 answers
181 views

Geometric intuition for pf in fppf-topology

I am trying to learn a bit about algebraic spaces and the various topologies on categories of schemes in general. And, as it seems to always be the case, I am struggling with understanding how exactly ...
Sergey Guminov's user avatar
0 votes
1 answer
169 views

Does the (Vistoli-)sheafification induce isomorphism?

Given a presheaf, in Angelo Vistoli's 2007 Notes on Grothendieck topologies, fibered categories and descent theory there is a construction of the sheafification (Proof for theorem 2.64). Note: In ...
Muster Maxfrau's user avatar
1 vote
0 answers
69 views

Reference Request: Cech cohomology of complexes on an arbitrary site

I am looking for a reference which is equivalent to this stacks project page [1], except formulated in the generality of an arbitrary site. I checked the "Cohomology on Sites" section of the ...
David Urbanik's user avatar
19 votes
2 answers
369 views

Why is $1$ not a dense sub-site in a group with the trivial Grothendieck topology?

A friend of mine had the following question while reading the section "C2.2 The topos of sheaves" in "Sketches of an Elephant". Let $G$ be a group (considered as a category with ...
Arshak Aivazian's user avatar
1 vote
0 answers
119 views

Extending Grothendieck topology from analytic manifolds to spaces?

Let $k\text{-Man}$ denote the Euclidean site of $k$-analytic manifolds where $k=\mathbb{R}, \mathbb{C}$. In words, $k\text{-Man}$ is the usual category of real/complex analytic manifolds considered ...
Andy Sanders's user avatar
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3 votes
1 answer
219 views

Compatibility of pullbacks with an equivalence relation

This question was originally posted last week in Math Stack Exchange (see here). I'm currently working on the proof of the existence of the sheafification in Angelo Vistoli’s 2007 Notes on ...
Muster Maxfrau's user avatar
5 votes
0 answers
186 views

Is there something similar to Lawvere-Tierney topologies for Abelian categories?

Lawvere-Tierney topologies generalize the notion of local operators on a Topos from Sheaf toposes over a Grothendieck site to arbitrary Toposes. However, while the special case of Sheaves of sets or ...
saolof's user avatar
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3 votes
0 answers
431 views

Flasque sheaves on a site

This is a cross-post from MathStackexchange. We define a flasque sheaf on a site as one whose first Čech cohomology vanishes for every covering of every object of the site. I know this definition is ...
Jehu314's user avatar
  • 153
9 votes
0 answers
278 views

Refinement of hypercovers by ordinary covers

I am asking for references and discussions of statements of the form Every bounded hypercover can be refined by an ordinary cover By "bounded" I mean "finite height". E.g., are ...
Konrad Waldorf's user avatar
4 votes
0 answers
194 views

Does Fpqc sheaf over category of rings imply representability

I am trying to read the article "Algebrization and Tannaka duality" by Bharghav Bhatt. In Corollary 1.2, he says that given a qcqs algebraic space $X$ (I am interested in the case when $X=\...
user45397's user avatar
  • 2,195
13 votes
3 answers
1k views

Resources for topos theory

I am trying to learn topos theory and I am finding a strong scarcity of resources. Is there any canonical textbook to refer someone to when learning this topic? So far, I have only been able to find ...
Sofía Marlasca Aparicio's user avatar
2 votes
0 answers
272 views

Small sheaves on big sites

Background: If one works with sheaves on small etale site over a fixed scheme (which is really an essentially large category), one can instead work with sheaves on the affine etale site (which turns ...
gregodom's user avatar
  • 319
1 vote
0 answers
172 views

Confusion in understanding the notion of $G$ Principal bundle where $G$ is a geometric group over a site

The first paragraph of the section Overview in the paper Principal infinity-bundles - General theory by Nikolaus, Schreiber and Stevenson https://arxiv.org/abs/1207.0248 precisely reads the following: ...
Adittya Chaudhuri's user avatar
2 votes
0 answers
605 views

Čech-Alexander complex in computing (crystalline/prismatic) cohomology

I have a naive question about Čech-Alexander complexes in prismatic cohomology (although I suspect that the situation is similar for crystalline cohomology). They seemed to be introduced as a method ...
student123's user avatar
8 votes
0 answers
335 views

What is a morphism of ∞-sites?

Recall that a morphism of sites is a covering-flat functor that preserves covering families. Morphisms of sites can be identified with those geometric morphisms of induced toposes for which the ...
Dmitri Pavlov's user avatar
25 votes
2 answers
3k views

Are there (enough) injectives in condensed abelian groups?

The question is very simple : does $Cond(\mathbf{Ab})$, the category of condensed abelian groups (as defined in Scholze's Lectures in Condensed Mathematics), have enough injectives ? Does it, in fact, ...
Maxime Ramzi's user avatar
  • 13.3k
3 votes
1 answer
284 views

What to call a morphism of sites inducing an equivalence on categories of sheaves?

Is there a standard term for a morphism of sites $(C,J)\to(C',J')$ which induces an equivalence on sheaf categories $\operatorname{Sh}(C',J')\xrightarrow\sim\operatorname{Sh}(C,J)$? The nLab entry ...
John Pardon's user avatar
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19 votes
1 answer
838 views

Has this "backwards" perspective on toposes been studied?

Topos theory can be seen as a categorification of topology via the following analogies. \begin{array}{|c|c|} \hline \text{locales}&\text{Grothendieck toposes}\\\hline \text{open sets}&\text{...
Oscar Cunningham's user avatar
3 votes
1 answer
244 views

"Covering-flat" part in definition of morphism of sites

Let $(\mathcal{C},\mathcal{I})$ and $(\mathcal{D},\mathcal{J})$ be sites where $\mathcal{C}, \mathcal{D}$ are categories and $\mathcal{I}$ and $\mathcal{J}$ are Grothendieck topologies on $\mathcal{C}...
Praphulla Koushik's user avatar
4 votes
1 answer
558 views

How to construct cup-product in a general site?

Let $\mathcal{C}$ be a site on the category of schemes for some Grothendieck topology, $X$ a scheme, and let $F,G$ be two sheaves of abelian groups on $\mathcal{C}$. Do we have cup-product as follows? ...
Heer's user avatar
  • 1,007
0 votes
0 answers
319 views

Grothendieck topology on a scheme equivalent to the circle

Suppose a class of morphisms of schemes is reasonable enough so that we can associate a small site to any scheme (just like small étale site). Is there a simple class of morphisms such that there is a ...
user avatar
12 votes
1 answer
738 views

Points of the big Zariski site

It's relatively simple to show that the geometric morphisms $ \mathbf{Set} \to \mathrm{Sh}(\mathbf{CRing}^\mathrm{op}_{\mathrm{fp}}, \mathrm{Zar})$ correspond to local rings. More precisely, since ...
user avatar
2 votes
0 answers
254 views

Pullback of a constant sheaf over arbitrary sites

Given a geometric morphism between arbitrary Grothendieck topoi, $f:\mathcal{Sh(D)}\to\mathcal{Sh(C)}$, does the pullback $f^{-1}$ (i.e, the left adjoint) take constant sheafs to constant sheafs?
Arun Kumar's user avatar
10 votes
2 answers
609 views

Equivalence of the definitions of a sheaf in SGA4 and in "Categories and Sheaves"

I asked this question on Mathematics Stack Exchange, but got no answer. I don't understand why the definition of a sheaf (Definition 17.3.1 (ii)) given in the book [KS] Categories and Sheaves by ...
Pierre-Yves Gaillard's user avatar
10 votes
0 answers
731 views

Differential Forms in Infinite Dimensions

In Kriegl/Michor's book "The convenient setting of global analysis", they define the space of differential $k$-forms on a possibly infinite-dimensional manifold $M$ as the space of smooth sections of ...
Matthias Ludewig's user avatar
4 votes
0 answers
169 views

Is the restriction of sheaves from the big to small etale sites an equivalence?

Let $X$ be a scheme. Let $(\mathrm{Sch}/X)_{ét}$ and $X_{ét}$ be the big and small étale sites, resp. Is the restriction functor from $\operatorname{Sh}((\mathrm{Sch}/X)_{ét})$ to $\operatorname{Sh}(...
Avi Steiner's user avatar
  • 3,031
6 votes
0 answers
175 views

Dense (∞,1)-subsites

So if $C$ is a 1-site and $D$ is a subsite (with the induced coverage), there are some conditions that ensure that the pre-composition and right Kan extension functors yield an equivalence of ...
Karthik Yegnesh's user avatar
4 votes
1 answer
321 views

Stacks for the extensive topology?

Recall that any extensive category can be canonically endowed the structure of a site via the extensive topology, which is the Grothendieck topology whose covering morphisms are the coproduct ...
Karthik Yegnesh's user avatar
1 vote
0 answers
66 views

Morphism of 2-sites

Let $X$ and $Y$ be Grothendieck sites. A $\textit{morphism}$ between $X$ and $Y$ is a functor on the underlying categories that is covering-flat and preserves covering families. See https://ncatlab....
user84563's user avatar
  • 915
5 votes
0 answers
397 views

Examples of nonstable ∞-categories in which sifted colimits commute with finite limits

What are some natural examples (if any) of nonstable ∞-categories in which finite limits commute with sifted colimits (or rather just colimits over Δ^op)? Stable ∞-categories do satisfy this property,...
Dmitri Pavlov's user avatar
5 votes
1 answer
389 views

Categorification of covering morphisms

Given a category $\mathsf{A}$, let $\mathsf{Fam}(\mathsf{A})$ be its free coproduct cocompletion (which is always extensive). This means every object has a unique up to iso presentation as a coproduct ...
Arrow's user avatar
  • 10.3k
1 vote
0 answers
179 views

$\mathbb E$-descent maps in topological spaces in terms of different sites?

The paper Facets of Descent I by Janelidze and Tholen defines $\mathbb E$-descent maps as those for which $\Phi^p:\mathbb EB\longrightarrow \mathsf{Des}_\mathbb{E}(p)$ is an equivalence of categories. ...
popo's user avatar
  • 11
0 votes
1 answer
98 views

A confusion about covering flatness

I'm reading this entry on nLab. But I'm confusing with the notion of covering-flatness. More precisely, I meet some trouble when I try to show that the $Sets$-valued flatness is a special case of ...
Syu Gau's user avatar
  • 415
12 votes
0 answers
446 views

Making the conceptual leap from locales to Grothendieck topologies?

I find the definition for locales and sheaves on locales to be straightforward, but I'm stumbling over the idea of a Grothendieck topology. Is there a nice way to see roughly how the latter ...
Harrison Smith's user avatar
3 votes
2 answers
499 views

Morphism on schemes induced by continuous morphism of sites

I am beginner in the theory of Grothendieck topologies and I have the following question. Let $X, Y$ be schemes over an algebraically closed field $k$. Denote by $X_{et}$ and $Y_{et}$ the Etale sites ...
user46578's user avatar
  • 823
8 votes
1 answer
334 views

Which dense inclusions of sites are ∞-dense?

An inclusion of sites f: D→C is dense if it induces an equivalence between the categories of sheaves on C and D. Likewise, f is ∞-dense is it induces an equivalence between the ∞-categories of ∞-...
Dmitri Pavlov's user avatar
3 votes
2 answers
502 views

1st cech cohomology groups on ringed sites

Let $(C, O)$ be a ringed site -- i.e., $C$ is a small category with a grothendieck topology $\tau$ and $O$ a sheaf of rings on the site $(C,\tau)$. In this context, for any object $U$ of $C$ one can ...
Andrew Stout's user avatar
20 votes
1 answer
2k views

Points in sites (etale, fppf, ... )

I asked a part of this in an earlier question, but that part of my question didn't receive precedence. Etale site is useful - examples of using the small fppf site? Let $X$ be a scheme (assume it ...
LMN's user avatar
  • 3,525
27 votes
2 answers
2k views

Etale site is useful - examples of using the small fppf site?

Edit: After the answers and comments, I'm hoping for a little bit of elaboration (in the comment to the answer below.) Also, question 2 was discussed here: Points in sites (etale, fppf, ... ) There, ...
LMN's user avatar
  • 3,525