# Questions tagged [sites]

For question 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 question need to be very specific to research-level mathematics to be on-topic.

**9**

votes

**1**answer

287 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 ...

**2**

votes

**0**answers

100 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?

**6**

votes

**1**answer

278 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 ...

**9**

votes

**0**answers

228 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 ...

**4**

votes

**0**answers

151 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}(...

**5**

votes

**0**answers

114 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 ...

**4**

votes

**1**answer

245 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 ...

**1**

vote

**0**answers

61 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....

**5**

votes

**0**answers

175 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,...

**5**

votes

**1**answer

294 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 ...

**1**

vote

**0**answers

171 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.
...

**0**

votes

**1**answer

81 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 ...

**9**

votes

**0**answers

299 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 ...

**3**

votes

**2**answers

369 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 ...

**8**

votes

**1**answer

231 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 ∞-...

**3**

votes

**2**answers

289 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 ...

**16**

votes

**1**answer

1k 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 ...

**20**

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, ...