# All Questions

Tagged with higher-category-theory sheaf-theory

10
questions

**5**

votes

**1**answer

338 views

### Universal property of sheaf category

Given a site $C$ with a Grothendieck topology and the category of presheaves $P(C)$ (either in the sense of presheaves of sets or in the $\infty$-sense), and the category $S(C)$ of sheaves with ...

**2**

votes

**0**answers

109 views

### Compact generation of quasicoherent sheaves on mapping stack

Let $k$ be a field of characteristic $0$, and let $\mathcal{C}= \mathbf{Vect}_k^{\leq 0}$ be the $\infty$-category of vector spaces concentrated in degrees $\leq 0$. Consider the category $\mathbf{Pr}(...

**16**

votes

**1**answer

669 views

### A sheaf is a presheaf that preserves small limits

There is a common misconception that a sheaf is simply a presheaf that preserves limits. This has been discussed here before many times and I believe I understand it well enough.
However when reading ...

**63**

votes

**2**answers

5k views

### What is Homology anyway?

Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid ...

**6**

votes

**0**answers

121 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 ...

**3**

votes

**0**answers

194 views

### “2-Sheafification” with Values in non $Cat$ categories?

Let $X$ be a 2-site and consider the category of 2-presheaves over $X$, which will be denoted as $Pshv(X;Cat)$. These are $Cat$-valued 2-functors, where $Cat$ is the 2-category of categories. There is ...

**12**

votes

**2**answers

538 views

### A bit of history of Verdier duality

I was wondering who originated the presentation of Verdier duality as an equivalence between categories of sheaves and cosheaves ?
I learnt it reading Jacob Lurie's Higher Algebra and Justin Curry's ...

**6**

votes

**1**answer

399 views

### Hypercovers of sheaves in classical and quasi-categories

I am interested in relating the definition of hypercovers in the $\infty$-topos of sheaves on an $\infty$-Grothendieck site to the classical definition of hypercovers of presheaves on a Grothendieck ...

**9**

votes

**2**answers

613 views

### Is the site of (smooth) manifolds hypercomplete?

By site of manifolds Man, I mean the category of manifolds (maybe submanifolds to obtain a small category) with continuous maps between them. A Grothendieck topology is given by open covers. Actually, ...

**1**

vote

**1**answer

415 views

### Descent of Morphisms of Sheaves

While reading Brylinski I am trying to understand the descent of morphisms of sheaves.
In trying to form a new definition of a presheaf $A$ over a space $X$, we associate to each surjective local ...