62
votes

Accepted

### What is homology anyway?

Let's take coefficients in a field $k$, for simplicity.
On 2): the singular cohomology of a topological space $X$ is the dual of its singular homology, almost by definition. But if $X$ is a space for ...

28
votes

Accepted

### Sheaf-theoretic approach to forcing

Yes, this is a model of ETCSR. Unfortunately, I don't know of a proof of this in the literature, which is in general sadly lacking as regards replacement/collection axioms in topos theory. But here'...

27
votes

### What is homology anyway?

I generally think about the relationship differently than Jacob, probably because I'm coming from an algebraic topology background rather than an algebraic geometry one. I would say that if $\mathcal{...

27
votes

### What is homology anyway?

For a long time (and still today), I very much shared the confusion of the OP. I think Jacob Lurie gives a very clear take on the standard perspective, but Mike Shulman does have a very valid ...

24
votes

Accepted

### Measuring a presheaf's failure to be a sheaf?

This answer is inspired by the Embedding Calculus (aka Manifold Calculus) of Weiss and Goodwillie. This is a framework for studying certain presheaves on manifolds. The idea is that sheafification of ...

24
votes

Accepted

### How to motivate constructible sheaves

Even if you're only interested in say cohomology with coefficients in the constant sheaf, working with constructible sheaves gives you extra flexibility and is more amenable to inductive proofs.
Here ...

24
votes

Accepted

### Are groups determined by their morphisms from solvable groups?

Let $G, G'$ be two non-isomorphic Tarski monsters of prime exponent $p$ or two non-isomorphic torsion-free Tarski monsters. Then for every solvable group $A$, $\mathbb{hom}(A,G)\cong \mathbb{hom}(A,G')...

22
votes

### Sheaf-theoretic approach to forcing

I think the language of classifying toposes is helpful in understanding this view of forcing.
Let $P$ be a poset.
The set theorists have the intuition that forcing over $P$ adjoins a generic filter of ...

21
votes

Accepted

### Why there is a Quot-scheme, not a Sub-scheme?

For standard universal properties, you need the scheme to behave well under base change, which in these cases would mean tensor products. Tensor product is right exact, so a quotient remain a quotient,...

21
votes

Accepted

### Is a direct sum of flabby sheaves flabby?

No, a direct sum of flabby sheaves need not be flabby.
Take $X=\{1,1/2,1/3,1/4,\dots\}\cup\{0\}$ with the subspace topology from $\mathbb R$, and let $\mathcal F$ be the sheaf whose sections over an ...

20
votes

Accepted

### Two points of view about Borel-moore homology

I'll have more time to write and provide a more thorough answer later, but I think the most straightforward proof (which I agree is hard to find) comes via sheaf theory: On the one hand, there is a ...

20
votes

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

This has nothing to do with $\infty$-categories, but with the fact that we look at the full topos and not an arbitrary site of definition:
Theorem: If $T$ is a (Grothendieck) ($1$-)topos, then a "...

20
votes

Accepted

### Are there (enough) injectives in condensed abelian groups?

Indeed, there are no nonzero injective condensed abelian groups.
Let $I$ be an injective condensed abelian group. We can find some surjection
$$ \bigoplus_{j\in J} \mathbb Z[S_j]\to I$$
for some index ...

20
votes

Accepted

### Understanding the definition of stacks

A canonical example of a sheaf of sets on a topological space $X$
is the sheaf that sends an open subset $U$ of $X$ to the set of continuous real-valued functions on $U$.
The gluing property then says ...

18
votes

### Sheaf-theoretic approach to forcing

Thanks for all the enlightening answers! Let me summarize my understanding now. (Please correct me if I'm saying something stupid!)
First, as explained by Mike Shulman in his answer, the answer to ...

18
votes

### What are the points (and generalized points) of the topos of condensed sets?

The category $\mathbf{Cond}$ of condensed sets is equivalent to the category of small sheaves over any of the following three large sites. (For small sheaves, see Mike Shulman's paper Exact ...

18
votes

Accepted

### Can one glue De Rham cohomology classes on a differential manifolds?

No.
Make $M$ by gluing three strips to two discs to form a thrice-punctured sphere. Take three open sets $U_\lambda$, each made by both discs and two of the strips. Then each $U_\lambda$ is ...

17
votes

Accepted

### Does inclusion from n-stacks into (n+1)-stacks preserve the sheaf condition?

You said it yourself in the question! The reason that sheaves of abelian groups are not $\infty$-sheaves in general, when considered as presheaves taking values in the $\infty$-category $\mathsf{Mod}_{...

16
votes

Accepted

### Characterize constant objects in the internal language of a topos?

For the first question: no, the notion of "constant object" doesn't depend on the site. The reason is in Anton's comment: every Grothendieck topos comes with a unique geometric morphism $p : \mathcal{...

16
votes

Accepted

### Splitting of exact triangles in derived category

In any triangulated category, the necessary and sufficient condition for a distinguished triangle $A\to B\to C\to A[1]$ to split is that the morphism $C\to A[1]$ in this distinguished triangle ...

16
votes

### Sheaf-theoretically characterize a Riemannian structure?

Suppose that $M$ is a smooth manifold and $g_0, g_1$ are Riemann metrics on $M$. $\newcommand{\eH}{\mathscr{H}}$ Denote by $\eH_{g_i}$, $i=0,1$ and the sheaf of $g_i$-harmonic functions. More ...

16
votes

Accepted

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

Yes, your counterexample seems to be correct — there’s an error in Def C2.2.1 as printed. This issue is mentioned in the n-lab’s article on dense sub-site (current revision permalink), which notes a ...

16
votes

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

To complement the answer by Peter, I think the "correct" statement of the comparison lemma for non-full subcategory (and in fact even non-faithful functor) can be found in a paper by (A.)...

15
votes

Accepted

### How is a Stack the generalisation of a sheaf from a 2-category point of view?

Let us start with what we know about sheaves, i.e. the "1-level". A sheaf on a (Grothendieck) site $\mathcal{C}$ is a contravariant functor $F : \mathcal{C}^\text{op} \to \textbf{Set}$ such that for ...

15
votes

### Measuring a presheaf's failure to be a sheaf?

Let $\mathcal{F}$ be a presheaf on $X$, and suppose $\mathcal{U}=\{U_i\}$ is an open cover of $X$. The Cech complex is the cochain complex whose degree $n$ piece is the direct sum of sections of $\...

15
votes

Accepted

### The Serre duality theorem intuition

First of all, dualizing sheaves are unfortunately not treated in EGA. The treatment in Hartshorne has some limitations. Perhaps some of them are related to your questions.
For pointers to more recent ...

15
votes

### Sheaf-theoretic approach to forcing

I can't really answer your question, since they are outside my field of expertise. But until Mike and others come to answer, let me make a long comment about the following sentence:
Note that in ...

15
votes

Accepted

### Is there a relation between Gelfand duality and the spectrum of a ring (with its Zariski topology)?

Yes, both Theorem A and Theorem B are special cases of a more general construction.
Denote by $R$ the category of commutative unital C*-algebras or the category of commutative rings.
Denote by $R'$ ...

15
votes

Accepted

### What's the point of a point-free locale?

A good answer to both questions is provided by the following variant of the Gelfand duality for commutative von Neumann algebras,
which shows that the following categories are equivalent:
The ...

15
votes

Accepted

### Is it true that $\operatorname{2-colim}_U \textsf{QCoh}(U) = \textsf{Vect}(K_X)$, as $U$ shrinks to the generic point?

Let $x$ be a point in a scheme $X$. There are two posets, namely the poset of affine opens containing $x$, $A(x)$, and the poset of opens containing $x$, $O(x)$.
The inclusion $A(x)^{op} \to O(x)^{op}$...

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