# Tag Info

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 ...
• 17.6k
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### 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'...
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• 1,802

### 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 ...
• 15k
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### 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,...
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### 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 ...
• 1,790
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### 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 ...
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### 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 "...
• 40.5k
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### 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 ...
• 20.8k
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### 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 ...
• 36.7k

### 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 ...
• 20.8k

### 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 ...
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### 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 ...
• 139k
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• 65.6k
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### 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 ...
• 15.3k

### 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 ...
• 34.2k
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### 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 ...

### 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.)...
• 40.5k
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### 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 ...
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