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62 votes
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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 ...
Jacob Lurie's user avatar
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28 votes
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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'...
Mike Shulman's user avatar
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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{...
Mike Shulman's user avatar
  • 65.6k
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 ...
Peter Scholze's user avatar
24 votes
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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 ...
Gregory Arone's user avatar
24 votes
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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 ...
Piotr Achinger's user avatar
24 votes
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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')...
markvs's user avatar
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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 ...
Zhen Lin's user avatar
  • 15k
21 votes
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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,...
Mohan's user avatar
  • 6,212
21 votes
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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 ...
Alexander Betts's user avatar
20 votes
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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 ...
Greg Friedman's user avatar
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 "...
Simon Henry's user avatar
  • 40.5k
20 votes
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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 ...
Peter Scholze's user avatar
20 votes
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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 ...
Dmitri Pavlov's user avatar
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 ...
Peter Scholze's user avatar
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 ...
Alexander Campbell's user avatar
18 votes
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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 ...
Will Sawin's user avatar
  • 139k
17 votes
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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}_{...
Dan Petersen's user avatar
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16 votes
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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{...
Mike Shulman's user avatar
  • 65.6k
16 votes
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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 ...
Leonid Positselski's user avatar
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 ...
Liviu Nicolaescu's user avatar
16 votes
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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 ...
Peter LeFanu Lumsdaine's user avatar
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.)...
Simon Henry's user avatar
  • 40.5k
15 votes
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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 ...
David Benjamin Lim's user avatar
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 $\...
Julian Rosen's user avatar
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15 votes
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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 ...
Leo Alonso's user avatar
  • 9,089
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 ...
Asaf Karagila's user avatar
  • 38.1k
15 votes
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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 $R'$ ...
Dmitri Pavlov's user avatar
15 votes
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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 ...
Dmitri Pavlov's user avatar
15 votes
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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 $A(x)^{op} \to O(x)^{op}$...
Maxime Ramzi's user avatar
  • 14.2k

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