59 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 ...
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  • 16.9k
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'...
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  • 58.8k
24 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{...
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  • 58.8k
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')...
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23 votes
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Do I know what "coherent sheaf" means if I know what it means on locally Noetherian schemes?

Take $A$ to be Brian Conrad's universal counter example, i.e., the infinite countable product of copies of $\mathbf{F}_2$. Then $A$ is absolutely flat and every finitely presented module is finite ...
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  • 371
23 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 ...
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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 ...
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21 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 ...
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  • 13.2k
20 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 ...
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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 "...
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19 votes
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Reference request: Book of topology from "Topos" point of view

"Topology via Logic" is only half way there. It is firmly rooted in classical mathematics and makes no connections with toposes. Mac Lane and Moerdijk is a good suggestion. As for reading it ...
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19 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,...
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19 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 ...
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19 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 ...
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18 votes
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Canonical Sheaf of Projective Space

det of the middle term of a short exact sequence is the tensor product of the dets of the left and right terms (det = top wedge). The canonical bundle is det of \Omega, det of O is O.
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  • 196
17 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 ...
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17 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 ...
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17 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 ...
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  • 114k
16 votes
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An apparent equivalence of the category of affine schemes over $S$ and the category of quasi-coherent $\mathcal{O}_S$-algebras

No, it's not true in general (EGA 2, (1.2.3)). The following example is taken from EGA 2, (1.3.3). Over a field $K$, let $S$ be the affine plane with a doubled origin. Then $S$ is the union of two ...
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16 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 ...
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16 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}_{...
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15 votes

What information is lost in $X \to \mathrm{Sh}(X)$?

Following Martin's suggestion, I will turn my comment into an answer. If $T$ is an Grothendieck topos, then the subobjects of the terminal object form a frame. If $X_T$ is the corresponding locale, ...
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15 votes

An apparent equivalence of the category of affine schemes over $S$ and the category of quasi-coherent $\mathcal{O}_S$-algebras

What is true is that there is an antiequivalence between the category of schemes affine over $S$ (that is $S$-schemes for which the preimage of an open affine of $S$ is an open affine) and quasi-...
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  • 15.6k
15 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{...
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  • 58.8k
15 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 ...
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15 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 ...
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14 votes
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The real numbers object in Sh(Top)

Following a suggestion of Thomas Holder, we can close the gap as follows: For each object $Y$ in $\mathbf{T}$, there is a pseudonatural local geometric morphism $\mathbf{Sh}(\mathbf{T}_{/ Y}) \to \...
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  • 13.2k
14 votes
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Meaning of the determinant of cohomology

It doesn't so much represent a dimension of a cohomology group as it does an Euler characteristic. More precisely, it's based on Grothendieck's generalization of the Riemann-Roch theorem to families. ...
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  • 114k
14 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 ...
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  • 35.3k
14 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 ...
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