58

Of course Weil did it (although I'm not able to give you the ref right away), and I even lectured on it in Paris 40 years ago. His method is very simple : (1) You first prove that your compact manifold X can be endowed with a Riemann structure (obvious locally, global result by using a smooth partition of unity). (2) By the general theory of Riemann spaces, ...


50

Definition: ''A smooth manifold is a locally ringed space $(M;C^{\infty})$ which satisfies the conditions: Each $x \in M$ admits a neighborhood $U$, such that $(U,C^{\infty})$ is isomorphic to $(\mathbb{R}^n,C^{\infty})$ as a locally ringed space. The global sections of $C^{\infty}(M)$ separate points. The structure sheaf $C^{\infty}$ is fine as a sheaf of ...


47

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 which singular cohomology is not the same as sheaf cohomology, then the sheaf cohomology of $X$ need not have a predual. For example, if $X$ is the Cantor set, ...


40

Let me expand on Yosemite Sam's comment. Pullbacks are indeed easier to define if you view a sheaf as a local homeomorphism. On the other hand, pushforwards are easier to define if you view a sheaf as a set-valued functor. Suppose we have a continuous map $f: X \to Y$ of topological spaces. Given a sheaf $F$ on $Y$, viewed as a local homeomorphism $\...


39

To me the obvious answer involves sheafification of a presheaf. If you look at the construction of the associated sheaf to a presheaf in, say, Hartshorne it goes through the étalé space construction without specifically telling you, and to me it makes the construction somewhat unmotivated. Namely, if $P$ is a presheaf on $X$, then taking the stalk $P_x$ ...


30

The so-called Oka-Grauert principle states that for any Stein space $X$ the holomorphic and the topological classification of complex vector bundles on $X$ coincide. The original reference is Hans Grauert, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135 (1958), 263-273. As a consequence, every locally free, coherent sheaf $\...


23

For another perspective, think about definitions of "complex manifold" or "real analytic manifold". Normally you use atlases for this, imposing the Hausdorff and paracompactness conditions separately. You can restate the atlas part of the definition in sheaf language, but you can't hope to get the rest of it that way, can you? I mean, you can't get the ...


23

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 a presheaf is analogous to the linearization of a function. From this point of view, sheafification is just the first in a sequence of approximation - for each $...


22

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 locally free. On the other hand, every element of $A$ is an idempotent. Hence if $B$ is a Noetherian ring whose spectrum is connected, then any ring map $A \to B$ ...


21

You can think about this as about a generalization of the adjunction between the extension and the restriction of scalars --- if $A \to B$ is a morphism of rings, $M$ is an $A$-module and $N$ is a $B$-module then $$ Hom_B(M\otimes_A B,N) = Hom_A(M,Res_A N), $$ where $Res$ is the restriction of scalars. This adjunction coincides with the one tou are ...


20

Here's a counterexample with additive functors on abelian categories. If $A$ is an abelian group, let $F(A)$ denote the subgroup of elements that are divisible by $2$. It is easy to see that $F:Ab\to Ab$ is an additive functor, and $F$ is a subfunctor of the identity. But $F$ is not left exact because it does not preserve kernels. For instance, $F$ sends ...


20

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 "sheaf of set" on $T$ is the same as a functor from $T^{op}$ to $Set$ sending colimits in $T$ to limits in Sets. Sketches of Proof: Sheaf of sets on T, mean ...


19

"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 backwards: that is pretty much the aim of my "Locales and Toposes as Spaces" (Chapter 8 in "Handbook of Spatial Logics" (ed. Aiello, Pratt-Hartman, van Bentham), ...


17

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, not left exact, so a sub may not remain a sub.


17

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 $\mathcal{E}$ is any $(\infty,1)$-topos, with $f:\mathcal{E}\to \mathcal{S}$ its unique geometric morphism to $\infty$-groupoids (homotopy spaces), then $f_*$ is ...


17

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 that a continuous function on a union of open subsets $U_i$ of $X$ is the same thing as a collection of continuous functions $f_i: U_i \to \mathbb{R}$ such ...


16

This is false even for $\mathrm H^0$: take $X$ to be $\mathbb A^2 \smallsetminus \{0\}$, and as $F$ the structure sheaf of $L \smallsetminus \{0\}$, where $L$ is a line through $0$.


16

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.


16

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 affine open subsets $Y_1$ and $Y_2$, each of them is isomorphic to the affine plane, glued along the complementary subset of their origin. In particular, $Y_1$ ...


15

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, then the topos $Sh(X_T)$ of sheaves on $X_T$ is the called localic reflection of $T$. One has that $T\mapsto X_T$ is adjoint to the functor that takes a ...


15

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 sheaf of locally finite singular chains whose hypercohomology is your $H^{lf}$. I work out the details in the setting of intersection homology on pseudomanifolds ...


15

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 precisely for any open set $U\subset M$ $$\eH_{g_i}(U)=\big\{\; f\in C^\infty(U):\;\;\Delta_{g_i} u=0\;\big\}, $$ where $\Delta_{g_i}$ denotes the scalar Laplacian ...


14

"Why" questions can often be answered in multiple ways; I'll give an answer that's different from the other good comments and answers that you've already had. Sheaves are the first rung on an infinite ladder of concepts. The next rung is "stack". A stack is something like a "sheaf of categories", but there are added complications. A typical example of ...


14

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-coherent $\mathcal{O}_S$-algebras. The anti-equivalence is realized by the pushforward of the structure sheaf and the relative spectrum (see Exercise 5.17 in ...


14

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 $p : \mathcal{E} \leftrightarrows \mathrm{Set}$, and the constant objects are those in the essential image of $p^* : \mathrm{Set} \to \mathcal{E}$. In fact, since every ...


13

Let's work over the complex numbers for simplicity. Let $X$ be a Riemann surface, fix a base point $x \in X$, and let $\Gamma$ be the fundamental group $\pi_{1}(X,x)$. The data of an $n$-dimensional local system on $X$, together with a trivialization at the point $x$, is equivalent to the data of an $n$-dimensional representation of $\Gamma$. This is true ...


13

The Weyl algebra construction can be done abstractly for any real vector space (even infinite-dimensional) endowed with an antisymmetric bilinear form, thanks to B. Blackadar's universal C*-algebra construction using generators and relations ("Shape theory for C∗-algebras", Math. Scand. 56 (1985) 249–275). However, since you asked for a concrete ...


13

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. Given a proper map $\pi: Y\to X$ and a line bundle $L$ on $Y$, we could of course expect Riemann-Roch or a generalize to give us a formula for the Euler ...


12

Actually, for simplicial sheaves, and to be more accurate infinity sheaves of infinity groupoids, you do not need hypercovers. Your "naive" idea about multiple intersections (actually fibered products) is correct. If you instead use hypercovers, you get the notion of a *hyper*sheaf. Both infinity sheaves and hypersheaves form an infinity topos, and the ...


12

I don't know any reference where this is proven in elementary terms (although this can be done, of course). This is part of folklore since years (in spirit, this goes back to Verdier's formula in SGA 4 (exposé V) and in Ken Brown's thesis), but the only explicit reference I know is Proposition 7.9 (for $n=0$) in the paper Dugger, Hollander and Isaksen, ...


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