Questions tagged [representable-functors]

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5
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1answer
259 views

Yoneda map for a composition of a representable functor and an arbitrary functor

Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $T : \mathcal{C} \rightarrow \mathcal{D}$ be a functor. Suppose that $F : \mathcal{D}^\mathrm{op} \rightarrow \mathrm{Set}$ is a functor. (So ...
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160 views

Reference request: Détailed explanation why the Grassmannian scheme represents the Grassmannian functor

Similar questions have been asked on this site, including by myself, but none of these have been given a satisfying answer. The question is: Why does the Grassmannian scheme represent the Grassmannian ...
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178 views

Moduli stacks and representability of diagonal by schemes

The answer to my question might very well be standard, but I have had trouble finding the right keywords to search for it, so I apologize if this is something well-known to the experts. I am learning ...
4
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0answers
144 views

Does Fpqc sheaf over category of rings imply representability

I am trying to read the article "Algebrization and Tannaka duality" by Bharghav Bhatt. In Corollary 1.2, he says that given a qcqs algebraic space $X$ (I am interested in the case when $X=\...
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1answer
408 views

Reference for the Brown representability theorem in the case of locally presentable (∞,1)-categories

The abstract Brown representability theorem, proved by Brown in 1964 (a missing assumption of a cogroup structure on generators was added later), states that a contravariant functor F from a category ...
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91 views

Local existence of (quasi)-universal family of sheaves

Let $p : X \to S$ be a projective morphism between two Noetherian $\mathbb{C}$-schemes of finite type with connected fibres. Let $O_X(1)$ be a very ample line bundle on $X$ relative to $S$. Given a ...
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278 views

Is the Kolmogorov-Arnold representation theorem an example of the Yoneda lemma?

From Wikipedia: https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold_representation_theorem In real analysis and approximation theory, the Kolmogorov–Arnold representation theorem (or ...
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53 views

A small lemma in Schlessinger's criterion paper

In the construction of a hull in Schlessinger's paper, one small lemma used is not clear in my opinion. That should be stated as follows: Let $(R,m)$ be a Noetherian complete local ring, $I_1\supset ...
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183 views

Reference for Grothendieck's theorem on representation of unramified functors

In the Exposé 294 of the Bourbaki Seminar of the year 1964-1965, Murre gives an outline of proof of a theorem of Grothendieck giving necessary and sufficient conditions of representability by a scheme ...
6
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1answer
232 views

(Pro-)representable functors and full subcategories in homotopy theory

$\DeclareMathOperator\Ab{Ab}\DeclareMathOperator\Ho{Ho}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Hotc{Hotc}\DeclareMathOperator\Sm{Sm}$Let $\mathcal{C}\overset{\iota}{\longrightarrow} \mathcal{...
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0answers
188 views

Algebraic groups as functors of points vs maximal points over algebraically closed field

So I only started learning about group schemes this summer, and I found two approaches. For the record I am interested in affine group schemes $G$ of finite type over a field $k$ (algebraic groups). ...
2
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92 views

Representability result

Let $X$ and $S$ be schemes over a field $k$. Reading this paper, there is a result on the representability of a morphism (proposition 3.1, page 4). Which result or reference on representability is ...
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75 views

Grassmannian of vector bundles with determinant being a square

I want to find a classfying space of vector bundles of rank $n$ generated by its global sections with determinant being a square of a line bundle. Stating in a formal way, suppose $X$ is a scheme, ...
7
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1answer
316 views

Any exact faithful functor is represented by a unique projective generator

In the book 'Tensor Categories' by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik on page 10 it says: 'Conversely, it is well known (and easy to show) that any exact faithful functor ...
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53 views

Representable functors and symplectic co-tangent bundles

I've been banging my head against something that I feel should follow from abstract non-sense, and I hope someone here can set me straight. Let $\mathcal{M}$ be the category of smooth manifolds, with ...
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276 views

Constructions that can be seen as objects representing a functor

Some constructions can be seen as objects representing a functor. For example, Consider a topological group $G$ and a functor $\mathcal{F}:\text{Top}\rightarrow \text{Gpd}$ defined as $M\mapsto \...
6
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1answer
275 views

Why does every chain complex have a map into its cone?

In Weibel's An introduction to homological algebra he defines a cone as an explicit chain complex associated to the given one -i.e. for a chain $C=(C_i, d)$ he defines $Cone(C)=\left(C_{i-1} \oplus ...
5
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1answer
197 views

Does every functor between Grothendieck categories have adjoints?

Let $F:\mathcal C\longrightarrow \mathcal D$ be an additive functor that preserves colimits. Suppose that $\mathcal C$ and $\mathcal D$ are Grothendieck categories. Does $F$ have a right adjoint? ...
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69 views

Finding a cover of the Hilbert functor, that corresponds to a cover of the Grassmannian

Let $\mathrm{Grass}_{k+1,n+1}:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$ be the Grassmann functor, which maps a scheme $S$ to the set: $$\left\{\mathscr{U}\subseteq\mathscr{O}_S^{n+1}:\,\,\mathscr{...
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129 views

Open subfunctor of Quot Functor induced by open immersion

Let $f: X \rightarrow S$ be a morphism of noetherian schemes and $\mathfrak{Q}uot_{\mathcal{E}/X/S}$ be the functor parametrizing families of quotients of $\mathcal{E}$ in the category of locally ...
3
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1answer
405 views

Representability of Grassmannian functor by a scheme

I am having some trouble following a proof that the Grassmannian functor is representable by a scheme. I am following the proof in EGA 9.7.4. It is only a small step that I am stuck on. For reference, ...
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1answer
428 views

Proving the representability of a functor that is covered by open subfunctors

I already posted this on math.stackexchange some while ago, but haven't received any answers yet. (https://math.stackexchange.com/questions/3221006/proving-the-representability-of-a-functor-that-is-...
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110 views

Induced map between Grassmannian

Definition : Let $S$ be a scheme, $\mathcal E$ a quasi coherent $\mathscr O_S$-module, and $e \geq 0$ an integer. For every $S$- scheme $h : T\longrightarrow S$ denote by $$ \operatorname{Grass}^e(\...
4
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0answers
257 views

EGA I (Springer), Proposition 0.4.5.4 [closed]

I do not understand one argument in the proof of Proposition 0.4.5.4. in the new version by Springer of EGA I. When proving that the functor $F$ is representable by $(X, \xi)$, where we obtained $X$ ...
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111 views

is the functor of finite order elements in grp representable

My question is whether the functor F:Grp->Set that sends a Group to its Set of elements with finite order is representable. I have the sense that it shouldn't be but I've so far failed to prove it in ...
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96 views

Explicit description of the scheme obtained by relative gluing data over a base scheme

I have recently been trying to get a better understanding of the projective space bundle of a quasi-coherent sheaf of graded algebras over a scheme $X$. The key idea is the following construction ...
6
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167 views

representability of $\mathrm{R}^1f_{*,\mathrm{fppf}}\mathscr{A}$

Let $f: X' \to X$ be a finite flat morphism of (nice) schemes and $\mathscr{A}/X'$ be a smooth commutative group scheme such that the Weil restriction $f_*\mathscr{A}/X$ is representable by a ...
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208 views

Quotient of a sheaf by group action and representabillity

Let $X$ be a scheme and $S$ be a sheaf of sets over the fppf topology of $X$. Let $G$ be a group scheme over $X$ and there is an action of $G$ on $S$. Now, I want to look at the quotient $G \setminus ...
3
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147 views

Representability of Flattening stratification functor

Let $f:X \to Y$ be a projective morphism between noetherian scheme. Is there any know condition under which there exists a functorial stratification of $Y$ i.e., there exists a filtration by locally ...
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148 views

Internal Mor of schemes

Let $S$ be a Noetherian scheme and $X,Y$ be $S$-schemes of finite type. Consider the functor $X^Y$ given by $T \mapsto Mor_S(Y \times_S T,X)$. When is this functor representable by an $S$-scheme of ...
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207 views

Representable $\text{Hom}$ functors

Let $X, Y, S$ be noetherian schemes, $X$ flat and quasi-projective over $S$, $Y$ projective over $S$. Is the hom-functor $T\mapsto\text{Hom}_T(X_T, Y_T)$ representable? If $X$ is flat and projective,...
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230 views

Is the relative moduli space of semi-stable sheaves on families of curves fine

Let $\pi:X \to B$ be a family of smooth, projective curves. Fix coprime integers $r,d$. Denote by $\mathcal{M}(r,d)$ the relative moduli functor corresponding to rank $r$, degree $d$, semi-stable ...
9
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1answer
899 views

Algebraic cycles, Chow spaces and Hilbert-Chow morphisms

In the sequel, let $S$ be a scheme, and $X$ a locally of finite type algebraic space over $S$. In his thesis ([R1-R4]), David Rydh introduces, among several others, the notion of relative cycles on $...
4
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3answers
326 views

About the Yoneda objects of a locally presentable category

This question is a follow-up of Extending functors defined on dense subcategories. Let $\mathcal{K}$ be a locally presentable category. An object $X$ of $\mathcal{K}$ is called a Yoneda object if ...
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254 views

When is the character group scheme of a group scheme representable? (Affine Case)

While reading Tate's article on Finite Flat Group Schemes in "Modular Forms and Fermat's Last Theorem" I was lead to this question. Let $S$ be a scheme, $G$ a group scheme over $S$, and $T$ an $S$-...
3
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1answer
95 views

Why does the variety of ideals in this quaternion type algebra have a non-reduced structure?

Let $A$ be the $\mathbb{C}$-algebra generated by elements $i,j$ with relations $i^2=j^2=0$ and $ij=-ji$, i.e. we have $A=\mathbb{C}\oplus\mathbb{C}i\oplus\mathbb{C}j\oplus\mathbb{C}ij$. Let $\mathcal{...
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106 views

How to describe the closure of a sub-group functor of a group scheme

Let $G$ be a group scheme over a field $k$, assume it has all the good conditions one may need (smooth, affine, algebraic, you name it). Let now $H:Aff/k\to Gp$ be a group functor and $i:H\to G$ be a ...
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0answers
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Infinite iterates of the contravariant hom endofunctors on sets

My recent answer to Is it possible to define higher cardinal arithmetics (about defining infinite tetrations) requires something I don't know. Here is the simplest case. Take a set $S$ and consider $$...
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1answer
81 views

representing base changes of the unit section

Let $S$ be a scheme and $G$ be a sheaf in groups on the big étale site over $S$. Let $e:S\rightarrow G$ be the unit section. Is it true that given an algebraic space in groups $H$, étale over $S$, and ...
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Yoneda embedding and Horn sentences

The following is taken from Borceux and Bourn's Mal'cev, Protomodular, Homological, and Semi-Abelian Categories. Metatheorem 0.1.3. Let $\mathcal P$ be a statement of the form $\varphi\implies \psi$, ...
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0answers
89 views

A question on uniformly corepresented functor

Let $\mathcal{F}$ be a functor from the category of $k$-schemes to sets, uniformly corepresented by $M$. Suppose $U$ is an open subscheme of $M$. I could not find a good reference for uniformly ...
2
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1answer
640 views

Is a left-exact limit-preserving functor $Ab \to Ab$ necessarily representable?

Let $Ab$ be the category of abelian groups, and let $F: Ab \to Ab$ be a covariant functor which is left-exact and limit-preserving. Is $F$ necessarily naturally equivalent to a functor of the form $\...
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History of the functor of points

Until now, I thought the functor of points approach was introduced by Grothendieck at the 1973 Buffalo seminar. However, in this note by Lawvere the author writes: "I myself had learned the ...
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1answer
671 views

Representable functors and direct limits

Let $\mathcal{F}:\mathrm{Sch}/S \to \{\mathrm{Sets}\}$ be a representable functor. Denote by $X$ the scheme representing $\mathcal{F}$. The question is whether the natural tranformation $\mathcal{F}(-)...
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1answer
207 views

On functors which are generically representable

Let $F$ be a set-valued (contravariant) functor on the category of schemes. Let $F_{\mathbb Q}$ be the associated functor on the category of schemes over $\mathbb Q$. Suppose that $F_{\mathbb Q}$ is ...
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1answer
447 views

When is the Hom-scheme connected?

Suppose that $A$ and $B$ are two algebras finite over a field $K$ (which may be assumed to be separably closed, if that helps), then we know that the functor $\mathrm{Hom}_K(\mathrm{Spec}(A),\mathrm{...
3
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1answer
388 views

What if the base change of an algebraic space is representable

Let $k\subset L$ be an extension of fields of characteristic zero. Suppose that $X/k$ is an algebraic space such that $X\otimes_k L$ is representable by a finite type $L$-scheme. I am sure there are ...
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0answers
166 views

How to determine a functor (natrually arising from geometry or homological algebra) to be locally of finite presentation?

How to determine a functor (natrually arising from geometry or homological algebra) to be locally of finite presentation? Is there any reference for such staff? My example of functors underlying this ...
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1answer
267 views

Are Brown representable functors determined by restriction to finite complexes?

Assume two $CW$ complexes $X,Y$ give two functors $h_X=[-,X], h_Y=[-,Y]$ on the homotopy category of $CW$ complexes whose restrictions to the full subcategory of finite $CW$ complexes are naturally ...
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1answer
683 views

Is Mumford's statement about the representability of some functor wrong?

I am having trouble proving a result in Mumfords book 'Lectures on Curves on an Algebraic surface. It is a statement about the representability of some functor. It is stated on page 108 and says the ...