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Questions tagged [representable-functor]

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1answer
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Showing that $F$ is not representable [closed]

As I'm trying to find (counter)examples of representable functors, I tried looking up some instructive examples. One of the counterexamples I'm having trouble with, is the following: Show that the ...
6
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1answer
95 views

Categories for which every contiuous sheaf is representable

I'm interested in locally small, cocomplete categories $\mathbf{C}$ such that every limit preserving functor $$\mathbf{C}^\mathrm{op}\to\mathbf{Set}$$ is representable. Is there a name for such ...
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242 views

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

I want to proof Theorem 8.9 from Algebraic Geometry I ( U.Görtz, T.Wedhorn), which reads as follows: Let $S$ be a scheme $F: Sch/S°\rightarrow Set $ a functor such that: F is a sheaf for the ...
3
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1answer
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Katz&Mazur's book: what is meant by “universal morphism” here?

I am currently reading Katz and Mazur's book "Arithmetic Moduli of Elliptic Curves", which can be found here. My question concerns the proof of proposition 1.6.2, page 23 of the book. First, the ...
3
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1answer
131 views

Equivalence of two definitions of local functors

All my functors are from commutative rings to sets. I've seen two different definitions of a local functor. In one definition we say $X$ is local if whenever $Y$ is a functor with open cover $Y_i$ ...
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95 views

Using the functor of points approach to show a scheme theoretic construction exists

T am trying to refram what I have learned in algebraic geometry in the context of the functor of points approach. In particular, I want to practice proving the existence of a scheme satisfying a ...
2
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2answers
67 views

Simple examples of non-representable functors

I am looking for examples of non-representable functors, to see how the Yoneda lemma works in these cases. Here is one: let $\mathbf{C}$ be the category of finite-dimensional Euclidean spaces, with ...
2
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1answer
45 views

Show that the functor that takes $R$ to the set of invertible elements of $R[X]/(X^2-a)$ is representable.

The following question is from the Fall 2016 UCLA algebra qualifying exam: Let $F$ be a field and $a\in F$. Show that the functor that takes $R$, commutative $F$-algebras to the invertible elements ...
2
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1answer
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Can it be useful to think of functors as representing themselves ?

Here's a thought I had and I wonder if it can be of any use, for instance has it ever helped proving any result (however minor the result). Say you're in a situation where you have some objects in ...
2
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1answer
223 views

Representations and representable functors

Wikipedia defines that a functor $F$ from a locally small category $\mathcal{C}$ to $\mathrm{Set}$ is representable if it is isomorphic to some $\mathrm{Hom}$ functor. On the other hand, nLab defines ...
2
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1answer
81 views

$\text{GL}_n:\mathbf{Alg}_k\rightarrow\mathbf{Set}:R\mapsto \text{GL}_n(R)$ is a representable functor

Following the definition of $\text{GL}_n:\mathbf{Alg}_k\rightarrow\mathbf{Set}:R\mapsto \text{GL}_n(R)$ found here: functor from $\mathbf{Alg}$ to $\mathbf{Set}$ I would like to show that it is a ...
2
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1answer
46 views

checking the functor $\texttt{Nil}_n$ is represented by $(\mathbb{Z}[x]/(x^n), \tau_R)$

This is the continuation of another question I did some days ago. Here. I have been working on it and I would like to know if my try to prove it is right or not. I would appreciate a lot any feedback ...
2
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1answer
165 views

Is : $ \mathrm{Gal} \ : \ F/ \mathbb{Q} \to \mathrm{Gal} (F / \mathbb{Q} ) $ represented by $ \overline{ \mathbb{Q} } $?

Let : $ \ \mathrm{Gal} : F/ \mathbb{Q} \to \mathrm{Gal} (F / \mathbb{Q} ) $ be the functor, which associate to a Galois extension $ F/ \mathbb{Q} $ of the field $ \mathbb{Q} $, the Galois group $ \...
2
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1answer
41 views

Characterization of a contravariant representable functor

A contravariant functor $\mathcal{F}:C\rightarrow Set$ is representable iff it has an universal object. Prove: [$\Rightarrow$] As $\mathcal{F}$ is representable, then $\tau:h_X\rightarrow \mathcal{F}...
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0answers
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Quot-like scheme for torsion sheaves

I am wondering if, as the Quot schemes parametrizes flat (quotients of) sheaves over schemes, there is anything similar for torsion sheaves. In first approximation, if $I$ is a sheaf of ideals over a ...
2
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1answer
63 views

Not able to make sense of gluing via the functor of points in EGA

I am trying to use a well known result of Grothendieck to show that if $S$ is a scheme, and $\mathcal{B}$ is a quasi coherent sheaf of $\mathcal{O}_{S}$-algebras, then there is a relative affine ...
2
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97 views

Representability criterion for Zariski sheaf in terms of open subfunctors

I've been trying to prove a fairly classical, well-known result, but am running into a lot of trouble following any of the proofs I have found. At the moment I am following EGA I 0.4.5.4. Let $F: \...
2
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1answer
109 views

Proof that a functor covered by open representable subfunctors is representable by a scheme

So I'm led to believe this is a fairly standard result. See, for example Lemma 25.15.4 here. I am trying to prove a criterion for the representability of a (contravariant) functor from schemes to sets....
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2answers
41 views

Why does $\mathbb{Z}$ represent the forgetful functor $U:\mathbf{Grp}\to\mathbf{Set}$

This is from Emily Riehl's Category theory in context The forgetful functor $U:\mathbf{Grp}\to\mathbf{Set}$ is represented by the group $\mathbb{Z}$ thanks to the natural isomorphism $\alpha:\mathbf{...
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3answers
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Is there only one isomorphism $h_C(X) \simeq h_C(Y)$ if it exists?

I'm trying to prove that if $F \simeq h_C(X)$ or "$X$ represents the functor $F$", then $X$ is unique up to unique isomorphism. I already know that if $h_C(X) \simeq F \simeq h_C(Y)$ that $s: X \...
1
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1answer
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Riehl's “Category Theory in Context” - Exercise 3.4.i

Let $\mathsf{I}$ be a small category, let $\mathsf{C}$ be a locally small category and let $F\colon\mathsf{I\to C}$ be a functor. Emily Riehl in her book "Category Theory in Context" defines a limit ...
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2answers
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Category theory - Prove that $\operatorname{Hom}$ preserves representations for quasi-inverse functors

Let $F: \mathcal C \to \mathcal D$ and $G: \mathcal D \to \mathcal C$ be quasi-inverse functors, and let $H : \mathcal C \to Set$ be a representable (contravariant) functor with representative $X \in \...
1
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1answer
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“Cats & Sheaves” defines $C_A$ w.r.t $F : C \to C'$ for object $A \in C'$, but then doesn't define $C_F$…

Let $F: C \to C'$ be a functor and let $A \in C'$. The category $C_A$ is given by $\text{Ob}(C_A) = \{(X, s); X \in C, s : F(X) \to A \}$ and $\text{Hom}_{C_A}((X,s), (Y, t)) = \{ f \in \text{Hom}...
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0answers
49 views

Ulrich, Torsten Proposition 8.4.2 (Closed Subscheme)

Let $S$ be a scheme and let $v : \mathcal E \longrightarrow \mathcal F$ be a morphism of quasi-coherent $\mathcal O_{S}$-modules. Let $\mathcal F$ be finite locally free. Then the locus $v=0$ is ...
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0answers
29 views

Irreducibility of the Jacobians of a curves.

I'm studying Jacobian varieties.I assume that the existence of the Jacobian variety for a curve and attempt to show irreducibility of the Jacobian for a curve according to Remark:IV.4.10.9 of ...
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0answers
30 views

How to arrive at unique factorization through the limit given naturality compatibility conditions?

If $\alpha : I \to C$ from a small category to any category $C$. Define a functor $\lim\limits_{\rightarrow} \alpha : X \mapsto \lim\limits_{\leftarrow} \text{Hom}_C(\alpha, X)$ from $C^{op}$ to $\...
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0answers
87 views

Nearly locally presentable categories

Here1, in the remark $2.3 (1)$ how from the fact that ${\cal K}(A,-)$ does not preserve coproducts it follows that ${\cal K}(A,-)$ sends special $\lambda$-directed colimits to $\lambda$-directed ...
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0answers
39 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 ...
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0answers
85 views

Expressing Representation of a Colimit as a Limit

$\newcommand{\C}{\mathcal{C}} \newcommand{\I}{\mathcal{I}} \newcommand{\L}{\mathcal{L}} \newcommand{\Hom}{\mathrm{Hom}_\C} \newcommand{\op}{\mathrm{op}} \newcommand{\colim}{\mathrm{colim}}$ Let $\C$ ...
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0answers
10 views

Reference request: Representability of multiplicative equivariant cohomology theories

Let $G$ be a topological group, say a compact Lie group, and $e^*_G$ a multiplicative $\mathbb Z$-graded $G$-equivariant cohomology theory defined on $G$–CW complexes. Is there some analogue result to ...
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67 views

Representability of zariski functors on schemes over a base by gluing

Suppose $F$ is a Zariski functor $(Sch/S)^{opp}\rightarrow Set$ where $S$ is a scheme. Suppose that I can show that for every affine $j:U\subseteq S$, I can show that the functor $F_U:(Sch/U)^{opp}\...
0
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1answer
61 views

The Yoneda lemma and a natural bijection

Let $S\colon\mathbf {Set}^{\cal A^{op}}\to \mathbf{ Set}$ be a functor. How does it follow from the Yoneda lemma that the following is a natural bijection: $\underline{\hom(A,-)\to SY \quad\quad\...
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2answers
30 views

Identifying region from equation

I am trying to identifying the region represented by the equation: $$x^2-y^2=9$$ I know that if it was a sum then it would be a circle but since it is a difference, how do I go about determining ...
0
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1answer
28 views

Universal elements of a functor and representability

A functor $F : \mathcal{C} \rightarrow \mathcal{Set}$ is said to representable if it is naturally isomorphic to $\mathcal{C}(A,–)$ for some object $A$ of $\mathcal{C}$. By the Yoneda lemma, we know ...
0
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1answer
33 views

How do you arrive at $\eta : \text{Hom}_{C'}(Y, Y') \to \text{Hom}_{C'}(LR(Y), Y')$ from $\eta : LR \to \text{id}_{C'}$?

On page 28 of "Categories and Sheaves" it says: $$ \eta : L R \to \text{id}_{C'} $$ is a functor but then they have in a commutative diagram right below that: $$ \text{Hom}_{C'}(Y, Y') \xrightarrow{...
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1answer
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Dualizing module and finiteness hypothesis

Serre, in his Galois Cohomology, states: Proposition 17. Let $n$ be an integer $\geq 0$. Assume: (a) $\text{cd}(G) \leq n$ (b) For every $A \in C^f_G$, the group $H^n(G, A)$ is finite. ...
0
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1answer
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Cats & Sheaves, understanding one paragraph related to Yoneda Lemma

Assume that $F \in C^{\wedge}$ is represented by $X_0 \in C$. Then $\text{Hom}_{C^{\wedge}}(h_C(X_0, F)) \simeq F(X_0)$ gives an element $s_0 \in F(X_0)$. Moreover, for any $Y \in C$ and $t \in ...
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0answers
23 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 $$ Grass^e(\mathcal E)(T) = \{...
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EGA I (Springer), Proposition 0.4.5.4.

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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40 views

Showing the Affine functor $\underline{\mathbb{A}}^r$ is representable by Affine Space $\mathbb{A}^r := Spec(\mathbb{Z}[X_1, \dots, X_r])$

Let $\underline{\mathbb{A}}^r$ be the functor from $\textbf{Schemes}$ to $\textbf{Sets}$ which associates to each scheme $S$ the set of morphisms $\bigoplus_{k=1}^rO_S \to O_S$ ($O_S$ is the structure ...
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Are the Following Endomorphism and General Linear Group Functors Representable?

Let $k$ be a unital commutative ring. Let $k$-Alg denote the category of commutative and unital $k$-algebras. Let Set denote the category of Sets Fix an arbitrary $k$-module $V.$ Consider the ...
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35 views

Brown representability and based homotopy classes

The following statement is the version of Brown's representability theorem I learned: Let $\mathbf{CW}$ be the category of based, connected CW complexes together with based homotopy classes of ...
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1answer
43 views

Brown representability for model categories

In Jardine's article on the subject: https://ncatlab.org/nlab/files/JardineBrownrep.pdf He shows a version of Browns representability, which asserts the representability of a functor out of a model ...
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1answer
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Given the adjunction morphisms $\epsilon, \eta$, then $(\eta \circ L) \circ(L\circ \epsilon) = \text{id}_L$ is easy to check.

Given the adjunction morphisms $\epsilon, \eta$, then $(\eta \circ L) \circ(L\circ \epsilon) = \text{id}_L$ is easy to check. This is from page 29 of "categories and sheaves". I have a major ...
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The isomorphism $F \simeq h_C(X)$ determines $X$ how?

I've asked something similar before, here. But I didn't quite understand their reasoning. So I'm breaking the problem down. First of all, how is $X$ determined? By yoneda $\text{Hom}_{C^{\wedge}}(...