# Questions tagged [representable-functor]

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45 questions
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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 ...
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### 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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### 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 ...
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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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### $\text{GL}_n:\mathbf{Alg}_k\rightarrow\mathbf{Set}:R\mapsto \text{GL}_n(R)$ 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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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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### 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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### 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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### 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 ...
### 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 ...
### 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}}(...