# Questions tagged [representable-functor]

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