# Questions tagged [representable-functor]

For questions about and related to Representable Functors, that are set-valued functors which can be "represented" by the hom-set of a single object from the domain category. Should be used together with the (category-theory) tag.

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### Why is the Grassmannian functor representable 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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### Details of the criterion for representability of a functor of S-schemes

I've come across a problem that's made me look back over representabiilty of scheme functors and I'm having a lot of trouble piecing together some categorical details that I used to think 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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### Representability of a functor in the category of schemes

I have read in some places that a functor of points of a scheme is representable if its defined by locally closed or open conditions. I would like to ask for some references about this fact. I don´t ...
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### $\mathbb{A}^2_{\mathbb{Z}} \backslash \{(0,0)\}$ represents a functor & a beautiful glueing story

Let $\mathbb{A}^2_{\mathbb{Z}}$ the affine plane and it is well known that $\mathbb{A}^2_{\mathbb{Z}}$ represents the contravariant functor \mathbb{A}^2: CRing^{op} \to Set, R \mapsto \{(r,s) \...
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### 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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### Gluing morphisms of sheaves on the big zariski site

I have been trying to get used to working with representable functors in place of getting my hands dirty with schemes. In particular I am working on a problem involving blowing up over an arbitrary ...
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### Universal properties in measure theory and probabilities

Obviously certains construction as for instance the push-forward (direct image) ${\cdot}_{\star}$ can be interpreted in the categories of measurable spaces, measured spaces or probability spaces as ...
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### Global sections functor is fully faithful if and only if it is logical

Let $\mathcal{E}$ be a non-degenerate topos. Question: Is it true that the "global sections" functor $\operatorname{Hom}_{\mathcal{E}}(1,-)$ is fully faithful if and only if it is logical? A non-...
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### Showing that a functor $Cat\to Set$ is representable

The problem is to show that the functor from the category of small categories to the category of sets that sends a category to its set of morphisms is representable. The major problem is to find a ...
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### Representability criterion for schemes

I am trying to understand. Lemma 25.15.4 Let $F$ be a contravariant functor the category of schemes with values in the category of sets. Suppose that $F$ satisfies the sheaf property. ...
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### 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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### 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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### Representable functor example

Consider a monoidal category $\mathcal{C}$ which is enriched over $\mathrm{vect_k}$ (=finite dimensional vector spaces over a field $k$). $\mathcal{C}$ is abelian and semi simple. If it helps one can ...
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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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### 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 ...
### 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}}(...