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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45 views

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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183 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 ...
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76 views

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 ...
3
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68 views

$\mathbb{A}^2_{\mathbb{Z}} \backslash \{(0,0)\}$ 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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82 views

Representability of functors in categories other than Set category

Let $\mathcal C,\mathcal D$ be locally small categories and also assume that $\mathcal D$ is small and that every morphism in $\mathcal D$ is a function between sets. Assume that for every $A,B \in \...
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100 views

On the functorial point of view in algebraic geometry.

Here's a question I've been thinking about lately. I hope it's not too vague - I apologize in advance if this should be the case. Suppose you want to do algebraic geometry using the $\textit{...
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70 views

Describing the universal property of the evaluation map

Emily Riehl's "Category Theory in Context", ${\rm Exercise}~2.3.{\rm iii}.$ The set $B^A$ of functions from a set $A$ to a set $B$ represents the contravariant functor ${\rm Set}(-\times A,B):{\rm ...
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57 views

Functor $G$ has right adjoint iff $Y \mapsto Hom(X, G(Y))$ is corepresentable

My professor in his notes claims that a functor $G$ has a right adjoint iff the functor $Y \mapsto Hom(X, G(Y)$ is corepresentable, i.e for each $X$ there is an object $F(X)$ and a nutural by $X$ ...
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30 views

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
104 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 ...
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158 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
141 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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1answer
23 views

Are these two functors same (isomorphic)?

Let $\mathscr{C}$ be the category of schemes (that is the category of schemes over $\mathrm{Spec}(\mathbb{Z})$), $X,Y$ be two schemes. Then we can have two functors: $\mathscr{C}^{op}\to \{\mathrm{...
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48 views

Why is the ring of local-ring valued points of a ring scheme a local ring.

I'm confused on a supposedly easy claim: Let $T$ be a base scheme, and let $\mathbf{R}$ be a ring scheme over $T$, i.e. a scheme $\mathbf{R} \to T$ such that for all $E \in \operatorname{Sch}_{/T}$ $\...
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102 views

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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47 views

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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32 views

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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52 views

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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58 views

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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55 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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59 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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35 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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91 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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54 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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103 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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17 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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79 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}\...
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42 views

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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203 views

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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68 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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55 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
53 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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67 views

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}}(...