# Questions tagged [representable-functors]

The representable-functors tag has no usage guidance.

94
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### Why are functor categories nice? [migrated]

I was looking at the Yoneda embedding and one motivation is that we are embedding the category into a functor category and "functor categories are nice". What does this mean? What nice ...

7
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2
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### Condition for an equivalence of functor categories to imply an equivalence of categories

Given small categories $\mathcal{C}$ and $\mathcal{D}$, we have that $[\mathcal{C}^\text{op},\textbf{Set}]\simeq[\mathcal{D}^\text{op},\textbf{Set}]$ if and only if the Cauchy-completions of $\...

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### Is this double quotient of $\operatorname{SL}_2$ representable by an algebraic space or a scheme?

$\DeclareMathOperator\SL{SL}$Let $B$ be a Borel subgroup (upper triangular matrices), and let $\Gamma := \langle \sigma\rangle$ be the group generated by a (hyperbolic) element of $\SL_2/\mathbb{Q}_p$ ...

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1
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### Yoneda lemma for one object categories

Let $G$ be a group and let $\mathbb{G}$ be the associated one object category. Is there an explicit presentation of representable functors from $\mathbb{G} \to $Set? If so how does the Yoneda lemma ...

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### Is this functor $\mathcal{F}: \text{Sch}/\mathbb{Q}\longrightarrow \text{Sets}$ a sheaf?

Consider the functor $\mathcal{F}: \text{Sch}/\mathbb{Q}\longrightarrow \text{Sets}$, defined by sending a scheme $X$ with coordinate ring $\mathcal{O}(X)$ to the set of orbits $B(\mathcal{O}(X))\...

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### The stack $\operatorname{GL}_2/B$

Let $F$ be the functor from the category of affinoid Tate algebras over $\mathbb{Q}_p$ to the category $\mathrm{Sets}$, which maps an affinoid $\operatorname{Spm} R$ to the set of orbits $\...

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### Representability of the sheaf $\mathrm{Hom}(G,\mathrm{SL}_2)$

$\DeclareMathOperator\Spm{Spm}\DeclareMathOperator\SL{SL}\DeclareMathOperator\Hom{Hom}$Let $T$ be the topos of $\Spm\mathbb{Q}_p$-rigid analytic spaces, $G$ an abstract group, and $\Hom(G,\SL_2)$ the ...

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### Representability of $\operatorname{Hom}(G_{\mathbb{Q}}, \operatorname{GL}_2)$

Let $G_{\mathbb{Q}}$ be the absolute Galois group of the rationals, and let $F: \mathrm{Aff}/\textbf{Q}_p\longrightarrow \mathrm{Sets}$ be the functor which associates to every affine $\mathbb{Q}_p$ ...

4
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1
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312
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### Construct morphisms of schemes on level of associated functors

I have a general question about techniques used in @Emerton's proof, sketched below, in the answer to $\mathbb{P}^n$ is simply connected.
Given a finite étale map $\pi: Y \to \mathbb P^n$ (we regard ...

4
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1
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### Can the functor of the points of a scheme be characterized by its values on subcategories of the affine schemes?

A scheme is equivalent to a functor $\mathcal{F}:\textbf{AffSchemes} \rightarrow \textbf{Set}$ such that it admits a cover of affine schemes and is a sheaf of rings on the Zariski site.
Suppose $\...

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133
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### Representablility of maps between classifying spaces

Assume that $G,H$ are two sheaves of groups (say in fpqc topology on the scheme $X$) and there is a map $G\to H$ which is representable by a closed immersion. Let us also assume that the quotient is ...

5
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381
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### Yoneda map for a composition of a representable functor and an arbitrary functor

Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $T : \mathcal{C} \rightarrow \mathcal{D}$ be a functor. Suppose that $F : \mathcal{D}^\mathrm{op} \rightarrow \mathrm{Set}$ is a functor. (So ...

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### Reference request: Détailed explanation why the Grassmannian scheme represents the Grassmannian functor

Similar questions have been asked on this site, including by myself, but none of these have been given a satisfying answer.
The question is: Why does the Grassmannian scheme represent the Grassmannian ...

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### Moduli stacks and representability of diagonal by schemes

The answer to my question might very well be standard, but I have had trouble finding the right keywords to search for it, so I apologize if this is something well-known to the experts.
I am learning ...

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### Does Fpqc sheaf over category of rings imply representability

I am trying to read the article "Algebrization and Tannaka duality" by Bharghav Bhatt. In Corollary 1.2, he says that given a qcqs algebraic space $X$ (I am interested in the case when $X=\...

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### Reference for the Brown representability theorem in the case of locally presentable (∞,1)-categories

Various generalizations of the Brown representability theorem
found in the literature identify additional conditions one can impose on a category $C$ so that functors $\def\op{{\sf op}}\def\Set{{\sf ...

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### Local existence of (quasi)-universal family of sheaves

Let $p : X \to S$ be a projective morphism between two Noetherian $\mathbb{C}$-schemes of finite type with connected fibres. Let $O_X(1)$ be a very ample line bundle on $X$ relative to $S$. Given a ...

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### Is the Kolmogorov-Arnold representation theorem an example of the Yoneda lemma?

From Wikipedia:
https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold_representation_theorem
In real analysis and approximation theory, the Kolmogorov–Arnold representation theorem (or ...

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### A small lemma in Schlessinger's criterion paper

In the construction of a hull in Schlessinger's paper, one small lemma used is not clear in my opinion. That should be stated as follows:
Let $(R,m)$ be a Noetherian complete local ring, $I_1\supset ...

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### Reference for Grothendieck's theorem on representation of unramified functors

In the Exposé 294 of the Bourbaki Seminar of the year 1964-1965, Murre gives an outline of proof of a theorem of Grothendieck giving necessary and sufficient conditions of representability by a scheme ...

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### (Pro-)representable functors and full subcategories in homotopy theory

$\DeclareMathOperator\Ab{Ab}\DeclareMathOperator\Ho{Ho}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Hotc{Hotc}\DeclareMathOperator\Sm{Sm}$Let $\mathcal{C}\overset{\iota}{\longrightarrow} \mathcal{...

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### Algebraic groups as functors of points vs maximal points over algebraically closed field

So I only started learning about group schemes this summer, and I found two approaches. For the record I am interested in affine group schemes $G$ of finite type over a field $k$ (algebraic groups).
...

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### Representability result

Let $X$ and $S$ be schemes over a field $k$.
Reading this paper, there is a result on the representability of a morphism (proposition 3.1, page 4).
Which result or reference on representability is ...

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### Grassmannian of vector bundles with determinant being a square

I want to find a classfying space of vector bundles of rank $n$ generated by its global sections with determinant being a square of a line bundle. Stating in a formal way, suppose $X$ is a scheme, ...

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### Any exact faithful functor is represented by a unique projective generator

In the book 'Tensor Categories' by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik on page 10 it says:
'Conversely, it is well known (and easy to show) that any exact faithful functor ...

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### Representable functors and symplectic co-tangent bundles

I've been banging my head against something that I feel should follow from abstract non-sense, and I hope someone here can set me straight.
Let $\mathcal{M}$ be the category of smooth manifolds, with ...

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### Constructions that can be seen as objects representing a functor

Some constructions can be seen as objects representing a functor.
For example,
Consider a topological group $G$ and a functor $\mathcal{F}:\text{Top}\rightarrow \text{Gpd}$ defined as $M\mapsto \...

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### Why does every chain complex have a map into its cone?

In Weibel's An introduction to homological algebra he defines a cone as an explicit chain complex associated to the given one -i.e. for a chain $C=(C_i, d)$ he defines $Cone(C)=\left(C_{i-1} \oplus ...

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### Does every functor between Grothendieck categories have adjoints?

Let $F:\mathcal C\longrightarrow \mathcal D$ be an additive functor that preserves colimits.
Suppose that $\mathcal C$ and $\mathcal D$ are Grothendieck categories.
Does $F$ have a right adjoint? ...

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### Finding a cover of the Hilbert functor, that corresponds to a cover of the Grassmannian

Let $\mathrm{Grass}_{k+1,n+1}:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$ be the Grassmann functor, which maps a scheme $S$ to the set:
$$\left\{\mathscr{U}\subseteq\mathscr{O}_S^{n+1}:\,\,\mathscr{...

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### Open subfunctor of Quot Functor induced by open immersion

Let $f: X \rightarrow S$ be a morphism of noetherian schemes and $\mathfrak{Q}uot_{\mathcal{E}/X/S}$ be the functor parametrizing families of quotients of $\mathcal{E}$ in the category of locally ...

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### Representability of Grassmannian functor 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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### Proving the representability of a functor that is covered by open subfunctors

I already posted this on math.stackexchange some while ago, but haven't received any answers yet. (https://math.stackexchange.com/questions/3221006/proving-the-representability-of-a-functor-that-is-...

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### 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 $$ \operatorname{Grass}^e(\...

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### EGA I (Springer), Proposition 0.4.5.4 [closed]

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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### is the functor of finite order elements in grp representable

My question is whether the functor F:Grp->Set that sends a Group to its Set of elements with finite order is representable.
I have the sense that it shouldn't be but I've so far failed to prove it in ...

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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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### representability of $\mathrm{R}^1f_{*,\mathrm{fppf}}\mathscr{A}$

Let $f: X' \to X$ be a finite flat morphism of (nice) schemes and $\mathscr{A}/X'$ be a smooth commutative group scheme such that the Weil restriction $f_*\mathscr{A}/X$ is representable by a ...

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### Quotient of a sheaf by group action and representabillity

Let $X$ be a scheme and $S$ be a sheaf of sets over the fppf topology of $X$. Let $G$ be a group scheme over $X$ and there is an action of $G$ on $S$. Now, I want to look at the quotient $G \setminus ...

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### Representability of Flattening stratification functor

Let $f:X \to Y$ be a projective morphism between noetherian scheme. Is there any know condition under which there exists a functorial stratification of $Y$ i.e., there exists a filtration by locally ...

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### Internal Mor of schemes

Let $S$ be a Noetherian scheme and $X,Y$ be $S$-schemes of finite type. Consider the functor $X^Y$ given by $T \mapsto Mor_S(Y \times_S T,X)$. When is this functor representable by an $S$-scheme of ...

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### Representable $\text{Hom}$ functors

Let $X, Y, S$ be noetherian schemes, $X$ flat and quasi-projective over $S$, $Y$ projective over $S$.
Is the hom-functor $T\mapsto\text{Hom}_T(X_T, Y_T)$ representable?
If $X$ is flat and projective,...

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### Is the relative moduli space of semi-stable sheaves on families of curves fine

Let $\pi:X \to B$ be a family of smooth, projective curves. Fix coprime integers $r,d$. Denote by $\mathcal{M}(r,d)$ the relative moduli functor corresponding to rank $r$, degree $d$, semi-stable ...

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### Algebraic cycles, Chow spaces and Hilbert-Chow morphisms

In the sequel, let $S$ be a scheme, and $X$ a locally of finite type algebraic space over $S$.
In his thesis ([R1-R4]), David Rydh introduces, among several others, the notion of relative cycles on $...

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### About the Yoneda objects of a locally presentable category

This question is a follow-up of Extending functors defined on dense subcategories.
Let $\mathcal{K}$ be a locally presentable category. An object $X$ of
$\mathcal{K}$ is called a Yoneda object if ...

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### When is the character group scheme of a group scheme representable? (Affine Case)

While reading Tate's article on Finite Flat Group Schemes in "Modular Forms and Fermat's Last Theorem" I was lead to this question. Let $S$ be a scheme, $G$ a group scheme over $S$, and $T$ an $S$-...

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### Why does the variety of ideals in this quaternion type algebra have a non-reduced structure?

Let $A$ be the $\mathbb{C}$-algebra generated by elements $i,j$ with relations $i^2=j^2=0$ and $ij=-ji$, i.e. we have $A=\mathbb{C}\oplus\mathbb{C}i\oplus\mathbb{C}j\oplus\mathbb{C}ij$.
Let $\mathcal{...

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### How to describe the closure of a sub-group functor of a group scheme

Let $G$ be a group scheme over a field $k$, assume it has all the good conditions one may need (smooth, affine, algebraic, you name it).
Let now $H:Aff/k\to Gp$ be a group functor and $i:H\to G$ be a ...

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### Infinite iterates of the contravariant hom endofunctors on sets

My recent answer to Is it possible to define higher cardinal arithmetics (about defining infinite tetrations) requires something I don't know. Here is the simplest case.
Take a set $S$ and consider
$$...

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1
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### representing base changes of the unit section

Let $S$ be a scheme and $G$ be a sheaf in groups on the big étale site over $S$. Let $e:S\rightarrow G$ be the unit section. Is it true that given an algebraic space in groups $H$, étale over $S$, and ...