The representable-functors tag has no usage guidance.

**16**

votes

**0**answers

372 views

### History of the functor of points

Until now, I thought the functor of points approach was introduced by Grothendieck at the 1973 Buffalo seminar.
However, in this note by Lawvere the author writes:
"I myself had learned the ...

**12**

votes

**1**answer

1k views

### Picard groups of non-projective varieties

As far as I know, the main representability result for the relative Picard functor $Pic_{X/k}$, for a noeth. sep. scheme of finite type over a field $k$ is:
If $X$ is proper then $Pic_{X/k}$ is ...

**10**

votes

**2**answers

450 views

### Why is Maps(X,Y) an open subfunctor of Hilb(X x Y)?

Let $X$ and $Y$ be projective schemes. Then we can define the mapping scheme between them, $\rm{Maps}(X,Y)$ as follows:
To any map $f:X\rightarrow Y$ we consider the graph $\Gamma_f$ as a closed ...

**10**

votes

**2**answers

489 views

### Do coarse moduli spaces respect Galois actions?

To explain, I will use the following concrete example: Let $\mathcal{M}_g$ be the functor for the moduli problem of classifying genus $g$ smooth projective curves (taking a scheme $S$ to the set of ...

**10**

votes

**1**answer

567 views

### Representing cohomology of a sheaf à la Eilenberg-Maclane

Suppose that we are given a nice space $X$ and a sheaf of abelian groups $F$ on $X$. Fix an integer $n$. Then We have a contravariant functor from nice spaces over $X$ to abelian groups; Namely, to a ...

**10**

votes

**0**answers

369 views

### Where is the representability of the moduli of curves with framed points proved?

There is a variant of the Knudsen-Mumford moduli problem $\mathcal{M}_{g,n}$ of pointed curves, where one endows the $n$ marked points with non-zero tangent vectors. It shows up in the theory of ...

**9**

votes

**3**answers

2k views

### What functor does Grassmannian represent?

As we know, the Projective space P^n represent the functor sending X to the set of line bundles L on X together with a surjection from the trivial vector bundle to L.
My question is, what functor ...

**9**

votes

**1**answer

196 views

### Are Brown representable functors determined by restriction to finite complexes?

Assume two $CW$ complexes $X,Y$ give two functors $h_X=[-,X], h_Y=[-,Y]$ on the homotopy category of $CW$ complexes whose restrictions to the full subcategory of finite $CW$ complexes are naturally ...

**9**

votes

**2**answers

479 views

### When is tensoring with a module representable by a scheme?

Consider the following: Let $A$ be a commutative ring, let $M$ be an $A$-module. When is the functor from $A$-algebras to Sets given by $R \mapsto R \otimes M$ representable by an $A$-scheme?
Unless ...

**8**

votes

**1**answer

366 views

### Comparing colimits in schemes with colimits in sheaves of sets

Suppose I have a diagram of schemes, and I know that the colimit exists in the category of schemes. How does this colimit compare with the colimit of the corresponding sheaves (I'm being nonspecific ...

**7**

votes

**4**answers

520 views

### An explicit description of Lawvere's segment in the category of simplicial sets

In any presheaf topos, there exists an object called Lawvere's segment, which can be described as the presheaf $L:A^{op}\to Set$ such that for each object $a\in A$, $L(a)=\{x\hookrightarrow\ h_a: x\in ...

**7**

votes

**1**answer

622 views

### When is a stack (NOT) geometric?

Following the terminology of $n$-Lab, a geometric stack $\mathcal{X}$ on a site $\mathcal{(C,J)}$ is a stack for which there exists a representable epimorphism $X \to \mathcal{X}$ from an object $X$ ...

**7**

votes

**1**answer

761 views

### What functor does a Schubert variety represent?

I'm inspired by Yuhao's question. The functor that takes a scheme S to the set of k-dimensional vector subbundles of C^n x S (understanding "subbundle" to mean that the quotient by it is another ...

**7**

votes

**1**answer

1k views

### construction of the Jacobian of a curve

I am trying to understand the construction of the Jacobian of a curve following the notes of J. S. Milne
The question is going to be about a particular step in the proof of Proposition 4.2b in ...

**6**

votes

**2**answers

521 views

### Is an algebraic space over a DVR, whose special fibre and generic fibre are schemes, actually a scheme?

Is an algebraic space over a DVR, whose special fibre (and all its infinitesimal neighborhood) and generic fibre are schemes, actually a scheme?

**6**

votes

**2**answers

891 views

### Relationship between Hilbert schemes and deformation spaces

Hi, I'm just starting to learn about deformation theory (via Hartshorne's Deformation theory, as well as Fantechi's section of FGA explained), and I feel like I'm confused about fundamental concepts. ...

**6**

votes

**1**answer

284 views

### Representability of sheaf of Ext^1 of a Néron model by $\mathbb{G}_m$

Let's work over a trait $S=\mathrm{Spec}R$, where $R$ is a dvr with fraction field $K$, residue field $k$. Given an abelian variety $A_K$ with semi-stable reduction, let $A$ over $S$ be its Néron ...

**6**

votes

**2**answers

593 views

### Do quotients of representable sheaves represent quotients?

Here's the context for the question: Proposition 4.6 of Freitag and Kiehl's book on etale cohomology shows that a sheaf (of sets) $\mathcal{F}$ (on the site Et(X)) is constructible if and only if it ...

**6**

votes

**1**answer

328 views

### formally smooth functor

Let $p$ be a prime number, $\mathcal{O}$ the integers of a finite extension of $\mathbb{Q}_p$ with residue field $k$. Let $\mathcal{C}$ be the category of complete, local, noetherian ...

**6**

votes

**1**answer

1k views

### When does a “representable functor” into a category other than Set preserve limits?

This might be a dumb question. If $C$ is an ordinary category, then for any $c \in C$ the covariant representable functor $\text{Hom}(c, -) : C \to \text{Set}$ preserves limits. However, it can ...

**5**

votes

**4**answers

364 views

### Is tensoring with a module representable iff it is locally free of finite rank?

Motivation:
It's nice when you can think of the elements of an $A$-module $M$ as sections some $A$-scheme $Y\to Spec(A)$. That is, maps $Spec(A)\to Y$ such that $Spec(A)\to Y \to Spec(A)$ is the ...

**4**

votes

**2**answers

540 views

### Is the functor of open subschemes representable?

First some simple observations in order to motivate the question:
The functor $Set^{op} \to Set, X \to \{\text{subsets of }X\}, f \to (U \to f^{-1}(U))$ is representable. The representing object is ...

**4**

votes

**2**answers

420 views

### Representability of sheaves of groups

There are lots of natural functors (that define sheaves in the fppf topology) that are not representable by schemes. For example, hilbert schemes of proper non-projective schemes in general need ...

**4**

votes

**1**answer

522 views

### When is a sheaf on a scheme extendable to a representable functor?

I'll start with example:
Let $X$ be a scheme, and $O_X$ be its structure sheaf. It is defined at the moment on open sets of $X$, and it takes them to $Sets$. However, it is extendable to a sheaf on ...

**4**

votes

**1**answer

292 views

### Is a group scheme determined by its category of representations?

More precisely, let $G$ be an affine group scheme over a field $k$, $Rep_k(G)$ be the category of finite dimensional representations of G, and $\omega_0$ be the forgetful functor from $Rep_k(G)$ to ...

**3**

votes

**1**answer

264 views

### What if the base change of an algebraic space is representable

Let $k\subset L$ be an extension of fields of characteristic zero.
Suppose that $X/k$ is an algebraic space such that $X\otimes_k L$ is representable by a finite type $L$-scheme.
I am sure there are ...

**3**

votes

**1**answer

143 views

### Is a left-exact limit-preserving functor $Ab \to Ab$ necessarily representable?

Let $Ab$ be the category of abelian groups, and let $F: Ab \to Ab$ be a covariant functor which is left-exact and limit-preserving. Is $F$ necessarily naturally equivalent to a functor of the form ...

**3**

votes

**1**answer

397 views

### Sheaf condition and representability in the category Top

This is a rather nice question I got from this user via private communication.
Let $\mathcal{C} = Top$ the category of topological spaces. Let $\mathcal{C}^\prime$ be the category ...

**3**

votes

**0**answers

152 views

### Examples of Sheafification via Hypercovers

For a presheaf $F$ on a category equipped with a pretopology, one has the sheafification $F^{\sharp}$ of $F$.
I know well the plus-construction of sheafification, which is presented in Artin's paper ...

**2**

votes

**1**answer

146 views

### Algebraic objects and lifts of their represented functors

I've seen the following theorem around in various forms:
To give an object $A \in \mathcal{C}$ the structure of a $\Omega$-algebra object in $\mathcal{C}$ is equivalent to giving a lift of the ...

**2**

votes

**1**answer

292 views

### Extending smooth irreducible representations

Hi,
Let $G_1, G_2$ be topological groups with $G_1 \subset G_2$ is closed. Let $\rho:G_1 \to Aut(V)$ be a smooth irreducible representation. Can anyone tell me if there is a criterion/example/idea ...

**2**

votes

**1**answer

227 views

### When is the Hom-scheme connected?

Suppose that $A$ and $B$ are two algebras finite over a field $K$ (which may be assumed to be separably closed, if that helps), then we know that the functor ...

**2**

votes

**0**answers

208 views

### Two functorial definitions of schemes

I have been reading a bit about the "functor of points" theory for schemes. There seem to be two ways of going about defining schemes this way:
Equip the category $\textbf ...

**1**

vote

**1**answer

411 views

### Schemes associated to vector spaces

Let $k$ be a field. Let $F$ be a covariant functor on the category of $k$-algebras to the category of sets. Assume that the opposite functor $F^{op}$ on the category of affine $k$-schemes is a sheaf. ...

**1**

vote

**1**answer

586 views

### Is Mumford's statement about the representability of some functor wrong?

I am having trouble proving a result in Mumfords book 'Lectures on Curves on an Algebraic surface.
It is a statement about the representability of some functor. It is stated on page 108 and says the ...

**1**

vote

**1**answer

157 views

### On functors which are generically representable

Let $F$ be a set-valued (contravariant) functor on the category of schemes. Let $F_{\mathbb Q}$ be the associated functor on the category of schemes over $\mathbb Q$.
Suppose that $F_{\mathbb Q}$ is ...

**1**

vote

**1**answer

218 views

### quotient of ind scheme

Is I consider an ind scheme such as $G(k((t)))$ for a reductive connected group over $k=\bar{k}$
I have the conjugacy action of $G(k[[t]])$.
In what category can I make the quotient ...

**1**

vote

**0**answers

42 views

### A question on uniformly corepresented functor

Let $\mathcal{F}$ be a functor from the category of $k$-schemes to sets, uniformly corepresented by $M$. Suppose $U$ is an open subscheme of $M$. I could not find a good reference for uniformly ...

**1**

vote

**0**answers

128 views

### How to determine a functor (natrually arising from geometry or homological algebra) to be locally of finite presentation?

How to determine a functor (natrually arising from geometry or homological algebra) to be locally of finite presentation?
Is there any reference for such staff?
My example of functors underlying this ...

**0**

votes

**1**answer

215 views

### Terminal object of category of elements of a representable functor

In Awodey's Category Theory (2nd edition), page 229, I read:
the category of elements $J$ of a representable $yC$ has a terminal
object, namely the element $1_C \in Hom_{\mathbf{C}}(C,C)$
...

**0**

votes

**1**answer

164 views

### Prorepresentable functors repres. by alg. spaces? Covering spaces by alg. spaces.

Let $X$ be a (reasonable) scheme. I'm curious about constructing the constructing the covering space of a scheme algebraically. The covering space functor $F$ (below) can be represented by a ...

**0**

votes

**0**answers

273 views

### restriction and pullback of representable etale sheaf along closed immersion

I find that the restriction and pullback of representable etale sheaf along closed immersion are very confusing. I think they are different in general, I hope some experts can confirm my ...

**-1**

votes

**1**answer

285 views

### Representable functors and direct limits

Let $\mathcal{F}:\mathrm{Sch}/S \to \{\mathrm{Sets}\}$ be a representable functor. Denote by $X$ the scheme representing $\mathcal{F}$. The question is whether the natural tranformation ...