# 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.

79 questions
Filter by
Sorted by
Tagged with
44 views

100 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 ...
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 ...
42 views

### Showing that $H_A\simeq H_{A'}$ implies $A\simeq A'$

I'm trying to solve this exercise: So we are given that for all objects $B$, there is a natural isomorphism $$H_A(B)\simeq H_{A'}(B)$$ Let's write this isomorphism as $f\mapsto \bar f$. The ...
68 views

70 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 ...
55 views

45 views

### Extension of a functor by colimits: Cisinski - Higher Categories and Homotopical Algebra - Remark 1.1.11

First, I state premilinary results. For a presheaf $X\colon A^{op}\to\mathsf{Set}$, it's category of elements, denoted by $\int X$, has pairs $(a,s)$ where $a \in A$ and $s \in X(a)$ as objects and ...
46 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 ...
67 views

### Two definitions of representable functor

Leinster Basic Category theory gives two equivalent definitions of representable functors: $X: \mathcal{A} \to Set$ is representable if $X \cong H^A = Hom(A,-)$ $X: \mathcal{A}^{op} \to Set$ is ...
30 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-...
40 views

### Equivalent functors carrying representability

Emily Riehl's "Category Theory in Context, ${\rm Exercise}~2.1.{\rm iii}.$ Suppose $F:{\rm C}\to{\rm Set}$ is equivalent to $G:{\rm D}\to{\rm Set}$ in the sense that there is an equivalence of ...
106 views

### Monomorphisms are preserved by Representable Functors

Emily Riehl's "Category Theory in Context", ${\rm Exercise}~2.1.{\rm ii}.$ Prove that if $F:{\rm C}\to{\rm Set}$ is representable, then $F$ preserves monomorphisms, i.e., sends every monomorphim in ...
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 $X$ ...
70 views

### Scheme $\text{Spec}(S^*\text{Hom}(V,W)^{\vee})$ from “The Geometry of Moduli Spaces of Sheaves”

I have a couple of questions about the notations & their meaning used in "The Geometry of Moduli Spaces of Sheaves" by Huybrechts & Lehn, in Example 2.2.2 (page 38): $V$ is assumed to be a be ...
68 views

85 views

### 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, ...
183 views

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 ...
103 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 ...
78 views

176 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 ...