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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44 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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Hom functor preserves limits [duplicate]

I was looking to prove that $Hom_C(Y,limX) \simeq lim Hom_C(Y,X)$. I saw something in https://ncatlab.org/nlab/show/hom-functor+preserves+limits, but I didn't really understood the proof. Can someone ...
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1answer
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Subobjects in the category of presheaves?

Suppose $\mathcal{C}$ is a locally small category, and $X$ be an element of $\mathcal{C}.$ A sub-object of $X$ is an isomorphism class of monomorphisms in to $X.$ Now suppose we embedd $X$ in $[\...
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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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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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1answer
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 ...
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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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1answer
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For which categories is its Yoneda embedding essentially surjective?

For a locally small category $\mathcal{C}$, you can embed $\mathcal{C}$ in the functor category $\mathrm{Set}^\mathcal{C}$ via the functor $X \mapsto \mathrm{Hom}_\mathcal{C}(X,{-})$. This embedding ...
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1answer
165 views

Representable functor is left-adjoint

$\newcommand{\catname}[1]{{\textbf{#1}}} \newcommand{\Set}{\catname{Set}}$ Let $\mathcal{C}$ be a locally small category. I am trying to show that if all coproducts exist in $\mathcal{C}$ and if $F:\...
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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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Alternate definition of Group Object, and how to 'play' with it?

I've come across an alternate definition of a group object, shown below: Suppose $F:C^{op} \rightarrow Grp$ is a functor such that its composition with the forgetful functor $?:Grp \rightarrow ...
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1answer
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Using Yoneda's lemma to “guess” the definition of exponential object in $SET$

Like what the title said, I want to use Yoneda's lemma to "guess" the definition of exponential object in $SET$. So basically I want to say that given any two sets $A, B$ in $SET$, the exponential ...
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Is there a canonical way to go back from $\hat{\mathbb{C}}$ (presheaves) to $\mathbb{C}$

Is there a "canonical" way to go back from $\hat{\mathbb{C}}$ (Category of presheaves) to $\mathbb{C}$. Here we define $\hat{\mathbb{C}}$ to be the category of functors from $\mathbb{C}^{op} \...
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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 ...
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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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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 ...
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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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1answer
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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 ...
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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 ...
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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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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 ...
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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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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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Why is the forgetful functor representable?

I'm reading Adowey's Category Theory, and I'm struggling with the last exercise of the second chapter, which is to show that the forgetful functor for monoids, $U : \mathbf{Mon} \to \mathbf{Sets}$, is ...
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Reflections in locally presentable categories-unclear step in the proof

Here in the paper by Rosicky Adamek, Reflections in locally presentable categories on the page 90 in the proof theorem on the page 89, I do not follow the 3rd line there: ... and hence is reflective ...
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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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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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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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On the proof that $1_{\textbf{Set}}$ is representable

Example 4.1.4 says: Consider $H^1:\textbf{Set}\to\textbf{Set}$, there $1$ is the one-element set. Since a map from $1$ to a set $B$ amounts to an element of $B$, we have $$H^1(B)\cong B$$ for each ...
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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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Why are coproduct objects corepresentations of cartesian products, rather than representations of disjoint unions?

One way we can define the product of two objects $A$ and $B$ in some category $\mathcal C$ is as a representation of the contravariant functor $(\to A) \times (\to B)$ in $[\mathcal C^{\mathrm{op}}, ...
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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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Every presheaf is a colimit of representables using point-wise computation of colimits

Let $C$ be a small category and let $F \in \text{Fun}(C^{op}, \text{Set})$ be a presheaf. I'm trying to show it is a colimit of representables using the fact that colimits in functor categories ...
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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 ...
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Simple examples of non-representable functors

I am looking for examples of non-representable functors, to see how the Yoneda lemma works in these cases. Here is one: let $\mathbf{C}$ be the category of finite-dimensional Euclidean spaces, with ...
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Proving the representability of a functor, which is covered by open subfunctors

I want to prove Theorem 8.9 from Algebraic Geometry I ( U.Görtz, T.Wedhorn), which reads as follows: Let $S$ be a scheme $F: Sch/S°\rightarrow Set $ a functor such that: F is a sheaf for the ...
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Universal elements of a functor and representability

A functor $F : \mathcal{C} \rightarrow \mathcal{Set}$ is said to representable if it is naturally isomorphic to $\mathcal{C}(A,–)$ for some object $A$ of $\mathcal{C}$. By the Yoneda lemma, we know ...
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The Yoneda lemma and a natural bijection

Let $S\colon\mathbf {Set}^{\cal A^{op}}\to \mathbf{ Set}$ be a functor. How does it follow from the Yoneda lemma that the following is a natural bijection: $\underline{\hom(A,-)\to SY \quad\quad\...
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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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1answer
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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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Why does $\mathbb{Z}$ represent the forgetful functor $U:\mathbf{Grp}\to\mathbf{Set}$

This is from Emily Riehl's Category theory in context The forgetful functor $U:\mathbf{Grp}\to\mathbf{Set}$ is represented by the group $\mathbb{Z}$ thanks to the natural isomorphism $\alpha:\mathbf{...
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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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Show that the functor that takes $R$ to the set of invertible elements of $R[X]/(X^2-a)$ is representable.

The following question is from the Fall 2016 UCLA algebra qualifying exam: Let $F$ be a field and $a\in F$. Show that the functor that takes $R$, commutative $F$-algebras to the invertible elements ...
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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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1answer
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Showing that $F$ is not representable [closed]

As I'm trying to find (counter)examples of representable functors, I tried looking up some instructive examples. One of the counterexamples I'm having trouble with, is the following: Show that the ...
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1answer
47 views

checking the functor $\texttt{Nil}_n$ is represented by $(\mathbb{Z}[x]/(x^n), \tau_R)$

This is the continuation of another question I did some days ago. Here. I have been working on it and I would like to know if my try to prove it is right or not. I would appreciate a lot any feedback ...
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162 views

Riehl's “Category Theory in Context” - Exercise 3.4.i

Let $\mathsf{I}$ be a small category, let $\mathsf{C}$ be a locally small category and let $F\colon\mathsf{I\to C}$ be a functor. Emily Riehl in her book "Category Theory in Context" defines a limit ...
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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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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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Can it be useful to think of functors as representing themselves ?

Here's a thought I had and I wonder if it can be of any use, for instance has it ever helped proving any result (however minor the result). Say you're in a situation where you have some objects in ...