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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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Yoneda lemma in a functor semicategory

I've read that the Yoneda lemma does not hold in general for semicategories (i.e., 'categories' possibly lacking identity morphisms)[1]. However, I'm wondering about a related situation, where there ...
Yaque's user avatar
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Proving that $f: T\to S$ factors through $V$ if and only if $f^{*}(v) = 0$ ( for some representable functor )

First note next Proposition ( Gortz's Algebraic Geometry book ): Proposition 8.4. Let $S$ be a scheme and let $v: \mathcal{E} \to \mathcal{F}$ be a homomorphism of quasi-coherent $\mathcal{O}_S$-...
Plantation's user avatar
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3 votes
1 answer
69 views

Orbit functor is not co-representable

Let ${1}\neq H\le G$ be groups. Denote by $G\textit{-}\mathsf{Set}$ the category of sets with a $G$ action, with $G$-equivariant maps as morphisms. Let $(-)/H: G\textit{-}\mathsf{Set}\to \mathsf{Set}$ ...
Robert's user avatar
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1 answer
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Tiny objects in a power set

I'm constructing a toy example in order to get into Cauchy completeness for categories. Suppose to have a set $X$ (=a discrete preorder) and compute its powerset $\mathcal{P}(X)$, which is its $\{0<...
whatisandwhatshouldneverbe's user avatar
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1 answer
108 views

Prove that any functor $F : \mathcal C \to \text{Sets}$ where $\mathcal C$ is small is a colimit of representable functors

Prove that for any small category $\mathcal C$ and any functor $F:\mathcal C^\text{op}\to\textbf{Set}$, $F$ can be written as a colimit of representable functors $h_x=\text{Hom}_{\mathcal C}(-,x)$. I ...
love and light's user avatar
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1 answer
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Condition that limit of a diagram $F$ exists if $\alpha: Mor(-,L) \to \lim Mor(X,F(i))$

Suppose $F : I \to \mathcal C$ is a diagram for a small category $I$ and a category $\mathcal C$. Suppose also that there is an object $L$ of $\mathcal C$ such that for all $X \in Obj(\mathcal C)$ ...
love and light's user avatar
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3 answers
150 views

Prove that the functor $P : \text{Set}^\text{op} → \text{Set}$ , $F(f)=f^{-1}:P(Y)\rightarrow P(X)$ for $f: X \rightarrow Y$ is representable

Let $P(A)$ denote the power set of a set $A$ For a map $f : X \mapsto Y$ of sets, I can define a map $P( f ): P(Y) → P(X)$ s.t that I obtain a functor $P : \text{Set}^\text{op} \to \text{Set}$ in ...
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How do I show that the forgetful functor $F : \text {Grp} \rightarrow \text {Set} $ is co-representable?

Let $F : \text {Grp} \rightarrow \text {Set} $ be the forgetful functor. I am trying to show that $F$ is co-representable My definition of co-representability reads: A functor $F: C \rightarrow \text{...
darkside's user avatar
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Representable morphisms of algebraic spaces

Let $X,Y$ be algebraic spaces, and let $X\longrightarrow Y$ be a representable morphism. Let $S$ be a scheme and let there be a morphism $S\longrightarrow Y$. The fibre product, $X\times_Y S$, is said ...
kindasorta's user avatar
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Representable presheaves on the slice category

$\def\sfC{\mathsf{C}} \def\op{\mathrm{op}} \def\set{\mathsf{Set}} \def\psh{\operatorname{PSh}} \def\ob{\operatorname{Ob}} \def\hom{\operatorname{Hom}}$Let $\sfC$ be a category. A presheaf over $\sfC$ ...
Elías Guisado Villalgordo's user avatar
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Existence of a right adjoint functor of the inverse image via a morphism of schemes between the categories of quasi-coherent modules

Let $f:X\to Y$ a morphism of schemes. It induces a covariant functor: $$f^*:Qcoh(Y)\to Qcoh(X)$$ Which happens to be the inverse image. Now, fixing any quasi-coherent $O_X$-module N we can define ...
Jorge A. Mateos's user avatar
2 votes
1 answer
111 views

Corepresentable functor definition.

I have been searching for the definition of a Corepresentable functor here https://ncatlab.org/nlab/show/small+object but still I did not get what exactly is its definition. I also love the definition ...
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Equivalence relation functor representable?

Let $E : \mathbf{Set} \to \mathbf{Set}$ be the contravariant functor taking a set $X$ to the set $E(X)$ of distinct equivalence relations on $X$. It takes (I assume) a function $f:X \to Y$ to a ...
Paul Orland's user avatar
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Understanding a proof of representability and exactness of tensor product functor

$\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\Mod}{Mod}$ $\DeclareMathOperator{\End}{End}$ $\DeclareMathOperator{\coker}{coker}$I'm going through the proof of the following theorem : Let $...
t_kln's user avatar
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1 vote
1 answer
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Example of non representable limits-preserving functor

Let $\mathcal{C}$ be a locally small category, and $F : \mathcal{C} \rightarrow \mathcal{Set}$ functor. Then : $$F \ \text{has left adjoint} \implies F \ \text{is representable} \implies F \ \text{ ...
simo210's user avatar
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Vector Spaces as Functors

This is a follow-up question for the answer: https://math.stackexchange.com/a/4214108/1140967. In that answer, we see that there are three ways to associate a $k$-vector space $V$ with a $k$-functor: ...
lzww's user avatar
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What to call the natural transformation in the Yoneda lemma

The Yoneda lemma is an isomorphism between f a and (a <-) ~> f where the Yoneda embedding is ...
Baldur's user avatar
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1 answer
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Definition of the evaluation map as a universal element of a representation

This is exercise Exercise 2.3.iii from Riehl's "Category Theory in Context": The set $B^A$ of functions from a set $A$ to a set $B$ represents the contravariant functor ${\rm Set}(-\times A,...
Alooffi's user avatar
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1 answer
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How to prove a map is epic using generalized elements only?

I have a map $\require{AMScd}f\colon X \to Y$ in some category $\mathcal E$ which I would like to show is epic. However the only description I have of $X$, $Y$, and $f$ is through the Yoneda ...
Olius's user avatar
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11 votes
2 answers
232 views

On representability of $\underline{Hom}_k(\mathbb A^1_k, \mathbb A^1_k)$

Let $F:k\text{-Sch} \rightarrow \text{Set}$ be the following functor: $$F(X)=\text{Hom}_{X-\text{sch}}(\mathbb A^1_X, \mathbb A^1_X).$$ I would like to show that such functor is not representable. I ...
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Representability of 2 dimensional $p$-adic Galois representations

Let $G = G_{\mathbb{Q}_p}$ be the absolute Galois group of $\mathbb{Q}_p$, and let $\overline{D}$ be a residual pseudorepresentation of $\mathbb{Z}_p[G]$ over $\mathbb{F}_p$. Denote by $\text{Rep}^d_{\...
kindasorta's user avatar
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65 views

Missing universal properties

Given a category $\mathcal{C}$, we can use copresheaves $H \in [\mathcal{C},Set]$ or presehaves $H \in [\mathcal{C}^{\mathrm{op}},Set]$ to state left or right universal properties. Existence of a ...
antoine_vm's user avatar
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109 views

Classification of principal G-bundles

Let $G$ be a group. We define the functor $$Bun_{G}^{*}:h(CW_{*}^{op})\rightarrow Sets_{*}$$ which takes a pointed CW complex and assigns to it the set of isomorphic classes of principal $G-$bundles ...
T. Wildwolf's user avatar
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1 answer
197 views

The functor $\operatorname{Isom}_k(X,Y)$ is representable

Let $X,Y$ projective schemes over $k$. Consider the contravariant functor $$\operatorname{Isom}_k(X,Y): \left(\operatorname{Sch}/k\right) \to (\operatorname{Set})$$ $$ S\mapsto \operatorname{Isom}_S(S\...
Menezio's user avatar
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The functor $\mathrm{Sch/R}\to\mathrm{Set}$ sending a scheme $X$ to its set of open subsets is not representable?

Here $R$ is a commutative ring with unity and $\mathrm{Sch}/R$ is the category of schemes over $\operatorname{Spec}R$. I know that the contravariant functor $\mathcal O$ of open subsets of a scheme ...
Menezio's user avatar
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5 votes
3 answers
727 views

Definition of Limits in Category Theory

I was reading Kashiwara, Schapira's book Categories and Sheaves, in that limit of a projective system, $$P:\mathcal{I}^{\text{op}}\to \textbf{Sets}$$ Is defined as follows, $$\lim P = \text{Hom}_{\...
MrPajeet's user avatar
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1 answer
165 views

Characterizing Representable / Hom Functors via Universal Property

I have been trying to think of a way to characterize the hom functors via universal property. I could not find any such thing elsewhere online. So I came up with a property, inspired by Yoneda lemma, ...
IsAdisplayName's user avatar
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What are the regular epimorphisms in $\mathrm{hTop}^{\mathrm{connectedCW}}_*$?

I was wondering how to determine what are the regular epimorphisms (i.e. those which are coequalizers) in $\mathrm{hTop}^{\mathrm{connectedCW}}_*$, the category of pointed connected CW complexes ...
I.A.S. Tambe's user avatar
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1 answer
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Unitary dualization functor continuous?

Let $G$ be a topological group and denote its unitary dual by $\hat{G}:=\{\pi:G\to\text{U}(\mathcal{H})\text{ irreducible unitary representation}\}/_\cong$. If $H$ is another topological group and $\...
kringelton4000's user avatar
3 votes
1 answer
106 views

When representables are adjoints

Let $\mathbf C$ be a complete category and let $U:\mathbf C\to\mathbf{Set}$ be a representable functor. Show that $U$ preserves limits. In general representables preserve limits, but the hypothesis ...
Ezio Greggio's user avatar
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4 votes
0 answers
128 views

Can (should?) the Yoneda embedding be formulated in terms of $0$-categories?

Assuming the Grothendieck axiom of universes (see also here or here), let $U_0$ denote the universe of "ZFC sets", i.e. of sets that can be constructed in ZFC alone without assuming axioms ...
hasManyStupidQuestions's user avatar
2 votes
2 answers
80 views

How does the duality functor with respect to $K$ behave on morphisms?

In A duality formalism in the spirit of Grothendieck and Verdier Boyarchenko and Drinfeld give the following definition of the terms dualizing object and duality functor: An object $K$ in a monoidal ...
Max Demirdilek's user avatar
4 votes
1 answer
376 views

When does a representable functor have a right adjoint?

In Wikipedia I saw the result that, when category $\mathcal{C}$ has all small copowers, a functor $\mathcal{C}\overset{K}{\rightarrow}\text{Set}$ has a left adjoint if and only if it is representable. ...
Richard Southwell's user avatar
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0 answers
26 views

Product of functors (A,_) being a generator and the arbitrary sum of left-exact functors

In Peter Freyd's book Abelian Categories he states a theorem saying that $\mathscr{L}(\mathscr{A})$ is complete and has an injective generator, with $\mathscr{L}(\mathscr{A})$ being the category of ...
Francisco's user avatar
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2 votes
0 answers
201 views

Why aren't sheaf categories locally small?

Given a scheme $S$ and reasonable topology, such as Zariski, étale, or fppf, is the category of sheaves ${Sh}(({Sch}/S)_{top})$ locally small? Or, if you like to assume that a category must be locally ...
jackson's user avatar
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2 votes
1 answer
205 views

Limits/Colimits as representing objects and size of functor category

Let $J,C$ be categories and $T\in C$, we define the functor $\Delta_J(T):J \to C, j \to T $ to be the constant $T$-valued $J$-diagram. Let $F:J \to C$ be a functor, my professor defined the limit of $...
Ben's user avatar
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3 votes
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Notation in Katz-Mazur Arithmetic Moduli of Elliptic Curves

A moduli problem $\mathcal{P}$ is a contravariant functor $\mathbf{Ell}\to\mathbf{Set}$. The objects of $\mathbf{Ell}$ are arrows $E\to S$ from an elliptic curve $E$ to a varying base scheme $S$. The ...
klein4's user avatar
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143 views

Representability of relative spectrum of a quasi-coherent sheaf of algebras

Let $X$ be a scheme and $\mathscr{R}$ a quasi-coherent $\mathscr{O}_{X}$-algebra. Let $$ F: (Sch/X)^{opp} \rightarrow (Sets), \hspace{1cm} (f:T \rightarrow X) \mapsto \text{Hom}_{(\mathscr{O}_X-Alg)}(\...
user267839's user avatar
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0 answers
73 views

Showing that a morphism is unique

Suppose we have two representables $H_A, H_{A'}: \mathscr A\to \mathbf {Set}$ and a natural transformation $H_A\to H_{A'}$ with components $\alpha_B:H_A(B)\to H_{A'}(B)$. It can be shown that each $\...
user557's user avatar
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1 vote
1 answer
125 views

Criterium for the representability of a functor

Let $F: (Sch/S)^{op} \to Sets$ be a functor that is both a sheaf in the Zariski topology and has an open covering $(f_i: F_i \to F)_{i \in I}$, where each of the $F_i$ is representable. In theorem 8.9 ...
user480840's user avatar
1 vote
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55 views

Limit cones and representations (Leinster's Proposition 6.1.1)

I can see two ways to read this proposition (it's probably the language barrier problem -- I'm not an English native speaker), and I suppose the first way below is what Leinster intended to convey. Is ...
user557's user avatar
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112 views

Observation about powers and copowers (in category theory)

Suppose that the category $\mathbf C$ has powers and copowers of every object. Fixed a set $x$ one can define the functor $F:\mathbf C\to \mathbf C$, whose object function is $c\mapsto \coprod_x c$ (...
Dr. Scotti's user avatar
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1 vote
2 answers
165 views

Dual version of "representables preserve limits"

The fact that $\mathscr A(A,-):\mathscr A\to \textbf{Set}$ preserves $D$-indexed limits translates to $$\lim\mathscr A(A,D(-))\simeq \mathscr A(A,\lim D)$$ I'm trying to prove that the dualized ...
user557's user avatar
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3 votes
1 answer
65 views

Universal properties for groups arising from representations

Consider the forgetful functor $U:\textbf{Grp}\to\textbf{Set}$ and its left adjoint $F$. The functor $X=\textbf{Set}(A,U(-))$ is represented by $F(A)$, i.e., $X\simeq H^{F(A)}$. By the Yoneda lemma, ...
user557's user avatar
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1 vote
1 answer
244 views

Relation between representable functors and adjunction

In class we proved that given two adjoint functors $F\dashv G:\mathbb C\to \bf{Set}$, where the category $\mathbb C$ is arbitrary, $G$ is (naturally) isomorphic to $H^{F*}$, where $*$ is the ...
Dr. Scotti's user avatar
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2 votes
0 answers
56 views

Why is the functor $S\mapsto$ isomorphism classes of curves of genus $g$ over $S$ not representable?

Whenever $R_{1}\hookrightarrow R_{2}$ is an injection of rings, \begin{equation*} \operatorname{Hom}(R,R_{1})\hookrightarrow\operatorname{Hom}(R,R_{2}) \end{equation*} is an injection for any ring ...
The Thin Whistler's user avatar
2 votes
1 answer
230 views

Zariski site is subcanonical?

I want to show that the zariski site is subcanonical as an exersice of the book "Sheaves in geometry and logic" and I need help with it... To be honest I didnot really understand the ...
user850424's user avatar
3 votes
4 answers
520 views

Is a subfunctor of a representable functor also representable?

I'm trying to learn about representable functors but I'm very new to this; even to categories... Suppose $G:C^{op}\rightarrow Set$ is a representable functor - as I understand this means that there ...
amateur's user avatar
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5 votes
1 answer
337 views

Deducing Cayley's theorem from the Yoneda lemma using Wikipedia's recipe [duplicate]

I'm following Wikipedia in trying to prove that Cayley's theorem emerges as a particular case of the Yoneda lemma. In case that article gets edited, here's the screenshot: A couple of aspects in the ...
user557's user avatar
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1 vote
0 answers
42 views

About Leinster's discussion of Corollary 4.3.10

Corollary 4.3.10 in Leinster's text says that $H_A\cong H_{A'}\iff A\cong A'\iff H^A\cong H^{A'}$. He writes: "the corollary tells us that two objects are the same if and only if they look the ...
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