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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Category Theory - Every functor $F:\mathcal D\to\mathcal{Sets}$ is a colimit of representable functor [closed]
Image
on page 77 of MacLane chapter 3 section 7, I didn't understand how he showed that $\theta$ was a natural transformation, I'm trying to see it via definition but I didn't understand what he did (...
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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 ...
- 1,178
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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 $\sfC$ ...
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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 ...
2
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1
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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 ...
- 1,767
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42
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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 ...
- 6,710
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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 $...
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1
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66
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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 :
$$F \ \text{has left adjoint} \implies F \ \text{is representable} \implies F \ \text{ ...
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Vector Spaces as Functors
This is a follow-up question for the answer: https://math.stackexchange.com/a/4214108/1140967.
In that answer, we see that there are three ways to associate a $k$-vector space $V$ with a $k$-functor:
...
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What to call the natural transformation in the Yoneda lemma
The Yoneda lemma is an isomorphism between f a and (a <-) ~> f where the Yoneda embedding is ...
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Definition of the evaluation map as a universal element of a representation
This is exercise Exercise 2.3.iii from Riehl's "Category Theory in Context":
The set $B^A$ of functions from a set $A$ to a set $B$ represents the contravariant functor ${\rm Set}(-\times A,...
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How to prove a map is epic using generalized elements only?
I have a map $\require{AMScd}f\colon X \to Y$ in some category $\mathcal E$ which I would like to show is epic. However the only description I have of $X$, $Y$, and $f$ is through the Yoneda ...
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On representability of $\underline{Hom}_k(\mathbb A^1_k, \mathbb A^1_k)$
Let $F:k\text{-Sch} \rightarrow \text{Set}$ be the following functor:
$$F(X)=\text{Hom}_{X-\text{sch}}(\mathbb A^1_X, \mathbb A^1_X).$$
I would like to show that such functor is not representable.
I ...
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Representability of 2 dimensional $p$-adic Galois representations
Let $G = G_{\mathbb{Q}_p}$ be the absolute Galois group of $\mathbb{Q}_p$, and let $\overline{D}$ be a residual pseudorepresentation of $\mathbb{Z}_p[G]$ over $\mathbb{F}_p$.
Denote by $\text{Rep}^d_{\...
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Missing universal properties
Given a category $\mathcal{C}$, we can use copresheaves $H \in [\mathcal{C},Set]$ or presehaves $H \in [\mathcal{C}^{\mathrm{op}},Set]$ to state left or right universal properties. Existence of a ...
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Classification of principal G-bundles
Let $G$ be a group. We define the functor $$Bun_{G}^{*}:h(CW_{*}^{op})\rightarrow Sets_{*}$$ which takes a pointed CW complex and assigns to it the set of isomorphic classes of principal $G-$bundles ...
- 647
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The functor $\operatorname{Isom}_k(X,Y)$ is representable
Let $X,Y$ projective schemes over $k$. Consider the contravariant functor $$\operatorname{Isom}_k(X,Y): \left(\operatorname{Sch}/k\right) \to (\operatorname{Set})$$
$$
S\mapsto \operatorname{Isom}_S(S\...
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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 ...
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3
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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}_{\...
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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, ...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 $...
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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 ...
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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)}(\...
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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 $\...
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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 ...
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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 ...
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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 $c\mapsto \coprod_x c$ (...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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The proof of the Yoneda lemma's corollary
Consider the functors $H_A, X:\mathscr A^{op}\to\mathbf {Set}$.
There's the following corollary of the Yoneda lemma:
There's a bijection $$\{ \text{natural isomorphisms } \alpha: H_A\to X\}\...
- 11.7k
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Why is the functor that associates to a scheme $S$ the set of $S$-isomorphism classes of elliptic curves over $S$ not representable?
A few days ago I heard an online presentation about elliptic curves and the presenter claimed that the functor which assigns to a scheme $S$ the isomorphism classes of elliptic curves over $S$ is not ...
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The functor $U\circ C:\mathbf{Top}^{op}\to \mathbf{Set}$ is representable
Consider the functor $C:\mathbf{Top}^{op}\to \mathbf{Ring}$ that sends an object $X$ to a continuous function $X\to \mathbb R$. Consider also the composition $U\circ C$ where $U:\mathbf{Ring}\to\...
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Calculating the geometric realization of a non-representable functor
Background
Let $\mathcal{F} : \textbf{CRing} \to \textbf{Set}$ be a functor and denote by $\textbf{P}_\mathcal{F}$ the category of points of $\mathcal{F}$ whose objects are pairs $(R , \rho)$ where $R$...
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If $M$ is an $R$-module, when is $A\mapsto Aut(M\otimes_{R}A)$ a representable functor?
Let $R$ be a ring (unitary, commutative, associative) and $M$ an $R$-module. Is the functor
\begin{eqnarray*}
F & : & {R}\mathbf{-Alg} & \longrightarrow & \mathbf{Grp}\\
&...
- 1,492
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Non-isomorphic locally ringed spaces which represent isomorphic functors $\mathsf{CommRing} \to \mathsf{Set}$.
It's well known that the restricted Yoneda functor $よ : \mathsf{Schemes} \to \operatorname{Fun}(\mathsf{CommRing},\mathsf{Set})$ is an embedding, so that (in particular) if $X$ and $Y$ are schemes ...
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72
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representable functors and finitely generated algebras
I'm bothered by a confusing statment on representable functors.
Let $k\rm Alg$ be the category of finitely generated $k$-algebra ($k$ is a field) and $F:k\rm Alg\to Set$ a functor. Let $B={\rm Nat}(F,...
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