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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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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Universal 2-property of $~Twisted(B)$

Given a category $B$ we can define $B^2$ abstractly by representation $$Cat(X,B^2) \simeq_{Cat} Cat(X,B)^2 \quad \quad \text{naturally in $X\in Cat$}$$ This extends to a general two-category $K$ $$K(X,...
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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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When does the representability of $ F \circ I $ imply that of $ F $?

Let $ I \colon \mathcal{C} \to \mathcal{D} $ and $ F \colon \mathcal{D} \to \mathbf{Set} $ be functors. For an object $ C \in \mathcal{C} ​$, consider the following two statements: $ F \circ I \colon ...
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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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3 votes
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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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1 answer
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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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3 answers
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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\}\...
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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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2 answers
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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}\\ &...
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13 votes
1 answer
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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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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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7 votes
1 answer
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Yoneda's lemma: group morphisms give Hopf-algebra morphisms

Let $k$ be a commutative ring. Let $\text{Alg}$ be the category of commutative $k$-algebras and $\text{CHopf}$ the category of commutative Hopf-algebras. Let us also write $[\text{Alg}, \text{Grp}]$ ...
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1 answer
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Does this functor commute with inverse limits?

Let $F$ a contravariant functor from the category $A$ of pointed connected CW-complexes up to homotopy to the category $B$ of pointed sets, with $F$ sending coproducts of $A$ to products of $B$. Let $...
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Definition of representable functor with values in $Ab$

I was reading these notes and I came up with a question about Definition 8.2.1 at page 283: what does it mean for a functor $F : C \to Ab$ to be representable? Is this definition related to that of a ...
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2 votes
1 answer
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Equivalent versions of the Mayer-Vietoris axiom in Brown theorem

In the hypotheses of Brown representability theorem there is a contravariant functor F from pointed connected CW complexes to pointed sets, which must respect two axioms, the second of which is the so-...
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1 vote
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Strong deformation retraction between CW-complexes, Brown theorem proof

I am currently reading Lemma 9.11 on Switzer, in the chapter dedicated to Brown representability theorem. However I am stuck on a few points of the proof. Let $F$ a contravariant functor from pointed ...
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7 votes
1 answer
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Relationship between two definitions of pro-representable functors

Edit: I'm pretty sure that my conjecture $$ \operatorname{Hom}(\varprojlim_i R/\mathfrak{m}^i, A) = \operatorname{colim}_i \operatorname{Hom}(R/\mathfrak{m}^i, A), $$ is true. To prove it, just use ...
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1 vote
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Presheaves are the Free Cocompletion - Proving that the functor preserves colimits

I am trying to understand a proof that, for any small category $\mathcal{C}$, the category $\widehat{C} = [\mathcal{C}^\mathrm{op}, \textbf{Set}]$ is the free cocompletion of $\mathcal{C}$. In ...
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4 votes
1 answer
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How to reduce to affine case to determine whether a given functor is representable

[Definition] These are the contents of Gortz and Wedhorn , Algebraic Geometry. $\widehat{(Sch)}$ is the category of functors $(Sch)^{opp} \rightarrow (Sets)$ For scheme $X$, define the functor $h_X :...
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2 votes
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The automorphism scheme of a fibered surface

Let $f:S \to B$ be a genus $g$ fibration ($B$ is a smooth projective curve, $S$ a smooth projective surface and the general fiber $F_b$ of $f$ is of genus $g \ge 2$). I would like to ask some ...
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0 votes
1 answer
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Colimits and $\text{Hom}(-,G)$

$\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\colim}{colim}$ Let $(A_i)_i$ be an inverse system of groups. If $G$ is a group, then we have a projective system of set $(\Hom (A_i,G))_i$. I ...
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1 answer
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What's the meaning of defining a functor in a natural way?

I have a question: Given a group $\mathbf{G}$, there is homomorphism $\rho$ $\colon$ $\mathbf{G}$ $\to$ $\mathbf{GL(V)}$. BTW, $\rho$ is a representation of a group $\mathbf{G}$ on a vector space. Now ...
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0 votes
1 answer
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Composition of representable functors

Let $\mathcal{C}$ be some small category and consider the representable functor $$\text{Hom}(,N):\mathcal{C}\rightarrow \text{Ab}.$$ The functor $$-\otimes_{\mathbb{Z}} M:\text{Ab}\rightarrow \text{Ab}...
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1 vote
1 answer
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Are these two functors same (isomorphic)?

Let $\mathscr{C}$ be the category of schemes (that is the category of schemes over $\mathrm{Spec}(\mathbb{Z})$), $X,Y$ be two schemes. Then we can have two functors: $\mathscr{C}^{op}\to \{\mathrm{...
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2 votes
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The connectedness property of representables

Exercise 6.2.21: Prove that representables have the following connectedness property: given a locally small category $\mathscr A$ and $A\in\mathscr A$, if $X,Y\in[\mathscr A^{op},\mathbf {Set}]$ with ...
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Expressing a presheaf as a colimit of representables

I don't understand how the highlighted isomorphism follows. And why is every object in $\mathbf {Set}\times\mathbf{Set}$ is a sum of copies of $(1,\emptyset)$ and $(\emptyset,1)$? Next, right after ...
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1 vote
2 answers
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Leinster's proof of "Representables preserve limits"

Leinster (p.148) gives the following proof of the fact that representatives preserve limits: I understand the argument, but why does this prove the claim? To prove that limits are preserved, one has ...
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1 vote
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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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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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1 vote
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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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4 votes
0 answers
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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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