# 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 vote
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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 $\sfC$ ...
56 views

### 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 ...
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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 ...
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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 ...
57 views

1 vote
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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 ...
1 vote
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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 $c\mapsto \coprod_x c$ (...
1 vote
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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, ...
1 vote
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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 ...
1 vote
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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\}\...
1 vote
### 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 ...