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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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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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 ...
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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 $c\mapsto \coprod_x c$ (...
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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, ...
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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 ...
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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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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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