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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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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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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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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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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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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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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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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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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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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Kan extensions and restrictions

Let $\mathcal{C},\mathcal{D}$ be two small categories and $\iota:\mathcal{C}\rightarrow \mathcal{D}$ a faithful functor. Then take a representable functor $$\text{Hom}(-,D):\mathcal{D}\rightarrow \...
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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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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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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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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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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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Hom functor preserves limits [duplicate]

I was looking to prove that $Hom_C(Y,limX) \simeq lim Hom_C(Y,X)$. I saw something in https://ncatlab.org/nlab/show/hom-functor+preserves+limits, but I didn't really understood the proof. Can someone ...
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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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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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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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Showing that $H_A\simeq H_{A'}$ implies $A\simeq A'$

I'm trying to solve this exercise: So we are given that for all objects $B$, there is a natural isomorphism $$H_A(B)\simeq H_{A'}(B)$$ Let's write this isomorphism as $f\mapsto \bar f$. The ...
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Describing the universal property of the evaluation map

Emily Riehl's "Category Theory in Context", ${\rm Exercise}~2.3.{\rm iii}.$ The set $B^A$ of functions from a set $A$ to a set $B$ represents the contravariant functor ${\rm Set}(-\times A,B):{\rm ...
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For which categories is its Yoneda embedding essentially surjective?

For a locally small category $\mathcal{C}$, you can embed $\mathcal{C}$ in the functor category $\mathrm{Set}^\mathcal{C}$ via the functor $X \mapsto \mathrm{Hom}_\mathcal{C}(X,{-})$. This embedding ...
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312 views

Representable functor is left-adjoint

$\newcommand{\catname}[1]{{\textbf{#1}}} \newcommand{\Set}{\catname{Set}}$ Let $\mathcal{C}$ be a locally small category. I am trying to show that if all coproducts exist in $\mathcal{C}$ and if $F:\...
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Representability of a functor in the category of schemes

I have read in some places that a functor of points of a scheme is representable if its defined by locally closed or open conditions. I would like to ask for some references about this fact. I don´t ...
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Alternate definition of Group Object, and how to 'play' with it?

I've come across an alternate definition of a group object, shown below: Suppose $F:C^{op} \rightarrow Grp$ is a functor such that its composition with the forgetful functor $?:Grp \rightarrow ...
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Using Yoneda's lemma to “guess” the definition of exponential object in $SET$

Like what the title said, I want to use Yoneda's lemma to "guess" the definition of exponential object in $SET$. So basically I want to say that given any two sets $A, B$ in $SET$, the exponential ...
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Is there a canonical way to go back from $\hat{\mathbb{C}}$ (presheaves) to $\mathbb{C}$

Is there a "canonical" way to go back from $\hat{\mathbb{C}}$ (Category of presheaves) to $\mathbb{C}$. Here we define $\hat{\mathbb{C}}$ to be the category of functors from $\mathbb{C}^{op} \...
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Extension of a functor by colimits: Cisinski - Higher Categories and Homotopical Algebra - Remark 1.1.11

First, I state premilinary results. For a presheaf $X\colon A^{op}\to\mathsf{Set}$, it's category of elements, denoted by $\int X$, has pairs $(a,s)$ where $a \in A$ and $s \in X(a)$ as objects and $...
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Universal properties in measure theory and probabilities

Obviously certains construction as for instance the push-forward (direct image) ${\cdot}_{\star}$ can be interpreted in the categories of measurable spaces, measured spaces or probability spaces as ...
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Two definitions of representable functor

Leinster Basic Category theory gives two equivalent definitions of representable functors: $X: \mathcal{A} \to Set$ is representable if $X \cong H^A = Hom(A,-)$ $X: \mathcal{A}^{op} \to Set$ is ...
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Global sections functor is fully faithful if and only if it is logical

Let $\mathcal{E}$ be a non-degenerate topos. Question: Is it true that the "global sections" functor $\operatorname{Hom}_{\mathcal{E}}(1,-)$ is fully faithful if and only if it is logical? A non-...
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Equivalent functors carrying representability

Emily Riehl's "Category Theory in Context, ${\rm Exercise}~2.1.{\rm iii}.$ Suppose $F:{\rm C}\to{\rm Set}$ is equivalent to $G:{\rm D}\to{\rm Set}$ in the sense that there is an equivalence of ...
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Monomorphisms are preserved by Representable Functors

Emily Riehl's "Category Theory in Context", ${\rm Exercise}~2.1.{\rm ii}.$ Prove that if $F:{\rm C}\to{\rm Set}$ is representable, then $F$ preserves monomorphisms, i.e., sends every monomorphim in ...
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Functor $G$ has right adjoint iff $Y \mapsto Hom(X, G(Y))$ is corepresentable

My professor in his notes claims that a functor $G$ has a right adjoint iff the functor $Y \mapsto Hom(X, G(Y)$ is corepresentable, i.e for each $X$ there is an object $F(X)$ and a nutural by $X$ ...
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Scheme $\text{Spec}(S^*\text{Hom}(V,W)^{\vee})$ from “The Geometry of Moduli Spaces of Sheaves”

I have a couple of questions about the notations & their meaning used in "The Geometry of Moduli Spaces of Sheaves" by Huybrechts & Lehn, in Example 2.2.2 (page 38): $V$ is assumed to be a be ...
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$\mathbb{A}^2_{\mathbb{Z}} \backslash \{(0,0)\}$ represents a functor & a beautiful glueing story

Let $\mathbb{A}^2_{\mathbb{Z}}$ the affine plane and it is well known that $\mathbb{A}^2_{\mathbb{Z}}$ represents the contravariant functor $$\mathbb{A}^2: CRing^{op} \to Set, R \mapsto \{(r,s) \...
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Representability of functors in categories other than Set category

Let $\mathcal C,\mathcal D$ be locally small categories and also assume that $\mathcal D$ is small and that every morphism in $\mathcal D$ is a function between sets. Assume that for every $A,B \in \...
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Why is the forgetful functor representable?

I'm reading Adowey's Category Theory, and I'm struggling with the last exercise of the second chapter, which is to show that the forgetful functor for monoids, $U : \mathbf{Mon} \to \mathbf{Sets}$, is ...
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Reflections in locally presentable categories-unclear step in the proof

Here in the paper by Rosicky Adamek, Reflections in locally presentable categories on the page 90 in the proof theorem on the page 89, I do not follow the 3rd line there: ... and hence is reflective ...
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On the functorial point of view in algebraic geometry.

Here's a question I've been thinking about lately. I hope it's not too vague - I apologize in advance if this should be the case. Suppose you want to do algebraic geometry using the $\textit{...
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Showing that a functor $Cat\to Set$ is representable

The problem is to show that the functor from the category of small categories to the category of sets that sends a category to its set of morphisms is representable. The major problem is to find a ...
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Representable functor example

Consider a monoidal category $\mathcal{C}$ which is enriched over $\mathrm{vect_k}$ (=finite dimensional vector spaces over a field $k$). $\mathcal{C}$ is abelian and semi simple. If it helps one can ...
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On the proof that $1_{\textbf{Set}}$ is representable

Example 4.1.4 says: Consider $H^1:\textbf{Set}\to\textbf{Set}$, there $1$ is the one-element set. Since a map from $1$ to a set $B$ amounts to an element of $B$, we have $$H^1(B)\cong B$$ for each ...
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Why is the Grassmannian functor representable by a scheme?

I am having some trouble following a proof that the Grassmannian functor is representable by a scheme. I am following the proof in EGA 9.7.4. It is only a small step that I am stuck on. For reference, ...
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Why are coproduct objects corepresentations of cartesian products, rather than representations of disjoint unions?

One way we can define the product of two objects $A$ and $B$ in some category $\mathcal C$ is as a representation of the contravariant functor $(\to A) \times (\to B)$ in $[\mathcal C^{\mathrm{op}}, ...
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Representability criterion for schemes

I am trying to understand. Lemma 25.15.4 Let $F$ be a contravariant functor the category of schemes with values in the category of sets. Suppose that $F$ satisfies the sheaf property. ...
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Every presheaf is a colimit of representables using point-wise computation of colimits

Let $C$ be a small category and let $F \in \text{Fun}(C^{op}, \text{Set})$ be a presheaf. I'm trying to show it is a colimit of representables using the fact that colimits in functor categories ...
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Quot-like scheme for torsion sheaves

I am wondering if, as the Quot schemes parametrizes flat (quotients of) sheaves over schemes, there is anything similar for torsion sheaves. In first approximation, if $I$ is a sheaf of ideals over a ...