Questions tagged [yoneda-lemma]

For questions related to Yoneda lemma in category theory, which *basically* says (among many other things it says) that every locally small category can be embedded nicely into it's functor category into the category of Sets. Use this tag along with (abstract-algebra) or (category-theory).

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1answer
48 views

Why is el(-)=$\int(-)$ a functor from functors to a slice cateory?

I am reading up a little on category theory. I am trying to solve Riehl's problem (Category Theory in Context) 2.4.vii on page 72. I believe that it should be quite easy, but it appears to me that ...
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1answer
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Possible adjoint to Yoneda embedding and Repeated Yoneda embedding?

While thinking about Yoneda embedding, I came up with following two questions (I should apologies, if those are too vague): Does the Yoneda embedding $y :\mathcal{C}\to\mathbf{Set}^{\mathcal{C}^{\...
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1answer
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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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Two questions on Yoneda's lemma

I used to be okay with the Yoneda lemma because in the material I studied it was only used in order to speed up a bit the discussion of universal objects that came up (tensor products or spectra for ...
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1answer
38 views

How does the $2$-Yoneda embedding for the category of categories act on 2-morphisms?

Let $\text{Cat}$ be the category of small categories. I am interested in the Yoneda embedding $$ Y : \text{Cat}^{op} \rightarrow [\text{Cat}, \text{Cat}]$$ $Y$ is a $2$-functor- it can be applied to ...
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Exponents in a slice category of presheaves

Let $\mathcal{C}$ be a category and denote $\hat{\mathcal{C}}$ for the category of presheaves on $\mathcal{C}$. For $k: K \to F$, denote $k^*$ for the pullback functor $\hat{\mathcal{C}}/F \to \hat{\...
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2answers
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Part of proof of Yoneda's Lemma from Vakil

I am trying to understand the second half of the proof of Yoneda's Lemma, which is given as a problem in Vakil's notes. So suppose we have two objects $A$ and $A'$ in a Category $D$, and morphisms $...
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1answer
51 views

Yoneda embedding

I have the next doubt about this problem: "Let $\mathfrak{U}$ be a small category and let $Y:\mathfrak{U}\rightarrow [\mathfrak{U}^{opp},\mathbf{Sets}]$ be the yoneda embedding $Y(A)=\mathfrak{U}(-,...
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1answer
67 views

Why is injectivity hard to prove in Yoneda lemma?

I'm following Emily Riehl's book Category theory in context. In Theorem 2.2.4 (Yoneda lemma) it is stated: For any functor $F:\mathscr{C}\to\textbf{Set}$ whose domain $\mathscr{C}$ is a locally small ...
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1answer
60 views

Show that $\hat{\mathbb{C}}$, category of presheaves for $\mathbb{C}$, is cartesian closed

So I want to show that $\hat{\mathbb{C}}$, which is the category of presheaves for $\mathbb{C}$ ([$\mathbb{C}^{op}\rightarrow Set$]), is cartesian closed. Using the yoneda lemma and adjunction, I ...
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Can every category be regarded as the functor category of another?

Given a category $\mathcal{C}$ is there necessarily another category $\mathcal{D}$ such that $\mathcal{C}$ is equivalent to the functor category $\mathrm{Set}^\mathcal{D}$? If so, is there a natural ...
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68 views

Is iteratively applying the Yoneda embedding interesting?

Since the Yoneda lemma is so important, I'm curious what happens if you iteratively take the Yoneda embedding: Let $\mathcal{C}_0 = \mathcal{C}$ be a locally small category, and define $\mathcal{C}_{i+...
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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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1answer
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Using Yoneda's Lemma to show that limits are unique

I recall my professor mentioning that Yoneda's Lemma can be used to show that limits of a functor are unique up to isomorphism. Here is my attempt: Let $F:J\to\mathcal{C}$ be a functor and let $X$ ...
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Left and right Kan extensions

Let $F:\mathcal{C}\to\mathcal{D}$ be a functor between small categories. We define the functor \begin{align*} f:\hat{\mathcal{D}}&\longrightarrow\hat{\mathcal{C}} \\ G&\longmapsto G\circ F^{\...
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Is the Yoneda completion of the rationals the extended real line?

A thing that I have often heard is that, when viewed as an enriched category, the Yoneda embedding of the poset of rationals $\mathbb{Q}$ into its category of presheaves is just the dense embedding of ...
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1answer
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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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1answer
33 views

By Yoneda lemma, kernel of morphism in preadditive category is unique

I'm reading https://stacks.math.columbia.edu/tag/09SE. It says How do we use Yoneda lemma to prove the uniqueness of the kernel of a morphism? I tried this: Please help, thanks.
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Cisinski, Higher Categories and Homotopical Algebra, Theorem 1.1.10

Let $X\colon \mathsf{C^{op}}\to \mathsf{Set}$ be a presheaf. It's category of elements, denoted by $\int X$, has pairs $(a,s)$ with $s \in X(a)$ as objects and $f \in \mathrm{Hom}_{\mathsf{D}}(a,b)$ ...
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1answer
79 views

How does this statement reduces to the Yoneda lemma?

Context: I am reading the following page in the nLab, which is about simplicial presheaves, i.e Functors $\mathscr{C}^{\mathrm{op}} \to [\mathbf{\Delta}^{\mathrm{op}}, \mathbf{Set}]$. Equivalently, ...
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Proving exponential arithmetic rules using the Yoneda lemma

Working in a Cartesian closed category we have have an exponential object $X^Y$ for object $X$ and $Y$. There are isomorphisms $1^X \cong 1$, $X^Y \times X^Z \cong X^{Y \times Z}$, $X^1 \cong X$ and a ...
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Why is the Yoneda embedding continuous?

I realize this is a duplicate of this question, but I do not understand the answer. I am talking about the contravariant Yoneda embedding and this is how far I got: Let $D: I \to \mathcal{C}$ be a ...
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1answer
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Not getting example to fit Yoneda

So apparently I'm doing something wrong, but can't figure out what. I've read a proof of the Yoneda lemma and understand it from Riehl's book, but to try it out, I tried an example. My input category:...
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Apply the Lemma of Yoneda

Let $\mathcal{A}\to\textrm{Sets}$ be a covariant functor, $X, X'\in\operatorname{Obj}(\mathcal{A})$ objects, and $(X, \Phi), (X', \Phi')$ be representations of $F$, i.e. $\Phi: F\to\operatorname{Hom}_{...
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A particular dualization of the Yoneda lemma fails to exist

Emily Riehl's "Category Theory in Context", ${\rm Exercise}~2.2.{\rm ii}.$ Explain why the Yoneda lemma does not dualize to classify natural transformations from an arbitrary set-valued functor to ...
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Natural transformation induced by adjoint functors. [duplicate]

My question is the same as here, but I can't understand why the transformation from the identity functor to the composition of the adjoint functors is natural. I've tried proving explicitly that the ...
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On functors agreeing with the powerset functor on objects and not being isomorphic to it

Recall the powerset functor $\mathcal{P}:\mathbf{Set} \to \mathbf{Set}$ defined as $\mathcal{P}(X) = 2^{X} = \{U\subseteq X\}$ on objects $\mathcal{P}(f: X \to Y): \mathcal{P}(X) \to \mathcal{P}(Y),\ ...
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Clarification in Yoneda Lemma exercise from Aluffi's Algebra Chapter $0$

I don't understand the sentence "Map $h_X$ to the image of $\mathrm{id}_X \in h_x(X)$ in $\mathscr F(X)$." $h_X$ is a functor, so, what can that sentence mean?
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Exercise 2.2. ii of Riehl's book “category theory in context”

Exercise 2.2.ii. Explain why the Yoneda lemma does not dualize to classify natural transformations from an arbitrary set-valued functor to a represented functor. I don't understand the meaning that "...
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The dual of the Yoneda lemma

One exercise in Leinster asks to state the dual of the Yoneda lemma. The original statement is: Let $\mathscr A$ be a locally small category. Then $$[\mathscr A^{op},\textbf{Set}](H_A,X)\cong X(A)$...
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About the naturality condition in the Yoneda lemma

After reading the statement of the Yoneda lemma (Theorem 4.2.1), I understand that it states that the functor $[\mathscr A^{op},\mathbf{Set}](H_\bullet,-)$ is naturally isomorphic to the functor $-(\...
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3answers
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Definition of $[\mathscr A^{\text{op}}, \textbf{Set}](H_A,X)\to [\mathscr A^{\text{op}}, \textbf{Set}](H_B,X)$ in the proof of the Yoneda lemma

Regarding the proof of the Yoneda lemma on p.98: How does one get that the map $$[\mathscr A^{\text{op}}, \textbf{Set}](H_A,X)\to [\mathscr A^{\text{op}}, \textbf{Set}](H_B,X)$$ is defined by $-\circ ...
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Why presheaves are generalized objects?

While self studying category theory (Yoneda lemma), I came across the statement that for any category $\mathsf{C}$ the functor category $\mathsf{Fun}(\mathsf{C}^{op}, \mathsf{Set})$ represents ...
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1answer
153 views

How to use Yoneda's lemma in proving: $F$ fully-faithful iff unit is an isomorphism

Let $F \dashv G$ be an adjunction with unit $e: Id \implies GF$. It is well known that $e$ is an isomorphism if and only $F$ is fully-faithful. I've a proof of this fact that doesn't use Yoneda's ...
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2answers
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Simple examples of non-representable functors

I am looking for examples of non-representable functors, to see how the Yoneda lemma works in these cases. Here is one: let $\mathbf{C}$ be the category of finite-dimensional Euclidean spaces, with ...
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1answer
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Is there a way to show Yoneda equality visually instead of compositional algebraic symbols?

The proof that $u \in F(A)$ is sufficient to define a natural transformation $\alpha : \text{Hom}(A, \cdot) \Rightarrow F(\cdot)$ goes like this: Let $\alpha_X : g \in \text{Hom}(X, Y) \mapsto F(f)(u)...
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1answer
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Proof of Yoneda's lemma on wikipedia.

https://en.wikipedia.org/wiki/Yoneda_lemma#Proof The part I'm not getting is the last one: Moreover, any element ${\displaystyle u\in F(A)}$ defines a natural transformation in this way. So we ...
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1answer
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Universal elements of a functor and representability

A functor $F : \mathcal{C} \rightarrow \mathcal{Set}$ is said to representable if it is naturally isomorphic to $\mathcal{C}(A,–)$ for some object $A$ of $\mathcal{C}$. By the Yoneda lemma, we know ...
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1answer
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The Yoneda lemma and a natural bijection

Let $S\colon\mathbf {Set}^{\cal A^{op}}\to \mathbf{ Set}$ be a functor. How does it follow from the Yoneda lemma that the following is a natural bijection: $\underline{\hom(A,-)\to SY \quad\quad\...
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A problem about colimits and the Yoneda functor

I encountered a problem when I was reading Chapter Two of the book Categories and sheaves. In this book, for $\alpha:I\rightarrow\mathcal{C}$, the colimit $\lim\limits_\rightarrow\alpha$ is defined ...
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1answer
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Trying to understand an application of Yoneda's lemma

Suppose I have a category where the functors are $$ (\text{reduced finitely generated commutative k-algebras}) \rightarrow (\text{groups}) $$ that are representable as set valued functors, and the ...
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Relation between Yoneda's Lemma and Tanaka Duality

This question was prompted since I read that category theory can be seen as a generalization of representation theory. Tanaka duality basically states that from the category of representations of a ...
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In the Yoneda Lemma, how is there an isomorphism $\text{Nat}[\text{Hom}(x,-), F(-)] \cong Fx$ when only $\text{Hom}(x,x)$ gets mapped to $Fx$?

I've been trying to understand the Yoneda Lemma. I have done all of the diagram chasing and understand that the naturality is dependent on one element $id_x$ and all arrows all naturally extended from ...
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1answer
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Clarification of the statement of the Yoneda lemma

Here is one version of the Yoneda lemma: Yoneda (Previously, for any category $\mathcal C$ and any $X\in \text{Ob} \mathcal C$, the guy $h_X$ was defined as the functor $\operatorname {Hom}_{\mathcal ...
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1answer
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What has gone wrong in this proof of the Yoneda Lemma

Here is a snippet of the first part of the proof of the Yoneda Lemma. The last line says the square commutes, but I don't get that it commutes. So what has gone wrong in the proof? Following one ...
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1answer
60 views

About proving a cokernel is not representable

I have been confused by an example from Dolgachev: Derived Categories, which aims to show that the cokernel of a morphism between two presheaves $h_A$ and $h_B$ may not be representable, even when $\...
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1answer
241 views

Describing $Hom(\mathbb{Z}^2, G)$ as a subset of $G \times G$

I am asked to describe $Hom(\mathbb{Z}^2, G)$ as a subset of $G \times G$. I interpret this as describing a relationship between the two (i.e. showing they are isomorphic) G is finite, not ...
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1answer
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If the Yoneda lemma essentially says that $\text{Hom}(\text{Hom}(\cdot, x), A) \simeq A(x)$, then what about higher iterates of $\text{Hom}$?

Assume that the $C$ in $\text{Hom}_C(x,y)$ can always be inferred from $x,y$ so that we can change our notation to $\text{H}(x,y) := \text{Hom}_C(x,y)$ Then the Yoneda lemma "looks at a single step ...
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1answer
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Category theoretic proof of Stone's representation theorem

Can Stone's representation theorem about Boolean algebras that every Boolean algebra $B$ is isomorphic to the algebra of clopen subsets of its Stone space $S(B)$, be proven categorically using Yoneda ...
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In the proof of the Yoneda Lemma in “Categories & Sheaves” by Kashiwara & Schapira.

Our goal is to show that $\text{Hom}_{C^{\wedge}}(h_C(X), A) \simeq A(X)$. We first want to show a map from left to right. The book says: $$ \text{Hom}_{C^{\wedge}}(h_C(X), A) \to \text{Hom}_{\text{...