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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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Yoneda lemma in a functor semicategory

I've read that the Yoneda lemma does not hold in general for semicategories (i.e., 'categories' possibly lacking identity morphisms)[1]. However, I'm wondering about a related situation, where there ...
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Understanding the particle accelerator analogy for the Yoneda lemma

In his Algebraic Geometry class a few years back, Ravi Vakil explained Yoneda's lemma like this: You work at a particle accelerator. You want to understand some particle. All you can do are throw ...
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Why naturality in Yoneda Lemma?

I understand the statement of Yoneda Lemma and its implications; however, at a very concrete level I have never seen how naturality condition is used in examples. Even when we consider a Yoneda ...
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Proof that the Yoneda embedding is cartesian closed

Is there a reference somewhere that the Yoneda embedding is cartesian closed? I tried showing this myself, but after staring at it for an afternoon, I did not yet see a proof. I saw an answer here ...
Tempestas Ludi's user avatar
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Category of Sets vs Category of Types (& Yoneda Lemma)

I've been wondering for a while why we discuss the category of sets, $\mathbf{Set}$, all the time, but hardly discuss something like the category of all types and functions between them ($\mathbf{Type}...
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Understanding flat functors

I am trying to understand the definition of internally flat functors. This is for convincing myself about the idea of Diaconescu’s theorem, so I would appreciate any remark in this particular ...
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Natural transformations of Hom-sets “transport” natural transformations from one pair of functors to another? (Reference)

Question 1: Does anyone know a name, or have a reference, for the following lemma? $\newcommand{\Hom}{\operatorname{Hom}}$$\newcommand{\F}{\mathscr{F}}$$\newcommand{\G}{\mathscr{G}}$$\newcommand{\op}{...
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Deducing the definition of subobject classifier from Sub functor being representable

As the title says. Let's consider some category $C$ and the contravariant functor $Sub : C^{op} \rightarrow Set$ that for each object gives its set of subobjects. The introductory book by Leinster ...
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Reproducing Kernel Hilbert Spaces and Yoneda Lemma

In the following articles: https://proceedings.mlr.press/v202/yuan23b/yuan23b.pdf(section 7.4), https://arxiv.org/pdf/2207.02917.pdf(Theorem 4), The Reproducing Kernel Hilbert Spaces(RKHS) are ...
FanFanKa's user avatar
1 vote
2 answers
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Definition of natural transformations

I am trying to understand Yoneda's lemma, and am trying to piece together the definition of a natural transformation. Wikipedia says that given two functors $F,G$ that map categories $C$ to $D$, a ...
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Density Theorem via Weighted Colimits: Confusing Variance

Here is a question concerning Emily Riehl's Categorical Homotopy Theory about weighted colimits. There is some $(-)^{\mathrm{op}}$-story confusing me. Let me begin by recalling the definition of (...
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Rephrasing the Yoneda lemma using the Yoneda embedding

The usual way that I see the Yoneda lemma stated is as something like $$\text{Nat}(\text{Hom}(-,X),F) \cong F(X) \,,$$ where $F: \mathcal{C} \to \text{Set}$ is a functor and $X$ is an object of $\...
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2 answers
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Yoneda lemma for representable profunctors?

For $F:C\rightarrow D$ a functor, consider the profunctor $h_F:=\operatorname{Hom}_D(-,F-):D^{op}\times C \rightarrow \mathsf{Set}$. For a functor $G:C\rightarrow D$, define $h_G$ similarly. Is it ...
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Why are functor categories nice?

I was looking at the Yoneda embedding and one motivation is that we are embedding the category into a functor category and "functor categories are nice". What does this mean? What nice ...
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Condition that limit of a diagram $F$ exists if $\alpha: Mor(-,L) \to \lim Mor(X,F(i))$

Suppose $F : I \to \mathcal C$ is a diagram for a small category $I$ and a category $\mathcal C$. Suppose also that there is an object $L$ of $\mathcal C$ such that for all $X \in Obj(\mathcal C)$ ...
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Prove that the functor $P : \text{Set}^\text{op} → \text{Set}$ , $F(f)=f^{-1}:P(Y)\rightarrow P(X)$ for $f: X \rightarrow Y$ is representable

Let $P(A)$ denote the power set of a set $A$ For a map $f : X \mapsto Y$ of sets, I can define a map $P( f ): P(Y) → P(X)$ s.t that I obtain a functor $P : \text{Set}^\text{op} \to \text{Set}$ in ...
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Automorphism group of the functor $P:G\text{-FinSet} \to \text{Set}$

I am trying to understand the proof of the following theorem from Qiaochu Yuan's answer Theorem: Let $G$ be a group and let $P : G\text{-FinSet} \to \text{Set}$ be the forgetful functor from the ...
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Stacky Yoneda lemma

I’m studying differentiable stacks from this paper (Gregory Ginot), and I’m trying to understand why the “Yoneda embedding” from the category of differentiable manifolds $\textbf{Man}$ to the category ...
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Left-adjoint to Yoneda embedding

Let $C$ be locally small. Consider the Yoneda embedding $Y:C\rightarrow [C^{op},Set]$. Since limits in functor categories are computed pointwise and since the hom-functor preserves limits, the Yoneda ...
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Two definitions of categorical limits

For $C$ a locally small category, $J$ an essentially small category and $F\colon J\rightarrow C$ a functor, the limit of $F$, if it exists, can be defined as a representation of the functor $\...
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Contravariant Yoneda lemma

I am learning category theory and trying to prove the contravariant Yoneda lemma: Let $\mathcal C$ be a locally small category and $F:\mathcal C\to\mathsf{Set}$ be a contravariant functor. Fix an ...
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Usage of Yoneda lemma to show ${\rm Lan}_K \dashv \mathcal{E}^K$

I am reading a proof showing that for $K:\mathcal{C}\rightarrow \mathcal{D}$ and $\mathcal{E}$ a category, the functor ${\rm Lan}_K$ is left adjoint to $\mathcal{E}^K$ and there is a thing I don't ...
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If the presheaf Hom$_\mathcal{C}(- \times A, B) : \mathcal{C}^\text{op} \to \textbf{Set}$ is representable, then $\mathcal{C}$ is ccc

We consider a small category $\mathcal{C}$ with binary products, and we consider, for any objects $A, B$ of $\mathcal{C}$, the assignments \begin{eqnarray*} F :\,\, &\mathcal{C}^\text{op} &\...
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1 answer
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A particular presheaf on a small category. What if it's representable?

We consider a small category $\mathcal{C}$ with binary products, and we consider, for any objects $A, B$ of $\mathcal{C}$, the assignments \begin{eqnarray*} F :\,\, &\mathcal{C}^\text{op} &\...
Jos van Nieuwman's user avatar
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1 answer
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Presheaf is terminal iff it maps every object to a singleton

I've read the statement in the title in a text as a side note without proof. One way is easy: using Yoneda, terminality of a presheaf $X \in [\mathcal{C}^\text{op}, \textbf{Set}]$ implies that for any ...
Jos van Nieuwman's user avatar
2 votes
2 answers
154 views

Dualising the Yoneda lemma.

Let $\mathcal{C}$ be a small category, and $A, B \in \mathcal{C}_0$ objects of $\mathcal{C}$. Suppose that for every $X ∈ \mathcal{C}_0$ we have bijections $f_X : \text{Hom}_{\mathcal{C}}(A, X) \tilde\...
Jos van Nieuwman's user avatar
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How does the 2-yoneda lemma exactly work

I understand the 1-Yoneda lemma well, and I know how to use it in proofs, but I have trouble understanding how the 2-Yoneda lemma exactly works. Say for simplicitly I have a strict 2-category $K$ and ...
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If $f:Y \to X$ is a mono in $\textsf{Set}^{\mathbb{C}^{op}}$, then $f$ factors uniquely through $Y \xrightarrow{g} A \stackrel{i}{\hookrightarrow} X$.

I have taken an introductory course in category theory and would like to learn more about presheaves. Currently I am working through "Generic figures and their glueings" by Marie La Palme ...
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1 answer
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How to go about proving theorems about category theory? [closed]

Prove that the functor $V : Cat → Set$ that sends a category $\mathcal{C}$ to its set of arrows, is represented by the category ${0 \rightarrow 1}$ with two objects and a single non identity morphism ...
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1 answer
122 views

How can we quickly verify naturality?

The Yoneda lemma is sometimes claimed to simplify proofs. For instance, the associativity of binary products can be proved by considering $\hom(X, (A \times B) \times C)$, and by the bijection of sets ...
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What to call the natural transformation in the Yoneda lemma

The Yoneda lemma is an isomorphism between f a and (a <-) ~> f where the Yoneda embedding is ...
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1 answer
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Finding terminal object in presheaf topos.

I am interested in finding the terminal objects of a presheaf topos, and in particular, ones constructed via Yoneda embedding $\hat{C} = \mathrm{Set}^{C^{\mathrm{op}}}$ from a category $C$. Starting ...
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1 answer
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Yoneda Lemma and isomorphisms

I have a problem understanding how Yoneda Lemma gives a natural isomorphism between functors. Let $F,G \colon \mathcal{A}\to \mathcal{B}$ two functors and I have a bifunctorial isomorphism $\mathrm{...
user34977's user avatar
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9 votes
1 answer
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Question on the “Yoneda perspective”

One of the consequences of the Yoneda embedding is that, given a category $C$ and two objects $A, B$ in $C$, we can obtain an isomorphism between $A$ and $B$ by finding a natural isomorphism between ...
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Explanation of an observation regarding Yoneda

Denote by $\mathsf{CAT}$ the category of all locally small categories, by $\mathsf{Set}$ the category of sets, and write $ [\mathsf{C}, \mathsf{D}]$ for $\mathsf{CAT}(\mathsf{C}, \mathsf{D})$. Also, $\...
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1 answer
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Definition of the evaluation map as a universal element of a representation

This is exercise Exercise 2.3.iii from Riehl's "Category Theory in Context": The set $B^A$ of functions from a set $A$ to a set $B$ represents the contravariant functor ${\rm Set}(-\times A,...
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Silly joke about Yoneda, but is it that silly..?

I stated this as a joke, but if you think about it, could this not be true in some sense..? This is the joke: "Does the Yoneda Lemma have the universal property of being the unique result in ...
Jos van Nieuwman's user avatar
3 votes
1 answer
106 views

Why the triangle diagram commutes because of naturality

I am new in category and I am reading Awodey's Category Theory. In the proof of Proposition 8.10 (See the picture of the proof here1, here2 and here3), after identifying $$x\in P(C)$$and $$x:yC\...
user914799's user avatar
1 vote
1 answer
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Cats in context Lemma 2.4.7.

I'm failing to understand the argument behind Lemma 2.4.7. in page 86 from Emily Riehl's Category Theory in Context. In particular along the following reasoning: Here $\int F$ is the category of ...
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Intuition for density formula for Pre-sheaves/Co-Yoneda lemma

I was reading Fosco's notes coend calculus, so far I am at Ninja-Yoneda lemma. The density formula for pre-sheafs says that, $${K} V\cong \int^{U\in C} K U\times \textstyle\text{Hom}_{C}(V,U)\cong\...
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1 answer
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How to prove a map is epic using generalized elements only?

I have a map $\require{AMScd}f\colon X \to Y$ in some category $\mathcal E$ which I would like to show is epic. However the only description I have of $X$, $Y$, and $f$ is through the Yoneda ...
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Does the Barr-embedding preserve coequalizers

Let $\mathbb C$ be a small regular category. Let $J$ be the Grothendieck topology generated by coverings $\{U'\twoheadrightarrow U\}$ consisting of precisely the regular epimorphisms in $\mathbb C$. ...
Nico's user avatar
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Yoneda lemma for endofunctors

The Yoneda lemma deals with functors to the category of functions. The Yoneda functor, say Y, for an object, say Z, is a specific example of such a functor. The lemma establishes a one-to-one ...
LucDupAtMathExchange's user avatar
2 votes
1 answer
255 views

Yoneda Lemma, Riehl's Category Theory in Context

Riehl writes about the Yoneda Lemma: "For any functor $F: C \rightarrow Set$, whose domain is locally small and for $c \in C, \exists$ bijection $Hom(C(c,-),F) \cong Fc$. Moreover, this ...
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Why do we take opposite category when stating the Yoneda Lemma?

For any function $F: C^{op} \to \text{Set}$ and any object $X$ in $C$, natural transformations $\text{Hom}(-,X) \to F$ are in bijection with the elements in the set $F(x)$. That is, $\text{Nat} (\text{...
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1 answer
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on the equivalence of notions of adjunctions

let $C$ and $D$ be categories and $F : C \leftrightarrows D : G$ be a pair of functors. as is well known, there is a natural correspondence of pairs of inverse isomorphisms $α : \hom_D (FX, Y) \...
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Symmetric monoidal categories and modules over the unit

Consider a symmetric monoidal category $(C, \otimes, I)$, where $I$ is the unit. Then there is a restricted Yoneda functor $$ C \rightarrow Hom(I,I)-mod $$ taking an object $X$ of $C$ to $Hom(I, X)$, ...
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1 vote
1 answer
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Difficulty with converting Yoneda's natural isomorphy into a group isomorphism in the proof of Cayley's theorem

$\newcommand{\A}{\mathscr{A}}\newcommand{\Gc}{\mathscr{G}}\newcommand{\G}{\mathcal{G}}\newcommand{\s}{\mathsf{Set}}\newcommand{\op}{^{\mathsf{op}}}\newcommand{\sym}{\mathsf{Sym}}$I am having ...
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3 votes
2 answers
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Coherence in closed monoidal categories

Let $(M, \otimes, I)$ be a left-closed (non-symmetric) monoidal category with left-internal hom $\underline{\operatorname{hom}}(-,-)$. Denote by $\sigma_{A,B,C}: M(A\otimes B, C) \xrightarrow{\sim} M(...
Max Demirdilek's user avatar
3 votes
0 answers
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The difference between totally (large) cocontinuous functors and small cocontinuous functors

$\newcommand{\cat}{\mathbf}\newcommand{\op}{\mathrm{op}}\newcommand{\Hom}{\operatorname{Hom}}\newcommand{\cSet}{\cat{Set}}$A category $\cat C$ is total if the Yoneda embedding $\cat C→[\cat C^{\op},\...
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