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
35 views

Quotient of a free group on a set

Suppose I have a free group $F_S$ on a set $S,$ with a nonempty proper subset $T.$ If $N$ is the smallest normal subgroup of $F_S$ containing $T$, I would like to show that $F_S/N$ is also free. There'...
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When left Kan extension preserve colimits

I am currently working with Kan extensions, and I have found a neat little fact, and I'd like to know if there is some literatture on it, and maybe some names and characterization I could use. ...
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1answer
97 views

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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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\}\...
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1answer
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Calculating the geometric realization of a non-representable functor

Background Let $\mathcal{F} : \textbf{CRing} \to \textbf{Set}$ be a functor and denote by $\textbf{P}_\mathcal{F}$ the category of points of $\mathcal{F}$ whose objects are pairs $(R , \rho)$ where $R$...
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Is there a connection between the Whitney embedding theorem and Cayley's theorem?

Background: I've been working through Guillemin and Pollack's "Differential Topology." They take the approach of defining smooth manifolds as "concrete" submanifolds of some ...
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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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1answer
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Yoneda embedding preserves finite coproducts

For the Yoneda Embedding, where $\mathcal{T}$ is an algebraic theory, $$ Y_\mathcal{T}: \mathcal{T}^\mathrm{op} \to Alg \mathcal{T} $$ it is claimed that $Y_\mathcal{T}$ preserves finite coproducts. ...
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1answer
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Proving an equality of functors in algebraic theories

Can somebody explain how are they claiming the following highlighted identity in the book Algebraic Theories by Vitale? The reason I am confused is because the image of the $F$ functor lands inside $\...
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1answer
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Exponentials in the category of graphs?

Let $\Gamma$ be the category $e\overset{s}{\underset{t}{\rightrightarrows}}v$ and $\mathbf{Graphs}=\mathbf{Sets}^{\Gamma}$ the category of (directed multi) graphs. For graphs $G$ and $H$, it's my ...
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1answer
83 views

Left (resp. right) adjoint functor fully faithful iff unit (resp. counit) isomorphism [duplicate]

Let $\mathcal{C}$ and $\mathcal{D}$ be categories and $\mathcal{F}:\mathcal{C}\longleftrightarrow\mathcal{D}:G$ functors such that $\mathcal{G}$ is right-adjoint to $\mathcal{F}$, ie. we have a ...
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2answers
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Are Yoneda lifts (twice) of fully faithful functors fully faithful?

$\newcommand{\C}{\mathcal{C}}$ $\newcommand{\D}{\mathcal{D}}$ $\newcommand{\A}{\mathcal{A}}$ $\newcommand{\S}{\mathcal{S}}$ $\newcommand{\Psh}{\mathrm{Psh}}$ $\newcommand{\Lan}{\mathrm{Lan}}$ $\...
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Every representable presheaf is projective

Let $\mathcal{C}$ be a small category and let $\hat{\mathcal{C}}$ be its category of presheaves. I want to show that every representable presheaf $y_C\in \hat{\mathcal{C}}$ (for some $C\in\mathcal{C}$)...
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1answer
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Topology: A Categorical Approach Chapter 0 Exercise 6 (Yoneda lemma)

I cannot figure out how to prove this lemma, at least partly because I'm still unfamiliar with the concepts and notation involved. Below I will write down my thoughts on how to go about it, and ...
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1answer
43 views

Is there a unique natural transformation of functors $t : F \to F'$ such that $t(U) = T(\text{Id}_F(U))$?

Let $C, D$ be categories, $F$ and $F'$ covariant functors from $C$ to $D$. For any natural transformation of bifunctors $T : \hom_D(−, F(−)) \to \hom_D(−, F'(−))$ there is a unique natural ...
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1answer
54 views

Are extensions of simplicial objects to functors $\mathsf{sSet} \to \mathsf{C}$ Kan extensions?

Suppose that we have a functor $F : \boldsymbol{\Delta}^\bullet \to \mathsf{C}$ with domain the full subcategory of simplicial sets given by representable functors. For example, for each $\Delta^n = \...
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1answer
106 views

Basic question about Yoneda's lemma

Out of curiosity, I am trying to self teach myself a bit of category theory and I have a question about Yoneda's lemma. I recall the notation that I am using. I hope to not get it (too) wrong. Given a ...
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1answer
120 views

Confusion about the Yoneda lemma

Let $C$ be a category and $F$ a presheaf on $C$. The Yoneda lemma states that the natural transformations $C(-, A)\Rightarrow F$ are in one-to-one correspondence with the elements of $F(A)$. To me, ...
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0answers
61 views

When is an isomorphism between representable functors natural?

Let $\mathcal{A},\mathcal{B}$ be two small categories and $\mathcal{C},\mathcal{D}$ two arbitrary categories. Let $F:\mathcal{A}\rightarrow\mathcal{B}$, $G:\mathcal{A}\rightarrow\mathcal{C}$ and $L:\...
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1answer
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A detail in the proof of the Yoneda lemma

In the proof of the Yoneda lemma, Leinster states this (p. 97): (The hat is the function $\alpha\mapsto \alpha_A(1_A)$ and the tilde is its inverse.) But I don't understand how he applies Lemma 1.3....
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1answer
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On the proof of the density theorem

I'm trying to understand Leinster's proof of the density theorem. Here's the terminology and the statement. Below is his proof. Here are some things that I don't understand: This must be silly, but ...
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1answer
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Using Yoneda to establish natural isomorphisms?

I know the Yoneda embedding can be used to easily establish isomorphisms between objects in categories. For example, in a locally small cartesian closed category $\mathbf{C}$ with coproducts, the &...
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1answer
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Replacing $\mathbf{Set}$ in Yoneda

The Yoneda lemma (or rather the existence of the Yoneda embedding) states, roughly, that for each category $C$ there's an embedding (a fully faithful functor) of $C$ into $\mathbf{Set}^{C^{op}}$. ...
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1answer
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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
149 views

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
38 views

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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2answers
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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
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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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1answer
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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
283 views

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
75 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
104 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
79 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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1answer
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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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1answer
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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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1answer
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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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1answer
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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
40 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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1answer
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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
122 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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1answer
107 views

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 $1^X \cong 1$, $X^Y \times X^Z \cong X^{Y + Z}$, $X^1 \cong X$ and a few ...
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1answer
120 views

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
92 views

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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1answer
149 views

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 ...