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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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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Yoneda lemma - Why is $\textrm{Nat}(\textrm{Hom}(A,-),F)$ a set?

Suppose $\mathcal{C}$ is a locally small category and $F:\mathcal{C}\to \textrm{Set}$ is a covariant functor. Since every class in bijection with a set is a set, and the Yoneda lemma establishes that $...
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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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Corollary of Yoneda Lemma

I have a question on one of the corolarries of the Yoneda lemma. Let $y$ be the yoneda embedding. Then how can one prove for two objects $c, d\in C$ that if $y(c) \cong y(d) \Rightarrow c\cong d$. I ...
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Proving that Yoneda Embedding preserves identity.

Here is the Yoneda embedding: $\mathscr{C} \xrightarrow{y} \operatorname{Func}(\mathscr{C}^{op}, \mathscr{S}et)$ where $$y(f: X \to X') = (\mathscr{C}(-, f):\mathscr{C}(-, X) \to \mathscr{C}(-, X'))$$ ...
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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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The Yoneda Lemma and coends

I am trying to understand the proof of Proposition 2.2 of nlab's page on the co-Yoneda lemma. I don't understand the last part of the argument, and I was hoping somebody could help. In particular, we ...
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Constructing counit in adjoint functor theorem for total categories

The theorem I am referring to is, Let $C,$ $D$ be locally small categories. Assume $C$ is a total category (i.e. the Yoneda functor $Y : C \to \operatorname{PreSh}(C)$ has a left adjoint $Y^L$). Let $...
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Three ways to to prove that projective modules are flat

I am trying to show that projective modules are flat using their defining property that $Hom(P,-)$ is an exact functor when $P$ is projective. The two ways I know of come down to the fact that ...
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Category theoretical proof that Abelianization commutes with products of group

Let $Grps$ and $Ab$ be the categories of groups and abelian groups respectively and lets denote $\times,\sqcup, \oplus$ the product and coproduct in Grps and the biproduct in $Ab$. If I want to prove ...
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Is the Yoneda bijection natural in the domain category?

In the Yoneda Lemma, we start with a locally small category (the domain category, sometimes called a site), together with some other data, and establish a certain bijection. What is the naturality (or ...
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Why $\Psi\circ\Phi=1$ in Yoneda lemma?

I am trying to understand the proof: How can one show that $\Psi\circ\Phi=1_{\textrm{Nat}(A(A,-),F)}$? Let $\tau:A(A,-)\rightarrow F$ be a natural transformation. Then $(\Psi\circ\Phi)(\tau)=\Psi(\...
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Proof verification: Yoneda preserves limits

$\newcommand{cop}{C^{op}}$ $\newcommand{cSet}{\mathsf{Set}}$ $\newcommand{Hom}{\operatorname{Hom}}$ $\newcommand{\eval}{\operatorname{eval}}$ Let $J$ be a small category, and let $C$ be locally small. ...
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Showing that a morphism is unique

Suppose we have two representables $H_A, H_{A'}: \mathscr A\to \mathbf {Set}$ and a natural transformation $H_A\to H_{A'}$ with components $\alpha_B:H_A(B)\to H_{A'}(B)$. It can be shown that each $\...
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Applying Yoneda Lemma

Let $\mathbb{C}$ be some category and let $G$ be a functor $\mathcal{C}^{op} \to \mathcal{C}$. Assume that I was able to show that there is a natural isomorphism $\mathcal{C}(-, G(A \oplus B)) \...
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Understanding the maps between $F(A)$ and the natural transformations used in the proof of the Yoneda lemma.

Understanding the Yoneda lemma maps. I'm trying to understand the maps between the natural transformations and $F(A)$ in the proof of the Yoneda lemma. I've been struggling for a bit to understand the ...
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Hilton/Stammbach Exercise 2.4.6: Yoneda Embedding and Functors

I'm working on the following exercise from Chapter 2 of Hilton/Stammbach's A Course in Homological Algebra: "Let $\mathfrak{A}$ be a small category and let $Y : \mathfrak{A} \to [\mathfrak{A}^\...
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Is there a more direct way of doing the Yoneda embedding into fibered category?

I want to figure out the Yoneda embedding for fibred categories but some of the higher categorical details are a bit hard to figure out. Right now I've been playing around with presheaves on the ...
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Reverse Yoneda: Nat(F, Hom(c, -)) in bijection with F(c)?

The Yoneda lemma states that for a locally small category $C$ and a functor $F: C \rightarrow Set$, for each object $c \in C$, there is a bijection between the set of natural transformations $[C, Set](...
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Specific question about adjunction of functors

Suppose to have three functors such that $F\dashv G, G':\mathbf D\to \bf C$. For every $c$ in $\mathrm{Ob}\bf C$, the adjunctions imply the existence of two universal arrows among the arrows of $F$ in ...
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Consequence of the Yoneda lemma

Take a functor $G:\mathbb D\to \mathbb C $, and call $Y:\mathbb D\to [\operatorname{\mathbb D,Set]}:D\mapsto\operatorname{Hom}_{\mathbb D}(-,D)$ the Yoneda functor. Consider the bifunctor $\...
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What are some interesting applications of Yoneda's lemma in (functional) analysis?

I recently took a class in homological algebra and there I saw Yoneda's lemma. This says that if we have a locally small category $C$ and a functor $F:C\to\text{Set}$, then the natural transformations ...
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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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1 answer
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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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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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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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8 votes
2 answers
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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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1 answer
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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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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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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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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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1 vote
2 answers
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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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2 votes
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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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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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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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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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4 votes
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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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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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1 vote
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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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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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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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2 votes
1 answer
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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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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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3 votes
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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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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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2 votes
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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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