# 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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### 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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### 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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### 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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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 $Y : \mathfrak{A} \to [\mathfrak{A}^\... 0 votes 0 answers 20 views ### Is there a more direct way of doing the Yoneda embedding into fibered category? I want to figure out the Yoneda embedding for fibred categories but some of the higher categorical details are a bit hard to figure out. Right now I've been playing around with presheaves on the ... 0 votes 0 answers 53 views ### Reverse Yoneda: Nat(F, Hom(c, -)) in bijection with F(c)? The Yoneda lemma states that for a locally small category$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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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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### 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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### 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, ... 1 vote
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 $L:\... 1 vote 1 answer 76 views ### A detail in the proof of the Yoneda lemma In the proof of the Yoneda lemma, Leinster states this (p. 97): (The hat is the function$\alpha\mapsto \alpha_A(1_A)$and the tilde is its inverse.) But I don't understand how he applies Lemma 1.3.... 0 votes 1 answer 213 views ### On the proof of the density theorem I'm trying to understand Leinster's proof of the density theorem. Here's the terminology and the statement. Below is his proof. Here are some things that I don't understand: This must be silly, but ... 2 votes 1 answer 124 views ### Using Yoneda to establish natural isomorphisms? I know the Yoneda embedding can be used to easily establish isomorphisms between objects in categories. For example, in a locally small cartesian closed category$\mathbf{C}$with coproducts, the &... 3 votes 1 answer 133 views ### Replacing$\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}}$. ... 3 votes 1 answer 95 views ### Why is el(-)=$\int(-)$a functor from functors to a slice cateory? I am reading up a little on category theory. I am trying to solve Riehl's problem (Category Theory in Context) 2.4.vii on page 72. I believe that it should be quite easy, but it appears to me that ... 3 votes 1 answer 297 views ### Possible adjoint to Yoneda embedding and Repeated Yoneda embedding? While thinking about Yoneda embedding, I came up with following two questions (I should apologies, if those are too vague): Does the Yoneda embedding$y :\mathcal{C}\to\mathbf{Set}^{\mathcal{C}^{\...
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 \$[\...