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Questions tagged [yoneda-lemma]

For questions related to Yoneda lemma in category theory. Use this tag along with (abstract-algebra) or (category-theory).

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Purely technical question on the definition of a sheaf : by Yoneda + restriction or by an equalizer

I have problems dealing with all the categorical language, even for very basic things like elementary limits calculus. I always understand the intuition but I seem to be unable to right down 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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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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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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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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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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Contravariant Yoneda lemma: clarifying confusion

A part of Yoneda lemma states that given a locally small category $\mathsf{C}$, a functor $F\colon\mathsf{C}\to\mathsf{Set}$ and an object $A \in \mathsf{C}$, there a bijection $\rho\colon\mathsf{...
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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{...
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Is Yoneda Lemma a characterization of isomorphism?

Let $C$ be a locally small category. Is it true that $$ X \cong_C Y \iff C(A, X) \cong_{\mathsf{Set}} C(A, Y) \iff C(X, A) \cong_{\mathsf{Set}} C(Y, A) $$ where we assume the last two isomorphism are ...
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Is there a Yoneda lemma for categories other than Set?

The Yoneda lemma says (in my understanding) that instead of studying a category directly, you can study that category's relationships between its relationships into Set. Is the function of Set unique ...
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Categories for which every contiuous sheaf is representable

I'm interested in locally small, cocomplete categories $\mathbf{C}$ such that every limit preserving functor $$\mathbf{C}^\mathrm{op}\to\mathbf{Set}$$ is representable. Is there a name for such ...
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Unpacking a comment about sets of morphisms

In a comment on this MathOverflow question, Theo Johnson-Freyd states: OTOH, in any category, if there are monomorphisms $A\to B$ and $B\to A$, then for any $X$ the sets $\operatorname{Hom}(X,A)$ ...
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Why is this universal map in a proof of the co-Yoneda lemma actually natural?

I'm attempting to prove that every presheaf is a canonical colimit of representable presheaves by constructing a limiting cocone directly (I'm aware that there are more elegant proofs, but this is ...
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Representability criterion for Zariski sheaf in terms of open subfunctors

I've been trying to prove a fairly classical, well-known result, but am running into a lot of trouble following any of the proofs I have found. At the moment I am following EGA I 0.4.5.4. Let $F: \...
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1answer
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Alternative Description Of The Isomorphism $[C_0,[C_1,\text{Set}]]\cong[C_1,[C_0,\text{Set}]]$

I have troubles wrapping my mind around this equation. It is trivial to prove, but I prefer to think about the objects in $[C,\text{Set}]$ as generalized objects over $C^\text{op}$, motivated by the ...
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1answer
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Detecting family of $[\mathcal{C},\mathbf{Set}]$

I am trying to prove that $\{C(A,-)\mid A\in \text{ob}\mathcal{C}\}$ is a detecting family for the functor category $[\mathcal{C},\mathbf{Set}]$. For that, I need to check that taking a natural ...
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1answer
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Additive Yoneda Lemma

I'm studying Abelian Categories in F. Borceux "Handbook of Categorical Algebra" Vol.2. In this reference we can find an additive version of the famous Yoneda Lemma : Let $\mathcal{C}$ be a ...
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When can functors fail to be adjoints if their hom sets are bijective?

Suppose we have categories $\mathcal{C}$ and $\mathcal{D}$ with some appropriate adjectives like local smallness along with a pair of functors $F: \mathcal{C} \rightarrow \mathcal{D}$ and $G: \mathcal{...
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Lattice subobjects of the power object in a topos

When proving that the subobject classifier of a topos is an internal Heyting algebra, we exploit the natural isomorphism $$Sub(X)\simeq Hom(X,\Omega)$$ Therefore the intersection of subobjects induces ...
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bivariate Yoneda lemma

$\newcommand{\Hom}{\operatorname{Hom}}$Any category is equipped with a covariant hom-functor $\Hom(A,-)$, by letting the second argument vary. The covariant Yoneda lemma says $\operatorname{Nat}(\Hom(...
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Do proofs using the Yoneda lemma work for Categories that are not locally small?

I often see proofs in category theory using the Yoneda lemma. I wonder if these apply to general categories(categories in which the hom-sets need not be sets) or there is a tacit assumption that the ...
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How do I apply the Yoneda lemma to this functor?

Let $\sigma : F \rightarrow F'$ be a natural transformation where $F, F' : X \rightarrow A$. If $\phi : A(Fx, a) \cong X(x, Ga)$, and $(\sigma_x)^*=A(\sigma_x, a) :A(F'x,a) \rightarrow A(Fx, a)$ and $...
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Extension of functor between small categories to presheaves - proof check

On Sheaves in Geometry and Logic MacLane and Moerdijk show (Theorem VII.2.2 pag. 359) that a functor $F:\mathbb{C}\rightarrow\mathbb{D}$ between small categories extends to an essential geometric ...
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1answer
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MacLane's proof in CWM that every set valued functor is a colimit of representable functors- why is his natural transformation well defined?

I'm using MacLane as a reference for some of the category theory sections in notes I'm writing on simplicial sets and came across this proof but have been unable to understand it (It is on page 76 of ...
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Yoneda lemma natural isomorphism proof question

From Mac Lane: Let $K: D \rightarrow \text{Set}$ be a functor with $D$ having small hom sets, the bijection: $y : \text{Nat}(D(r,-), K) \cong Kr$ by $\alpha : D(r,-) \rightarrow K$ to $\...
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1answer
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Understanding the proof of Yoneda Embedding

Definitions Recently I was going through the proof Yoneda Embedding from Steven Roman's An Introduction to the Language of Category Theory. But before I go into the problem that I am having with the ...
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Cats & Sheaves, understanding one paragraph related to Yoneda Lemma

Assume that $F \in C^{\wedge}$ is represented by $X_0 \in C$. Then $\text{Hom}_{C^{\wedge}}(h_C(X_0, F)) \simeq F(X_0)$ gives an element $s_0 \in F(X_0)$. Moreover, for any $Y \in C$ and $t \in ...
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What does Yoneda's Lemma tell us about a representable functor?

Someone told me that a functor being representable is good because we can use Yoneda's Lemma. But I'm not sure how Yoneda's Lemma tells us anything new given that a functor is representable. For ...
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Is there only one isomorphism $h_C(X) \simeq h_C(Y)$ if it exists?

I'm trying to prove that if $F \simeq h_C(X)$ or "$X$ represents the functor $F$", then $X$ is unique up to unique isomorphism. I already know that if $h_C(X) \simeq F \simeq h_C(Y)$ that $s: X \...
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The isomorphism $F \simeq h_C(X)$ determines $X$ how?

I've asked something similar before, here. But I didn't quite understand their reasoning. So I'm breaking the problem down. First of all, how is $X$ determined? By yoneda $\text{Hom}_{C^{\wedge}}(...
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“Cats & Sheaves” defines $C_A$ w.r.t $F : C \to C'$ for object $A \in C'$, but then doesn't define $C_F$…

Let $F: C \to C'$ be a functor and let $A \in C'$. The category $C_A$ is given by $\text{Ob}(C_A) = \{(X, s); X \in C, s : F(X) \to A \}$ and $\text{Hom}_{C_A}((X,s), (Y, t)) = \{ f \in \text{Hom}...
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Representability $h_C(X) \simeq F$ of functors determines rep. $X$ up to unique isomorphism (using Yoneda Lemma).

A functor $F \in C^{\wedge}$ from $C^{op}$ to $\text{Set}$ is representable if there is an isomoprhism $\varphi \in \text{Hom}_{C^{\wedge}}(h_C(X), F)$ for some $X \in C$, we can also write that as $...
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How does $h_C(f) : h_C(X) \to h_C(Y)$ an isomorphism imply $f$ is (Yoneda Lemma)?

This is from Categories & Sheaves by Kashiwara & Schapira. Let $C$ be a category, $f : X \to Y$ a morphism in $C$. Assume that for each $W \in C$, the morphism $\text{Hom}_C(W, X) \...
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Yoneda Lemma (again)

Let $C^{\wedge} = \text{Fct}(C^{\text{op}}, \text{Set})$. Let $\text{h}_C : C \to C^{\wedge}, \ X \mapsto \text{Hom}_C(\cdot, X)$. Then the Yoneda lemma is: For $A \in C^{\wedge}$ and $X \in C$, $...
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Yoneda lemma, wrong direction, weighted colimit

Here in the definition of weighted colimit, it seems to me that the r.h.s. $C(W⋅F,c)≅Set^{J^{op}}(W,C(F−,c))$ has a wrong direction for the application of the Yoneda lemma.In fact, the Yoneda lemma ...
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Yoneda's Lemma and $K(G,n)$

Let $G$ be an abelian group. Using the universal coefficient theorem and the Hurewicz theorem we can prove: $$ H^n(K(G,n),G) \cong Hom(H_n(K(G,n),G),G) \cong Hom(\pi_n(K(G,n)),G) \cong Hom(G,G)$$ ...
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Yoneda lemma - Examples of locally small categories [closed]

I am looking for examples of locally small categories where one can talk about Yoneda lemma and produce some interesting(personal choice) results. One such example is : Let $G$ be a group. We ...
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Yoneda lemma as a generalisation of Cayley's theorem

I have seen answers in questions asking the same question. They have first described what is Yoneda lemma and then deduced Cayley's theorem from that. I am not asking for that. I am planing to ...
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Using Yoneda lemma, the pair $(U, s)$ representing presheaf $F$ are unique up to unique isomorphism.

I'm quoting from the Stacks Project: Definition 4.3.6. A contravariant functor $C \to \text{Sets}$ is said to be representable if it is isomorphic to the functor of points $h_u$ for some object $u$ ...