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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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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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} &\...
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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 ...
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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\...
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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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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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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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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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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{...
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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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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 ...
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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
$$x:yC\...
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Cats in context Lemma 2.4.7.
I'm failing to understand the argument behind Lemma 2.4.7. in page 86 from Emily Riehl's Category Theory in Context.
In particular along the following reasoning:
Here $\int F$ is the category of ...
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Intuition for density formula for Pre-sheaves/Co-Yoneda lemma
I was reading Fosco's notes coend calculus, so far I am at Ninja-Yoneda lemma.
The density formula for pre-sheafs says that,
$${K} V\cong \int^{U\in C} K U\times \textstyle\text{Hom}_{C}(V,U)\cong\...
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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 $\mathbb C$. ...
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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 ...
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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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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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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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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(...
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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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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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