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# 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} &\...
1 vote
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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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### 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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### 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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1 vote
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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 $\mathbb C$. ...
1 vote
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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 ...
1 vote
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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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### 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 $Hom(I, X)$, ...
1 vote
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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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### 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 ...
1 vote
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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 ...
1 vote
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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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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 $[C, Set](... 1 vote 2 answers 72 views ### Specific question about adjunction of functors Suppose to have three functors such that$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$in ... 3 votes 1 answer 319 views ### Consequence of the Yoneda lemma Take a functor$G:\mathbb D\to \mathbb C $, and call$Y:\mathbb D\to [\operatorname{\mathbb D,Set]}:D\mapsto\operatorname{Hom}_{\mathbb D}(-,D)$the Yoneda functor. Consider the bifunctor$\...
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 ...
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'...