# Yoneda-Lemma as generalization of Cayley' s theorem?

I found the declaration that Yoneda-lemma is a generalization of Cayley' s theorem which states, that every team is isomorphic to a team of permutations.

Just how specifically does generalises Yoneda-lemma Cayley' s theorem? Can Cayley's theory be reasoned from Yoneda lemma, is it a generalization of a certain instance of Yoneda, or is this rather, a thoughtful declaration?

To me, it appears that Yoneda embedding is extra approved than Cayley's theory due to the fact that in the last you need to pick whether the team acts from the left or from the exactly on itself. Yet possibly this is a visual fallacy.

Just as Cayley's theory states that every team is a subgroup of a symmetrical team, Yoneda's lemma states that every (in your area tiny) group $C$ installs right into a group of functors specified on $C$.

Especially, Yoneda's lemma states that if $F:C\to Set$ is an approximate set - valued functor, after that $F(A) = Nat(\hom(A,-), F)$, to make sure that all-natural makeovers from a hom - set functor remain in bijective document with the components in the functor photo.

For this to be a straight generalization, we might take into consideration a team $G$ as a group $C_G$ (in fact a groupoid) with a solitary object $\ast$, and also each team component a morphism. After that, there is just a solitary object, so a set - valued functor coincides point as a set $S$ with a map from $G$ to $Bij(S,S)$, the set of bijections from $S$ to $S$. We can see this by unfolding the definition of set - valued functor : $\ast$ mosts likely to $S$, and also each component of $G$ mosts likely to some map $S\to S$ ; every one of which maps need to be bijections given that or else the team buildings endure.

Intend since we have some such $G$ - set $S$, Yoneda's lemma informs us that its components are bijective with all-natural makeovers from $G$ to $S$ ; so what is an all-natural makeover below? Our 2 functors are $S$, that takes $\ast$ to $S$, and also $\hom(\ast,-)$ that takes $\ast$ to $G$ ; both as collections. An all-natural makeover of these is a set - valued map from one photo to the various other, such that the 'noticeable' square of generated maps from morphisms in $C_G$ commutes - hence, $Nat(\hom(\ast,-),S)$ is the collection of $G$ - set maps from $G$ as a left $G$ - depiction to $S$ itself.

Among things Yoneda brings is a bijection $Nat(\hom(a,-),\hom(b,-)) = \hom(b,a)$ : the Yoneda embedding. Applied to the team scenario, this informs us that $\hom_G(G,G)=G$. Absolutely, left reproduction by a component is a team endomorphism ; and also what this informs us is that these are all there is. This bijection is specifically what is made use of in the evidence of Cayley's theory on the wikipedia web page.

I can additionally add that both Yoneda and also Cayley are outcomes which adhere to from the basic ideology of exploring algebraic frameworks by allowing them act upon themselves.

1) If you allow a team $G$ act upon itself, you understand it as a subgroup of $\mathfrak{S}_n$.

2) If you allow a ring with device act upon itself, you understand it as a subring of $E$, where $E$ is the underlying additive team.

3) Similarly if you allow a $k$ - algebra act upon itself, you understand it as a subalgebra of $\mathcal{M}_n(k)$. Specifically this offers the timeless understanding of $\mathbb{C}$ as a matrix algebra over $\mathbb{R}$ and also of the quaternions as a matrix algebra over $\mathbb{C}$ or $\mathbb{R}$.

4) You can allow a Lie algebra act upon itself, yet however this activity need not be loyal (Lie algebra do not have devices.). So you just get the very easy very first step of Ado's theory concerning installing Lie algebras right into mtraix algebras.

5) If you allow a group $\mathcal{C}$ act upon itself, you get an embedding right into $Fun(\mathcal{C}^{op}, Set)$, which is the web content of Yoneda's lemma.

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