I'm new to category theory.
I am trying to understand how Yoneda Lemma is a generalization of representations theorems in algebra.
Cayley theorem can be interpreted as a instance for the case where the category $\mathcal{C}$ is the one with only one element $*$ and arrows are iso arrows that correspond to the elements of $G$ (i.e., it is a grupoid. I am using this wiki-page as reference) . From the Yoneda Lemma we can establish the Cayley theorem.
But there is another result from algebra that says that there isn't a representation theorem for Rings in general. (I don't understand why this is the case, I just heard it). There is only a representation theorem if we restrict to the context of Boolean algebras (the Stone Theorem).
Why we can't just apply Yoneda lemma in some way to get a general result for Ring theory? (Not the Stone theorem) I mean, in the context of groups we've used the grupoid category. Why we can't consider something like it in Rings and apply the Yoneda? Some category like a “ringoid'' (sorry for this word).
Maybe there is to much happening here, but I think some intuition of this might help understand better the nature of Yoneda Lemma.
Edit: the discussion in the comments and in this link pointed me that the result I mentioned is false: there is a Cayley on Rings. I will not edit the original text question. So now I think the question is:
How do we get this version of Cayley on Rings with Yoneda?
I still very confused with all this new information, and I didn't understand the answer there.
Best Answer
A preadditive category is a category $\mathcal{A}$ together with the structure of an abelian group of each set $\mathcal{A}(A, B)$, in such a way that that all composition functions $$ \mathrm{Hom}_{\mathcal{A}}(A, B) × \mathrm{Hom}_{\mathcal{A}}(B, C) \longrightarrow \mathrm{Hom}_{\mathcal{A}}(A, C) \,, \quad (f, g) \longmapsto g ∘ f $$ are $ℤ$-bilinear. More explicitly, $$ g ∘ (f_1 + f_2) = g ∘ f_1 + g ∘ f_2 \,, \quad (g_1 + g_2) ∘ f = g_1 ∘ f + g_2 ∘ f \,. $$
For any object $A$ in a preadditive category $\mathcal{A}$, the set $\mathrm{End}_{\mathcal{A}}(A)$ becomes a ring: its underlying abelian group is $\mathrm{Hom}_{\mathcal{A}}(A, A)$, and its multiplication is given by composition of endomorphisms.
There are two important examples of preadditive categories are the following:
The category $\mathbf{Ab}$ of abelian groups.
Every ring $R$ can be regarded as a preadditive category $\mathcal{R}$ consisting of only a single object $\ast$, with $\mathrm{End}_{\mathcal{R}}(A) = R$.
For any preadditive category $\mathcal{A}$, its opposite category $\mathcal{A}^{\mathrm{op}}$ is again preadditive.
A functor $F \colon \mathcal{A} \to \mathcal{B}$ between preadditive categories $\mathcal{A}$ and $\mathcal{B}$ is called additive if for any two objects $A$ and $B$, the map $$ \mathrm{Hom}_{\mathcal{A}}(A, B) \xrightarrow{\enspace F \enspace} \mathrm{Hom}_{\mathcal{B}}( F(A), F(B) ) $$ is a homomorphism of groups. If $F$ is such an additive functor, then for every object $A$ of $\mathcal{A}$, the map $$ \mathrm{End}_{\mathcal{A}}(A) \xrightarrow{\enspace F \enspace} \mathrm{End}_{\mathcal{B}}(F(A)) $$ is a homomorphism of rings.
An important example of additive functors are represented functors: for every object $A$ of a preadditive category $\mathcal{A}$, both $$ \mathrm{Hom}_{\mathcal{A}}(A, -) \colon \mathcal{A} \longrightarrow \mathbf{Ab} $$ and $$ \mathrm{Hom}_{\mathcal{A}}(-, A) \colon \mathcal{A}^{\mathrm{op}} \longrightarrow \mathbf{Ab} $$ are additive.
Given two functors $$ F, G \colon \mathcal{A} \longrightarrow \mathcal{B} $$ with $\mathcal{B}$ preadditive, the set of natural transformation from $F$ to $G$, denoted by $\mathrm{Nat}(F, G)$, becomes an abelian group with addition given by $$ (α + β)_A ≔ α_α + β_A $$ for every two natural transformations $α$ and $β$ and every object $A$ of $\mathcal{A}$.
It follows that the functor category $[\mathcal{A}, \mathcal{B}]$ inherits a preadditive structure from $\mathcal{B}$.
One can now state a version of Yoneda’s lemma for preadditive categories:
As a consequence, one gets a Yoneda embedding for preadditive categories.
As a consequence, one find that for every object $A$ of a preadditive category $\mathcal{A}$, the map $$ \mathrm{End}_{\mathcal{A}}(A) \longrightarrow \mathrm{End}_{[\mathcal{A}^{\mathrm{op}}, \mathbf{Ab}]}( \mathrm{Hom}_{\mathcal{A}}(-, A) ) $$ is an isomorphism of rings.
Given a ring $R$, we can form the corresponding preadditive category $\mathcal{R}$ consisting of only a single element $\ast$ with $\mathrm{End}_{\mathcal{R}}(\ast) = R$. A functor from $\mathcal{R}^{\mathrm{op}}$ to $\mathbf{Ab}$ is then “the same” as a right $R$-module. The right $R$-module corresponding to the represented functor $\mathcal{R}(-, \ast)$ is just $R$ itself.
Yoneda’s lemma gives us isomorphisms of rings $$ R = \mathrm{End}_{\mathcal{R}}(\ast) ≅ \mathrm{End}_{[\mathcal{R}^{\mathrm{op}}, \mathbf{Ab}]}( \mathrm{Hom}_{\mathcal{R}}(-, \ast) ) ≅ \mathrm{End}_{\mathrm{Mod}\text{-}R}(R) \,. $$ This overall isomorphism sends any element $r$ of $R$ to the map $$ R \longrightarrow R \,, \quad x \longmapsto r x \,, $$ which is indeed a homomorphism of right $R$-modules.
The ring $\mathrm{End}_{\mathrm{Mod}\text{-}R}(R)$ is a subring of $\mathrm{End}_{ℤ}(R)$. Therefore, every ring $R$ can be realized as a subring of $\mathrm{End}_ℤ(A)$ for some abelian group $A$. This is Cayley’s theorem for rings.