I was reading the book of McLane about Categories. In Yoneda's lemma he shows that a functor $F$ evaluated in a object $O$ can be written as the set of natural transformations between the representable functor of $O$ and $F$. My question is about a particular application of this theorem. Let $C$ be the category of finite rings with the limits of subcategories of $C$. Let $A$ be a finitely generated ring over the integers. Then evidently the tensor product with $A$ generates naturally a functor from $C$ to the category of sets. Let $F$ be this functor. My question is the following:

Is posible reconstruct $A$ as the set of natural transformacions from $F$ to itself?.

If this is not possible then:

Does there exist a category generated by finite objects and a functor $F$ such that we can reconstruct $A$ as a set of natural transformation from $F$ to itself?.

  • $\begingroup$ I don't understand the sentence "Let $C$ be the category of finite rings with the limits of subcategories of $C$". Is $C$ the category of finite rings? Anyway, the first thing that comes to mind is to let $C'$ be the category of finitely-presentable rings. Then for any ring $A$, $Hom(-,A): {C'}^{op} \to Set$ is a functor from which $A$ can be reconstructed; in fact, the functor $Rng \to Fun({C'}^{op}, Set)$, $A \mapsto Hom(-,A)$ is fully faithful. Note that finite rings have certain restrictions, e.g. they are torsion, so they are not so great for probing rings of characteristic zero. $\endgroup$ – Tim Campion Jan 21 '19 at 17:28
  • $\begingroup$ In principle I'm interested in categories $C$ such that given two objects $A$, $B$ in $C$, then $Hom(A,B)$ is a finite set. Not necesary ring or even groups. In my question I thought about $C$ as the category of finite rings. For example $Z_p$ the p-adics are a limit of finite rings, but this ring has to much elements. My question is about if there exist a way to glue the finite rings to reduce the elements in the limit. In fact I want to know if the algebraic structures can be descomposed as a game of Lego in finite structures. $\endgroup$ – camilo Jan 21 '19 at 21:04

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