Tiny objects

Annoying Precision

The starting observation of Morita theory is that the abelian category $latex text{Mod}(R)$ of (right) modules over a (not necessarily commutative) ring $latex R$ does not uniquely determine $latex R$, since for example we always have Morita equivalences of the form

$latex displaystyle text{Mod}(R) cong text{Mod}(M_n(R))$.

Determining $latex R$ is equivalent to isolating the module $latex R in text{Mod}(R)$ (regarded as a module over $latex R$ via right multiplication), from which we can recover $latex R$ as its endomorphism ring. In some sense what this tells us is that $latex R$ cannot always be isolated in $latex text{Mod}(R)$ by a categorical property.

The next best thing we can try to do is to classify all of the rings $latex S$ such that $latex text{Mod}(R) cong text{Mod}(S)$ by isolating the corresponding modules $latex S in text{Mod}(R)$ by some categorical property. The crucial property turns out to be that the hom functor

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