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Generators

Previously we proved a theorem due to Gabriel characterizing categories of modules as cocomplete abelian categories with a compact projective generator, where “generator” meant “every object is a colimit of finite direct sums of copies of the object.”

But we also used “generator” to mean “every object is a colimit of copies of the object,” and noted that these conditions are not equivalent: as this MO question discusses, the abelian group $\mathbb{Z}$ satisfies the first condition but not the second. More generally, as Mike Shulman explains here, there are in fact many inequivalent definitions of “generator” in category theory.

The goal of this post is to sort through a few of these definitions, which turn out to be totally ordered in strength, and find additional hypotheses under which they agree. As an application we’ll restate Gabriel’s theorem using weaker definitions of “generator” and give a more explicit description of all of the rings Morita equivalent to a given ring.

Tiny objects

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

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

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

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

$\displaystyle \text{Hom}(S, -) : \text{Mod}(R) \cong \text{Mod}(S) \to \text{Ab}$

is faithful and preserves colimits. An object with this second property is called a tiny object, and in this post we’ll discuss how this condition behaves with an eye towards better understanding Morita equivalences. Along the way we’ll prove a theorem due to Gabriel characterizing categories of modules among abelian categories.