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.

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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.

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Compact objects

If R is a noncommutative ring, then Morita theory tells us that R cannot in general be recovered from its category \text{Mod}(R) of modules; that is, there can be a ring R', not isomorphic to R, such that \text{Mod}(R) \cong \text{Mod}(R'). This means, for example, that “free” is not a categorical property of modules, since it depends on a choice of ring R, or equivalently on a choice of forgetful functor.

It’s therefore something of a surprise that “finitely presented” is a categorical property of modules, and hence that it does not depend on a choice of ring R. The reason is that being finitely presented is equivalent to a categorical property called compactness.

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Projective objects

The goal of this post is to summarize some more-or-less standard facts about projective objects. A subtlety that arises here is that in abelian categories there are several conditions equivalent to being projective (we’ll list seven of them below) which are not equivalent in general. We’ll pay more attention than might be usual to this issue.

In particular, several times below we’ll give a list of conditions and a hypothesis under which they’ll be equivalent, and these conditions won’t all be equivalent in general. In these lists we’ll adopt the following convention: whenever we give a list of conditions and prove implications between them, the list will be organized so that proofs downward are easier and require fewer hypotheses, while proofs upward are harder and require more hypotheses. We’ll also prove more implications than we strictly need in order to see this more explicitly.

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The problem of finding solutions to Diophantine equations can be recast in the following abstract form. Let R be a commutative ring, which in the most classical case might be a number field like \mathbb{Q} or the ring of integers in a number field like \mathbb{Z}. Suppose we want to find solutions, over R, of a system of polynomial equations

\displaystyle f_1 = \dots = f_m = 0, f_i \in R[x_1, \dots x_n].

Then it’s not hard to see that this problem is equivalent to the problem of finding R-algebra homomorphisms from S = R[x_1, \dots x_n]/(f_1, \dots f_m) to R. This is equivalent to the problem of finding left inverses to the morphism

\displaystyle R \to S

of commutative rings making S an R-algebra, or more geometrically equivalent to the problem of finding right inverses, or sections, of the corresponding map

\displaystyle \text{Spec } S \to \text{Spec } R

of affine schemes. Allowing \text{Spec } S to be a more general scheme over \text{Spec } R can also capture more general Diophantine problems.

The problem of finding sections of a morphism – call it the section problem – is a problem that can be stated in any category, and the goal of this post is to say some things about the corresponding problem for spaces. That is, rather than try to find sections of a map between affine schemes, we’ll try to find sections of a map f : E \to B between spaces; this amounts, very roughly speaking, to solving a “topological Diophantine equation.” The notation here is meant to evoke a particularly interesting special case, namely that of fiber bundles.

We’ll try to justify the section problem for spaces both as an interesting problem in and of itself, capable of encoding many other nontrivial problems in topology, and as a possible source of intuition about Diophantine equations. In particular we’ll discuss what might qualify as topological analogues of the Hasse principle and the Brauer-Manin obstruction.

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The Picard groups

Let R be a commutative ring. From R we can construct the category R\text{-Mod} of R-modules, which becomes a symmetric monoidal category when equipped with the tensor product of R-modules. Now, whenever we have a monoidal operation (for example, the multiplication on a ring), it’s interesting to look at the invertible things with respect to that operation (for example, the group of units of a ring). This suggests the following definition.

Definition: The Picard group \text{Pic}(R) of R is the group of isomorphism classes of R-modules which are invertible with respect to the tensor product.

By invertible we mean the following: for L \in \text{Pic}(R) there exists some L^{-1} such that the tensor product L \otimes_R L^{-1} is isomorphic to the identity for the tensor product, namely R.

In this post we’ll meander through some facts about this Picard group as well as several variants, all of which capture various notions of line bundle on various kinds of spaces (where the above definition captures the notion of a line bundle on the affine scheme \text{Spec } R).

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Let \Sigma_g be a closed orientable surface of genus g. (Below we will occasionally write \Sigma, omitting the genus.) Then its Euler characteristic \chi(\Sigma_g) = 2 - 2g is even. In this post we will give five proofs of this fact that do not use the fact that we can directly compute the Euler characteristic to be 2 - 2g, roughly in increasing order of sophistication. Along the way we’ll end up encountering or proving more general results that have other interesting applications.

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