A common theme in mathematics is to replace the study of an object with the study of some category that can be built from that object. For example, we can
- replace the study of a group with the study of its category of linear representations,
- replace the study of a ring with the study of its category of -modules,
- replace the study of a topological space with the study of its category of sheaves,
and so forth. A general question to ask about this setup is whether or to what extent we can recover the original object from the category. For example, if is a finite group, then as a category, the only data that can be recovered from is the number of conjugacy classes of , which is not much information about . We get considerably more data if we also have the monoidal structure on , which gives us the character table of (but contains a little more data than that, e.g. in the associators), but this is still not a complete invariant of . It turns out that to recover we need the symmetric monoidal structure on ; this is a simple form of Tannaka reconstruction.
Today we will prove an even simpler reconstruction theorem.
Theorem: A group can be recovered from its category of -sets.
The idea of the proof is that we want to find a categorical property that allows us to isolate the subcategory of transitive -sets. The -set itself can be uniquely identified among transitive -sets as the “largest” one (more precisely, it is the unique weak initial object among transitive -sets), and then can be recovered as the opposite group of the group of automorphisms of (as a -set).
The categorical property we want is the following.
Definition: An object in a category is connected if the representable functor preserves coproducts.
The idea behind the definition is that if one thinks of coproducts as a disjoint union, then a morphism from a connected object into a disjoint union of objects must land entirely in one of the objects, or else it will be “disconnected” by the fact that it’s spread out over a disjoint union.
Example. In , the connected objects are precisely the one-element sets. Note that the empty set is not connected.
Example. In (which we’ll take to be the category of simple graphs), the connected objects are precisely the connected graphs in the usual sense. Note that the empty graph is not connected.
Example. In , the connected objects are precisely the connected topological spaces in the usual sense. Note that the empty space is not connected.
Example. In , any affine scheme such that has nontrivial idempotents (in other words, such that is not a connected ring) is not connected in the categorical sense. To see this, let contain a nontrivial idempotent and consider the map induced by the ring homomorphism
This map factors through neither of the projections , from which it follows that is not connected. (One interpretation of this argument is that is the free commutative ring on an idempotent.)
The converse statement – that the spectrum of a connected ring is connected in the categorical sense – is false. For example, let be a homomorphism induced by a non-principal ultrafilter on an infinite set . Then the corresponding morphism does not factor through any of the inclusions, so is not connected in the categorical sense. More generally, any ring which can be obtained as a nontrivial ultraproduct does not have connected spectrum in the categorical sense. However, it is true that if is a connected ring, then preserves finite coproducts.
Note that the spectrum of the zero ring is not connected. In general, the initial object of a category is never connected; it is too simple to be simple.
Intuitively, a nontrivial coproduct should not be connected. The following result shows that this is true under reasonable hypotheses. These hypotheses don’t hold for but do hold for , which is enough; I don’t know how much they can be relaxed.
Proposition: Let be two objects, neither of which is the initial object, of a concrete category with finite coproducts such that the forgetful functor preserves finite coproducts. Then is not connected.
Proof. We prove the contrapositive, namely that under the hypotheses above, if is connected then one of is the initial object. Recall that “preserves coproducts” means the following, for binary coproducts. If are objects, the natural inclusions induce natural maps which in turn induces a natural map
and to say that preserves binary coproducts means that this map is always a bijection for all . In particular, every morphism factors through one of the inclusions .
Now let . Suppose that is connected; then WLOG the identity map factors through . It follows that the inclusion map is a split epimorphism. However, by assumption, the map on underlying sets is an injection in , and since faithful functors reflect monomorphisms, it follows that the inclusion map is a monomorphism, hence an isomorphism. But looking at the corresponding natural isomorphism of representable functors , this is possible if and only if is the initial object.
(Note that a sufficient condition for to preserve finite coproducts is that it has a right adjoint. This is true of the forgetful functors from , and to .)
Proposition: The connected objects of are precisely the transitive -sets (the empty -set is not transitive).
Proof. Every -set can be expressed uniquely as a coproduct of transitive -sets (the orbits of the group action). It follows that a connected object of is necessarily transitive. Conversely, if is a transitive -set, then the image of any homomorphism from into another -set is necessarily also a transitive -set, hence contained in an orbit of .
Proposition: The -set is the unique (up to isomorphism) transitive -set which admits a morphism to all other transitive -sets.
Proof. Every transitive -set has the form for some subgroup of , so in particular admits a quotient map . On the other hand, if is not the trivial subgroup, then there exist no morphisms , since no map of sets can respect the -action (nontrivial elements of preserve the identity coset of but can’t preserve its image in ).
Theorem: A group can be recovered from its category of -sets.
Proof. We know that from the category of -sets we can recover the subcategory of transitive -sets, and we know that from the transitive -sets we can recover the -set itself as the unique weak initial object. The automorphism group of , as a -set, is , from which we can recover by taking the opposite group.