objects such that commutes with certain colimits
In this article, we collect some results about representable functors (where is some base of hom-enrichment) that preserve coproducts.
When is an extensive category regarded as enriched in , we call a connected object, and this terminology matches well one’s intuition about connectedness from familiar cases such as , or the category of graphs, etc. Some basic results with proofs may be found at connected object, including
A connected colimit (i.e., a colimit over a connected diagram of) connected objects is connected.
If is connected and is epic, then is connected.
For non-extensive categories (e.g., categories of modules), the relation to “connectedness” tends to be less intuitive1. Nevertheless, the concept of (arbitrary) coproduct-preserving representable remains important, and it is useful to collect some basic information.
In an Ab-enriched category, finite coproducts are absolute colimits, hence are preserved by every representable. Thus, the interest is in representables which preserve infinite coproducts as well (which is what most of this page is about).
Let denote the category of right modules over a ring , and suppose an object induces a coproduct-preserving hom-functor . Various names for such appear in the literature, “compact” and “dually slender” among them. In any event, a question arose on MathOverflow as to whether or to what extent this condition coincides with the condition of being finitely generated, for various rings .
The information provided below was mostly culled from the answers given to that question, especially those by Pierre-Yves Gaillard and Fernando Muro. Here and there some minor details and background information have been filled in, and some key results have been slightly rearranged.
We begin with some easy preliminary remarks. Given a family of objects in an -enriched category and a functor , there is a canonical arrow
and if this arrow is an isomorphism for every family , we say preserves coproducts. Turning to the case of representable functors on modules, let
be the obvious projection (, else ), and given , put
Then preserves the particular coproduct if for each , we have for all but finitely many .
Clearly preserves coproducts, and if , are coproduct-preserving functors , then so is . It follows that
If preserves coproducts and is epic, then preserves coproducts.
Given , we have for all but finitely many , whence for all but finitely many since is epic.
Combining the two preceding observations, we infer that
Here is a sharper description of coproduct-preserving representables, based on subobject lattices.
preserves coproducts if and only if the union of every countable chain of proper submodules of is a proper submodule.
(As adapted from Gaillard’s answer.) Let be a chain of proper submodules of whose union is , and put . Since for each we have for all but finitely many , the map
corresponding to the tuple of quotient maps , factors through the inclusion . However, since each is nonzero, does not belong to the subgroup
and thus the canonical map is not an isomorphism.
In the other direction, if does not preserve coproducts, then we can find some map
not belonging to the subgroup . This means that infinitely many components are nonzero. Choose a countable subset such that is nonzero for every , and put
Each is a proper submodule of , and the form a nondecreasing chain, but the union of the is (because for each , only finitely many can be nonzero).
Let be a Noetherian ring, and suppose is a monomorphism of -modules. Then if is coproduct-preserving, so is .
(As adapted from Muro’s answer.) Consider a family of modules, and a map . Since there are enough injectives, there exists an embedding in an injective module, for each . Next, as explained here, the Noetherian assumption allows us to infer that is injective. Thus, there exists such that the diagram
commutes. Because preserves coproducts, we have for all but finitely many . Since the diagram
commutes and is injective, we see for all but finitely many , whence preserves coproducts.
Let be Noetherian. If preserves coproducts, then is finitely generated.
Next, we construct an example of a ring and an -module such that preserves coproducts but is not finitely generated.
A module is finitely generated if and only if the union of a totally ordered family of proper submodules of is a proper submodule.
The following proof is a practically verbatim transcription from Gaillard’s answer, modulo some notational changes. Assume that is not finitely generated. Let be the set of those submodules of such that is not finitely generated, ordered by inclusion. Clearly the poset is nonempty and has no maximal element. By (the contrapositive of) Zorn’s Lemma, there is a nonempty totally ordered subset which has no upper bound. Letting be the union of the submodules occurring in , we see that is finitely generated. There is thus a finitely generated submodule of which generates modulo . Then the collection
is a totally ordered set of proper submodules whose union is .
Thus, our task is to construct a ring and an -module such that every countable chain of proper submodules of is bounded above by a proper submodule of (cf. Theorem 2), but admitting an uncountable chain of proper submodules whose union is (so that is not finitely generated, by the preceding lemma 1). The solution as presented below is essentially from the answers of Gaillard and Brandenburg, with a few extra glosses.
The task is more or less straightforward if we just remember that valuation rings can model arbitrarily complicated rates of growth, i.e., in the present case, we just want to build a valuation field with uncountable supply of rates of growth (or of degrees of infinite/infinitesimal elements). Thus, consider for example the free abelian group generated by the first uncountable ordinal, and make this a totally ordered group by imposing the lexicographic order. This is the value group of a valuation field whose elements are Hahn series, formally described as functions
whose support is well-ordered when considered as a subset of the opposite order . (More suggestively, we think of as a formal series
where and is an indeterminate viewed as a generic infinite element, with obvious rules for adding and multiplying. The well-ordering condition is used to ensure that the rules for addition and multiplication are well-founded, and the value of such a series is the least lying in the support of . That is to say, the greatest that indexes a non-zero coefficient .)
Now let be the valuation ring consisting of bounded elements of , i.e., those in where whenever is greater than the identity element of . Then is the field of fractions of , and we may regard as an -module.
For each , let be the -submodule of consisting of all those in such that . These are “principal fractional ideals”, and they form a system of proper submodules which is cofinal in the lattice of -submodules of , ordered by inclusion.
By cofinality, any countable chain of submodules of is bounded above by some , so that preserves coproducts. However, the union of all the is , so that is not finitely generated.
The divide between extensive categories and categories of modules is somewhat analogous to the divide between the classical particles and quantum particles. In the classical picture, if an elementary particle (which we can think of as “connected”) is in a state described by a union , then it is either in state or in state . Whereas in the quantum picture, one has to consider superpositions of states , , and the usual sort of classical logic of disjunctions breaks down. Notice that classical logic (meaning here, non-quantum logic) is largely derived from our experience with extensive categories such as toposes. ↩