Compact objects

If is a noncommutative ring, then Morita theory tells us that cannot in general be recovered from its category of modules; that is, there can be a ring , not isomorphic to , such that . This means, for example, that “free” is not a categorical property of modules, since it depends on a choice of ring , 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 . The reason is that being finitely presented is equivalent to a categorical property called compactness.

Definition and examples

Let be a category which has filtered colimits.

Definition: An object in is compact if preserves filtered colimits.

The terminology stems from the fact that a topological space is compact in the usual sense iff it is compact as an object in the category of open subsets of . In general, compactness can be thought of as a finiteness condition on an object : it means that cannot “spread itself out” too much. One way to make this precise is that if is compact, then any morphism from to a filtered colimit factors through some .

As expected of a finiteness condition, compactness is preserved by finite constructions.

Proposition: A finite colimit of compact objects is compact.

Proof. Maps out of a finite colimit is a finite limit, and finite limits commute with filtered colimits in .

Since infinite limits do not in general commute with filtered colimits in , there is no reason to expect that compactness is preserved by infinite colimits. On the other hand, since filtered colimits are precisely the colimits which commute with finite limits in , compactness is the strongest colimit-preserving condition on that is itself preserved by taking finite colimits.

Example. In , the one-point set is compact since is the identity functor. The finite colimits of copies of are precisely the finite sets, so finite sets are also compact. These are in fact all of the compact objects. The reason is that every set is the filtered colimit of its finite subsets, so if is a compact set then the identity factors through a finite subset of . It follows that is a retract of a finite set, hence is itself finite.

Example. In , the free abelian group is compact since represents the forgetful functor from groups to sets, which preserves filtered colimits. Starting from and iterating finite colimits, we get precisely the finitely presented abelian groups, so these are also compact. These are in fact all of the compact objects. The argument is nearly identical to the previous one: every abelian group is the filtered colimit of its finitely generated (hence finitely presented, since is Noetherian) subgroups, so if is a abelian group then the identity factors through a finitely presented subgroup of . It follows that is a retract of a finitely presented abelian group, hence is itself finitely presented.

Finitely presented objects

The argument given above is clearly very general; here is a very general setting in which to put it.

Let be a cocomplete category for which there exists an essentially small full subcategory of compact objects in such that every object in is a colimit of objects in . One might call a compactly generated category if it admits such an , although this isn’t standard and is used to refer to a different but related condition in the setting of model categories. I believe this condition is equivalent to the condition that is locally finitely presentable, although we won’t need this.

Example. Let be a small category. Then the category of presheaves on is compactly generated. can be taken to consist of the objects , or more precisely the corresponding representable presheaves; by the Yoneda lemma, morphisms out of a representable presheaf correspond to evaluation of presheaves on objects, and since colimits in functor categories are computed pointwise preserves not only filtered colimits but all colimits. And by the universal property of the Yoneda embedding, every presheaf is a colimit of representable presheaves.

Example. Let be the category of models of a Lawvere theory (so could be the category of groups, rings, or modules over a ring). Then is compactly generated. can be taken to consist of the finitely generated free objects . These are compact because they represent the finite powers of the forgetful functor , which preserve filtered colimits (see this previous post). And every object is the colimit of the canonical diagram of finitely generated free objects mapping into it; this diagram produces a presentation of in which every element of is used as a generator and every relation between elements of is used as a relation. Intuitively, the reason we can restrict our attention to finitely generated free objects rather than all free objects is that the operations in a Lawvere theory only involve finitely many elements at a time.

Fix a particular choice of .

Definition: An object of is finitely presented if it is an iterated finite colimit of objects in .

This terminology requires some justification.

Proposition: When is the category of models of a Lawvere theory and is the finitely generated free objects, then the finitely presentable objects in the above sense are precisely the finitely presentable objects in the usual sense.

Proof. “The usual sense” means an object which can be written as a coequalizer

where denotes the free object on generators. Here the are the generators and the pair of maps from are the relations between generators. By definition, such an object is therefore finitely presentable in the above sense. Using the coequalizer-of-coproducts construction of colimits, being finitely presentable in the usual sense is equivalent to being expressible as a single (not iterated) finite colimit of s.

From here it remains to show that the condition of being finitely presented in the usual sense is closed under finite colimits. Since the coproduct of coequalizers is the coequalizer of coproducts, being finitely presented in the usual sense is closed under finite coproducts. It remains to show that it is closed under coequalizers, and this reduces to two observations.

First, recall that when computing the coequalizer of two maps we can precompose with an epimorphism and get the same coequalizer. In the category of models of a Lawvere theory, this tells us that when computing a coequalizer where the source is finitely presented in the usual sense, we can precompose with an epimorphism from an , and hence we may assume WLOG that the source is an .

Second, suppose are a pair of maps and is finitely presented in the usual sense, hence is itself the coequalizer of a pair of maps . Because free objects are projective (in the sense of this previous post, and because the forgetful functor preserves epimorphisms), lift to a pair of maps . Now the claim is that can be computed as the coequalizer of the maps

.

This is just a matter of verifying that the two coequalizers have the same universal property. A map out of this coequalizer is a map out of whose pullback to resp. coequalizes the maps resp. . The second condition implies that this map factors through , and since are lifts of , the first condition implies that this map, when regarded as a map out of , coequalizes . But this new coequalizer, unlike the original one, is finitely presented in the usual sense, and the conclusion follows.

By construction, the finitely presented objects are compact and closed under finite colimits. We’d like to show that in fact, with the above hypotheses, they are precisely the compact objects. The key result will be the following.

Theorem: With the above hypotheses, every object of is a filtered colimit of finitely presented objects.

Proof. Let be written as a colimit where is a (small) diagram category and is the corresponding diagram, and moreover each . The coequalizer-of-coproducts description of this colimit is

.

This captures exactly the universal property that a map out of is a collection of maps out of each such that for every morphism in we have

.

The point of the proof is that this condition can be checked one morphism at a time, and so it can be checked on the finite subgraphs of . More precisely, let be the diagram of inclusions among finite subgraphs of . Then is not only a filtered category but in fact a directed poset, since the union of two finite subgraphs is another finite subgraph. (Note that we can’t replace “finite subgraph” with “finite subcategory” here, since some categories may have no interesting finite subcategories.)

If is a finite subgraph, write for the same coequalizer as above, but only involving objects and morphisms in . This is a slight abuse of notation since is not a category, but it is still true that this is a coequalizer of a pair of maps between finite coproducts of objects in , and hence is finitely presented. We want to show that is canonically isomorphic to

.

But this is clear: a morphism out of this colimit is a compatible family of morphisms out of for each finite subgraph . Compatibility implies that this family of morphisms assigns a well-defined morphism out of each (that is, that does not depend on which finite subgraph we regard as living in), and also that for every morphism in (which occurs in some finite subgraph containing and ) we have

and no other conditions are imposed, so this colimit has the same universal property as the original colimit.

Note that in groups, rings, and modules there is a more obvious version of this result, which is that every object is a filtered colimit of finitely generated objects: in fact every object is the filtered (even directed) colimit of its finitely generated subobjects. A finitely generated subobject need not be finitely presented, e.g. in the case of modules over non-Noetherian rings, but intuitively, in the above proof we get around this by imposing the necessary relations one at a time.

Corollary: With the above hypotheses, the compact objects of are precisely the finitely presented objects. In particular, with the above hypotheses, the notion of a finitely presented object does not depend on the choice of .

Proof. Let be a compact object. Write as a filtered colimit of finitely presented objects . Then the identity morphism factors through some . It follows that is a retract of a finitely presented object, hence is itself finitely presented.

Corollary: Let be a ring. Then the compact objects of are precisely the finitely presented modules. In particular, the notion of a finitely presented module does not depend on the choice of .