Previously we looked at several examples of -ary operations on concrete categories . In every example except two, was a representable functor and had finite coproducts, which made determining the -ary operations straightforward using the Yoneda lemma. The two examples where was not representable were commutative Banach algebras and commutative C*-algebras, and it is possible to construct many others. Without representability we can’t apply the Yoneda lemma, so it’s unclear how to determine the operations in these cases.
However, for both commutative Banach algebras and commutative C*-algebras, and in many other cases, there is a sense in which a sequence of objects approximates what the representing object of “ought” to be, except that it does not quite exist in the category itself. These objects will turn out to define a pro-object in , and when is pro-representable in the sense that it’s described by a pro-object, we’ll attempt to describe -ary operations in terms of the pro-representing object.
The machinery developed here is relevant to understanding Grothendieck’s version of Galois theory, which among other things leads to the notion of étale fundamental group; we will briefly discuss this.
Operations on finite groups
A general recipe for finding a pair where is not quite representable is to start with an example where is representable by some object and then restrict to a subcategory not containing that object. We’ll focus first on the example of the category of finite groups (the subject of the puzzle in the previous post) for concreteness.
The obvious forgetful functor is not representable: indeed, if is any finite group then we can find a nontrivial finite group such that there are no nonzero homomorphisms (for example by taking , so cannot represent the forgetful functor. However, we can come close: for any positive integer , the cyclic group represents the functor sending a group to its set of -torsion elements, and in some sense “as ” this gives us the forgetful functor. More precisely, the forgetful functor is the colimit
where the diagram containing the is the diagram of finite quotients of together with the natural quotient maps; this is because every element of a finite group is -torsion for some . (Recall that colimits in functor categories are computed pointwise, so the above notation denotes both a colimit of sets and a colimit of functors to .) It follows that at least the unary operations can be computed, by the universal property of colimits and the Yoneda lemma, as
where denotes the profinite integers, which are isomorphic to the direct product of the -adic integers over all primes (essentially by the Chinese remainder theorem). What this tells us is that is a unary operation on finite groups not only for every integer , but for every profinite integer .
What do these strange new operations actually look like? Roughly speaking, the profinite integers are the answer to the question “what happens if I try to specify an integer by specifying its remainder modulo all positive integers, but my requirements aren’t satisfied by any actual integer?” Such a formal object can still act sensibly on any given finite group , since to define we only need to know the value of , but it may act on a collection of finite groups in a way that an actual integer can’t. For an explicit example, consider the element of whose component at every prime other than is and whose component at is the -adic integer
The corresponding unary operation on finite groups acts as follows. Every element of a finite group generates a cyclic subgroup of some order . This cyclic group is the product of its Sylow -subgroups over all primes . On the Sylow -subgroups, , our new operation acts as the identity, but on the Sylow -subgroup our operation acts as exponentiation “to the power” in the sense that on -power torsion the map is invertible and the operation we have described computes its inverse. This follows from the observation that is the sum of the -adically convergent infinite series .
More generally, inherits a topology, the coarsest topology such that all projection maps are continuous (where the latter is given the discrete topology). A sequence converges with respect to this topology if and only if it converges -adically for all primes , and with respect to this topology is dense. Thus, for example, the series
converges -adically with respect to all primes , and consequently it defines an operation
which corresponds to an element of .
What about -ary operations? Here the diagram analogous to the diagram of the finite cyclic groups above is more complicated: it consists of all finite groups generated by elements, together with quotient maps preserving a particular choice of generators. Equivalently, the diagram is the diagram of all finite quotients of the free group , hence
where runs over all subgroups of of finite index. The same argument as before shows that the -ary operations can naturally be identified with
which gives the profinite completions of the free groups . The resulting Lawvere theory might be called the profinite completion of the Lawvere theory of groups. There is again a topology on , and we can write down elements of by writing down convergent sequences of elements of , such as
which corresponds to an element of .
A digression on representability
In the next several examples we’ll be showing that various forgetful functors fail to be representable. To do this it will be helpful to record the following corollary of the Yoneda lemma.
Proposition: Suppose is represented by some object . Then the element has the property that, in the natural identification , the element of associated to a given morphism is the image of under the induced map .
Intuitively, is the free object of on one element, and is that element. We’ll be showing that various are not representable by showing that there does not exist any object and element of capable of mapping to any element of some other .
The example of finite groups is unusually complicated in that the diagram representing doesn’t give a diagram representing in a straightforward way, the problem being that most coproducts don’t exist in the category of finite groups. In the following examples finite coproducts will exist and the situation simplifies.
Example. Let be the category of finite abelian groups. The unary operations are the same as for finite groups, but for -ary operations with we find that we can use a simpler diagram to represent , roughly speaking the coproduct of copies of the diagram for . More precisely, we have
since any -tuple of elements of a finite abelian group is -torsion for some . It follows that the -ary operations can be determined from the unary ones in a straightforward way: they are just given by the elements of .
Example. Let be the category of finite-dimensional -modules. (This example is closely analogous to .) The forgetful functor is the colimit over a rather large diagram, namely
where ranges over all nonzero polynomials and the maps are the obvious maps preserving . Equivalently, we’re looking at the diagram of all finite-dimensional quotients of as a -module. It follows that the unary operations can naturally be identified with the limit
or what one might call the “pro-finite-dimensional completion” of . The -ary operations are given by as above. Analogous to the case of , the Chinese remainder theorem shows that
is the ring consisting of families of formal power series expansions at each point in the complex plane. Note in particular that any holomorphic function has such a family of power series expansions; in other words, the ring of holomorphic functions is naturally a subring of , hence we recover abstractly the concrete fact that on a finite-dimensional -module we can not only apply polynomials in but appropriate power series in , in particular . This construction is relevant to the passage from finite-dimensional representations of a Lie algebra to finite-dimensional representations of a suitable Lie group with Lie algebra .
Example. Let be the category of finite -sets for a group. If is finite, it represents the obvious forgetful functor to and determining the -ary operations is straightforward. If is infinite, the forgetful functor will sometimes but not always be representable, but it will always be expressible as
where ranges over all normal subgroups of of finite index (the point being that if acts on a finite set then it necessarily acts through some finite quotient). It follows that the unary operations can naturally be identified with the limit
which is just the profinite completion of . In particular, if is infinite, then cannot be representable. (In particular, if is both infinite and embeds into , then cannot be representable. A group embeds into its profinite completion iff it is residually finite.) More generally, taking coproducts we have
since acts on any -tuple of elements of a finite -set through a finite quotient, and we find that the -ary operations are given by : more explicitly, by a unary operation on one factor.
Example. Let be the category of commutative nil rings; that is, commutative rings (necessarily non-unital) in which every element is nilpotent. The obvious forgetful functor is not representable: if is a nil ring and is any element, then for some , so cannot map to an element satisfying . The forgetful functor can be expressed as a colimit as follows: let be the free non-unital ring on a -step nilpotent element. Then
The same argument as before shows that the unary operations can naturally be identified with the limit
which gives the non-unital ring of formal power series with no constant term. Note that this is precisely the condition required for composition of power series to be well-defined. More generally, taking coproducts we have
since any -tuple of elements of a nil ring is -step nilpotent for some , and we find that the -ary operations can naturally be identified with the non-unital ring of formal power series in variables with no constant term.
If we think of the Lawvere theory of commutative rings as the category of affine spaces over (since this is the opposite of the category of free commutative rings on finitely many elements), then the Lawvere theory we obtain here is in some sense the category of formal neighborhoods of the origin in the affine spaces over .
Example. We return now to the case of commutative Banach algebras. The obvious forgetful functor is not representable: if is a Banach algebra and an element, then has some fixed norm , hence can map to elements of spectral radius at most . The forgetful functor can be expressed as a colimit of representable functors as follows. For each , let denote the algebra of functions on the closed disk of radius which are holomorphic in the interior and continuous on the boundary. These conditions are preserved by uniform limits, so equipped with the uniform norm is a Banach algebra. The holomorphic functional calculus shows that any element of a Banach algebra of spectral radius less than can be associated to a homomorphism so that the function given by the inclusion of the disk of radius into is sent to , although the converse doesn’t seem to hold (see the discussion at MO). Nevertheless, this is enough to ensure that
where the diagram over the is given by the natural restriction maps . It follows that the unary operations can be identified with
which can in turn be identified with the holomorphic functions . More generally, taking coproducts we have
since any -tuple of elements of a Banach algebra has spectral radius less than for some (where denotes the Banach algebra tensor product), and we find that the -ary operations can be identified with the holomorphic functions (thinking of as a suitable algebra of functions on the product of copies of the closed disk of radius ).
This result provides us with a definition of holomorphic function that makes no explicit mention of differentiability! However, the derivatives of a given operation can be determined by applying it to nilpotent elements: for example, the first derivatives of an operation can be determined by applying it to , once the latter is equipped with a suitable Banach algebra norm (which can be obtained by taking the norm induced from any embedding into a matrix algebra).
Example. The case of commutative C*-algebras is nearly identical to the above, except that the algebras should be replaced with the algebras of continuous functions on the closed disk of radius .
Directed colimits and pals
In the above examples, we dealt with forgetful functors that weren’t representable by writing them as a colimit of representable functors. It turns out that this can always be done.
Theorem: Let be a category. Any functor is canonically a colimit of representable functors.
“Canonically” means the following: if is any category, any set of objects in , and any object, there is a canonical diagram consisting of all morphisms from an object in to and all morphisms between objects in respecting these. (We constructed similar diagrams, but going the other direction, to describe the profinite completion, where was the category of groups and a set of objects containing at least one copy of each finite group.) is canonically the colimit of objects in if the morphisms exhibit as the colimit of this diagram.
Proof. It suffices to verify the universal property. The canonical diagram consists of all natural transformations together with all natural transformations between the respecting these. By the Yoneda lemma, these natural transformations can be identified with elements of and the natural transformations respecting these can be identified with morphisms sending some element of to some element of . (The shape of this diagram is the opposite of the category of elements of .) More use of the Yoneda lemma shows that a cocone over this diagram is precisely a natural transformation , and the initial such natural transformation is the identity . The conclusion follows.
This result doesn’t directly get us what we want (although it does lead to the useful intuition that a presheaf over a category is like a formal colimit of objects in that category; see free cocompletion); we would like to write functors as colimits of representable functors in order to compute natural transformations, but the colimit we get above doesn’t help us compute natural transformations at all (it just spits back the usual definition of a natural transformation). In practice, what we would like to do is write a functor as a colimit over a smaller and more manageable diagram of representable functors, and the above result at least assures us that there isn’t an obvious obstruction to doing this.
The diagrams we actually used above all had the following additional property. A directed set is a non-empty poset such that every pair of elements has an upper bound. Equivalently, thinking of such a poset as a category, for any two objects there exists an object together with morphisms . In particular, any non-empty poset with either suprema or a greatest element is directed. A directed colimit or inductive limit (this is older terminology and conflicts with the modern use of the term limit, so we will avoid it) is a colimit over a diagram over a directed set .
Directed colimits in most familiar categories are “increasing unions”: in particular, many forgetful functors preserve directed colimits even when they don’t preserve other colimits such as coproducts.
Example. is the directed colimit of its initial segments . This is true more generally for any limit ordinal.
Example. Every set is the directed colimit of its finite subsets.
Example. The group is the directed colimit of its finite subgroups . The former group may be regarded as the group of roots of unity and the latter as the subgroups of roots of unity.
Example. More generally, every group is the directed colimit of its finitely generated subgroups. Similar statements are true in other categories of algebraic objects, e.g. every ring is the directed colimit of its finitely generated subrings.
Example. A topological space is the directed colimit of its compact subspaces if and only if it is compactly generated. In particular, locally compact spaces have this property. Above we implicitly used the fact that is the directed colimit of the closed disks of radius for every .
Taking duals, a codirected set is a non-empty poset such that every pair of elements has a lower bound. A codirected limit or projective limit or inverse limit (the latter two are older terminology and, as before, we will avoid them) is a limit over a codirected set. We saw several such limits when computing operations above.
Example. The -adic integers are the codirected limit over the quotients of .
Example. The ring of formal power series over a base ring is the codirected limit over the quotients of .
Example. The profinite completion of a group is the codirected limit over the finite quotients of .
Example. Stone spaces, or profinite sets, are precisely the codirected limits of the finite discrete topological spaces in .
Codirected limits are just directed colimits in the opposite category, and this has some surprising applications.
Example. Part of Pontrjagin duality asserts that the functor induces an equivalence between the opposite of the category of abelian groups and the category of compact (Hausdorff) abelian groups. Since every abelian group is the directed colimit of its finitely generated subgroups, which are direct sums of and the finite cyclic groups, it follows that every compact abelian group is a codirected limit of products of the Pontrjagin duals of and the finite cyclic groups, namely and the finite cyclic groups. (More generally, in the nonabelian case, the Peter-Weyl theorem has as a corollary that every compact (Hausdorff) group is a codirected limit of compact Lie groups.)
Subexample. is the directed colimit of its subgroups . It follows that its Pontrjagin dual can be identified with the codirected limit of a suitable diagram involving countably many copies of indexed by the positive integers. There is a morphism in this diagram whenever , and it is given by . The corresponding group is an example of a solenoid; see this math.SE question.
Subexample. Similarly, can be identified with the codirected limit of a suitable diagram involving the finite cyclic groups, in fact the same diagram whose limit is the profinite integers . Hence the latter is the Pontrjagin dual of the former. (This makes explicit something we ignored earlier, which is that has a natural (compact Hausdorff) topology given by taking the codirected limit in the category of topological groups rather than the category of groups.)
The intuition that directed colimits behave like increasing unions can be made precise for sets as follows.
Proposition: Let be a directed set and be a diagram of sets of shape . Then the directed colimit of is the quotient of the disjoint union by the following equivalence relation: an element is equivalent to another element if and only if there exists an object and two morphisms in such that .
Intuitively, two elements are equal if and only if they are “eventually equal” as we keep mapping them “forward” with respect to the directed set .
Proof. We will build the directed colimit out of coproducts and coequalizers in the usual way, namely as the coequalizer of
where denotes the set of all morphisms in , denotes the domain of a morphism , and act by the identity and by the induced map on components respectively, where denotes the codomain.
In , the coequalizer of a pair of morphisms is the quotient of by the equivalence relation generated by the relation for every . Applied to the above diagram, we conclude that for every , where is some morphism in , we want to identify with . Composition of morphisms guarantees that this relation is transitive, but to make it an equivalence relation we also need to make it symmetric. This means that two objects which are identified under the equivalence relation generated by this relation are related by a zigzag of morphisms
in general, and we want to show that when is directed this zigzag can be reduced to a cospan
by removing spans from any zigzag. Indeed, if is such a span, then since is directed we can find and maps such that the diagram
commutes (since any diagram commutes in a poset). This allows us to replace the span in any zigzag with (if there are parts of the zigzag to the left of or to the right of mapping into them then we can compose with the maps ), and the conclusion follows.
(What we used above is sometimes called the diamond property. The diamond property figures prominently in various versions of the diamond lemma, one version of which has applications such as the Poincaré–Birkhoff–Witt theorem; see, for example, this blog post by Ben Webster.)
Arguably the most important property of directed colimits is the following.
Proposition: Directed colimits in commute with finite limits in .
Proof. By “commute” we mean the following. Let be a diagram in , where is a directed set and is finite. There are natural maps (where ) hence natural maps . These maps are compatible with the morphisms in , hence give a natural map
and we want to show that this map is an isomorphism. More abstractly, taking directed colimits of shape defines a functor , and we want to show that this functor preserves finite limits. It suffices to show that preserves binary products and equalizers.
First we will show that preserves binary products. be two diagrams of shape . Following the intuition that directed colimits in behave like increasing unions, we will abuse notation a little and write the colimits as unions and . There is a natural map
and we want to show that it is an isomorphism. In fact we can exhibit its inverse: if is an element of the RHS, so and for some , then by directedness we can find some equipped with maps which exhibit as elements of , hence is exhibited as an element of . This map doesn’t depend on the choice of by inspection, and it defines an inverse also by inspection. The conclusion follows.
The proof that preserves equalizers is essentially identical.
Directed colimits are not the most general class of colimit that commute with finite limits in . These would be colimits over filtered categories, or filtered colimits. A filtered category is a category in which not only does every pair of objects have an upper bound, but every pair of parallel morphisms is coequalized by some morphism. Everything we’ve said above about directed sets remains true for small or more generally essentially small filtered categories, and it seems to me that in practice there’s not much of a difference between directed colimits and filtered colimits: in particular, a category has all directed colimits iff it has all small filtered colimits, and a functor preserves all directed colimits iff it preserves all small filtered colimits. Below we will be saying “filtered” but thinking “directed.”
We can now justify the statement we made above that directed colimits behave like increasing unions in many familiar categories.
Corollary: Let be a Lawvere theory and let be its category of models. Then has all small filtered colimits, and the forgetful functor preserves them.
Proof. By definition, models consist of product-preserving functors . In the functor category , colimits are computed pointwise, hence all small colimits exist, and the forgetful functor is just evaluation on a particular object of , but the colimit of a diagram of product-preserving functors may not be product-preserving. However, the small filtered colimit of a diagram of product-preserving functors continues to be product-preserving (since we can commute the product past the filtered colimit), so the conclusion follows.
(It may be more satisfying to prove this result by hand, at least for specific examples like groups. Given a small filtered diagram of groups, take its filtered colimit as a diagram of sets, then define the group operation on this set, then verify that it satisfies the universal property. What makes everything work out is that the group axioms only ever involve finitely many elements of the filtered colimit at a time, so we can map any such finite collection of elements into a common group and work there.)
Back to operations
Say that a functor is pro-representable if it is a small filtered colimit of representable functors. (This is not entirely standard, but I think it’s the best option; see this math.SE question for discussion.) This was the case for all of the forgetful functors we examined above. The terminology comes from the idea that the thing that represents such a functor is not an object of but in fact a cofiltered diagram of objects in , and after converting these into functors we get a filtered diagram of objects in . Such a diagram is called a pro-object in , since we can think of it as a formal cofiltered limit of objects in . The following proposition generalizes the work we did in all of the examples above except .
Proposition: Let be a concrete category with finite coproducts such that is pro-representable by a diagram . Then is pro-representable by the coproduct of copies of , hence -ary operations are given by elements of the cofiltered limit
Proof. Write . Then
Since the colimit is filtered, it commutes with finite products, hence is also naturally isomorphic to
and the conclusion follows from the universal property of the colimit and from the Yoneda lemma.
How can we tell if a given functor could be pro-representable?
Proposition: If is pro-representable, then it is left exact (preserves finite limits).
Proof. Write as above where is a cofiltered diagram. Let be a diagram with finite. Then
Since representable functors and and filtered colimits both commute with finite limits, this is
and the conclusion follows.
Under mild hypotheses on , the converse holds.
Theorem (Grothendieck): Let be an essentially small category with finite limits. Then every left exact functor is pro-representable.
Proof. Since is canonically the colimit over a diagram of shape the opposite of the category of elements of , it suffices to show that is essentially small and filtered, or equivalently that is essentially small and cofiltered. Essential smallness follows straightforwardly from the essential smallness of . Furthermore, our assumptions imply that has finite limits, and any category with finite limits is cofiltered. More explicitly, if are two objects of , then they are mapped to by the element , and similarly if are two parallel arrows inducing parallel arrows in , then they are equalized by the equalizer .
The following corollary generalizes the work we did determining the operations on .
Corollary: Let be a category, an object in , and be an essentially small full subcategory of . Assume that is closed under finite limits in . Then the functor is pro-representable.
Proof. By assumption, any finite diagram in has a limit in which is contained in ; by fullness this is also a limit in , from which it follows that is left exact, and the result follows by the above.
We can also give a more direct proof as follows. Namely, consider the diagram of all morphisms where , which has finite limits and is therefore cofiltered, and which is essentially small since is. Then take the corresponding functors to get a diagram of representable functors which is filtered and whose colimit in is by inspection.
Note that when the above construction specializes to the profinite completion.
Pro-objects and pals
Pro-objects in a category themselves form a category , sometimes called the pro-completion of . Explicitly, this is the category whose objects are small cofiltered diagrams in . If and are two such diagrams, then morphisms between them are given by
which exhibits as the opposite category of the category of pro-representable functors . Intuitively, is obtained from by freely adjoining small cofiltered limits.
The corollary above can be interpreted as associating a functor to a suitable inclusion of categories . This functor may be regarded as the left pro-adjoint of the inclusion in the following sense. A functor has a left adjoint if and only if the functor is representable for all ; the representing object is denoted and we can show that this extends to a functor. The corollary above identifies hypotheses guaranteeing that is pro-representable for all , and sending to the corresponding pro-object in gives a functor as before.
Dually, we can define an ind-object in a category to be a pro-object in the opposite category, or a small filtered diagram of objects in , and this gives us a notion of ind-completion . Intuitively, is obtained from by freely adjoining small filtered colimits.
Example. A profinite group is an object in . We’ve encountered several examples above, given as profinite completions of various groups. ( is, as it turns out, a full subcategory of topological groups.)
Example. is the ind-completion of the category of finitely generated groups. Similar statements are true for other categories of algebraic objects.
Example. is the ind-completion of . Since the latter is the opposite of the category of finite Boolean algebras, we conclude that is the pro-completion of the category of finite Boolean algebras. This turns out to be the category of complete atomic Boolean algebras.
Example. The category of Boolean rings is the ind-completion of the category of finite Boolean rings. Since the latter is the opposite of the category of finite sets, we conclude that is the pro-completion of the category of finite sets, which is one way to define the category of profinite sets. This is essentially Stone duality, except that we have not mentioned topological spaces at all.
Example. is the ind-completion of the category of finitely-generated abelian groups. By Pontrjagin duality, it follows that the category of compact (Hausdorff) abelian groups is the pro-completion of its subcategory of finite products of and cyclic groups (precisely the compact (Hausdorff) abelian Lie groups).
Example. The category of torsion abelian groups is the ind-completion of the category of finite abelian groups, which is Pontrjagin self-dual. It follows that is the pro-completion of the category of finite abelian groups, or equivalently the category of profinite abelian groups.
Example. is the ind-completion of , which is self-dual. It follows that is the pro-completion of , what we might call the category of pro-vector spaces, or equivalently the linearly compact vector spaces.
Grothendieck’s Galois theory
Profinite groups naturally appear in Galois theory as follows. Given a field , define its absolute Galois group to be the Galois group of a separable closure of . The separable closure of is its largest Galois extension, and infinite Galois theory (as opposed to finite Galois theory, where we specialize to studying subextensions of a fixed finite Galois extension ) relates the study of all Galois extensions of to the study of .
The absolute Galois group is naturally a profinite group. To see this, note that is the colimit, in the category of separable -extensions, of all of its finite Galois subextensions. Note further that any endomorphism of a separable -extension is an automorphism. We may write
where runs over all finite Galois subextensions of . But for such a subextension, the image of any homomorphism is necessarily itself, and conversely it is a standard property of that any homomorphism extends to a homomorphism , hence may be naturally identified with . We find that
is canonically a cofiltered limit of finite groups. This exhibits the absolute Galois group as the group which acts on all finite Galois extensions of in a way compatible with inclusions between such extensions. Together with the fact that we found various profinite objects above when studying operations, this suggests that the absolute Galois group should have a natural interpretation in terms of operations, which it does as follows.
Let denote the category of finite separable extensions of . If is a separable closure, then defines a functor
We will call this a fiber functor for reasons to be explained later. The fiber functor is pro-representable; the pro-object which represents it, which is an ind-object in , consists of all of the finite Galois subextensions of as before. The unary operations can therefore be identified with the absolute Galois group . This gives us a functor
from to the category of finite -sets. Actually we can modify the target category slightly to the category of finite, transitive, and continuous -sets, where “continuous” means either such that the corresponding map is continuous if the source is given the profinite topology and the target is given the discrete topology or, equivalently, that it factors through a finite quotient .
Fundamental theorem of Galois theory: With the above modification, the above functor is an equivalence of categories.
See, for example, Szamuely’s Galois Groups and Fundamental Groups. The fundamental theorem of Galois theory can therefore be regarded as analogous to the reconstruction theorem for categories of models of a Lawvere theory: indeed the absolute Galois group defines a special kind of a Lawvere theory, namely a Lawvere theory which is generated in degree in the sense that it is generated under composition and finite products by unary operations. However, it is somewhat unsatisfactory from this point of view because of the restriction to transitive -sets. It would be nice to be get an equivalence between some generalization of and the category of finite continuous -sets with no assumption of transitivity; this would be more or less the category of models of the corresponding Lawvere theory in , except for the continuity assumption.
We can get a hint as to what this category should be by noticing that transitive -sets don’t have coproducts; similarly, doesn’t have products. To fix this, we introduce the following notion. A finite-dimensional -algebra is étale if it is a finite direct product of separable extensions of . The functor naturally extends to finite-dimensional étale algebras in a way that sends products to coproducts, since any homomorphism from a finite direct product to an integral domain factors uniquely through one of the factors. Hence we get a fiber functor
As before, the unary operations can be identified with elements of the absolute Galois group , and we get a functor
to the category of finite -sets. Once again, we should really work with the continuous such -sets.
Grothendieck’s fundamental theorem of Galois theory: With the above modification, the above functor is again an equivalence of categories.
The absolute Galois group of a field therefore contains a huge amount of information about it (since in particular it classifies the finite separable extensions of ). A large chunk of number theory can be said to be about the study of absolute Galois groups of number fields, in particular . For example, class field theory studies the -dimensional representation theory, or equivalently the abelianization, of absolute Galois groups, and elliptic curves can be used to study their -dimensional representation theory. More generally, the Langlands program tries to understand their -dimensional representation theory; see, for example, this MO question.
Finite étale algebras over should be thought of as the correct notion of finite covering space of in the following sense. Recall that for “nice” (locally path-connected, semi-locally simply connected) topological spaces there is a Galois theory of covering spaces of , which in the formalism we have described here takes the following general form. For any point there is a fiber functor
from the category of (not necessarily connected) covering spaces of to the fiber over of the covering map (hence the name “fiber functor”). (It may not seem like we “chose a point” when talking about Galois theory, but we did: there the analogous choice is the choice of separable closure.) If is path-connected, this functor is representable by a universal cover of . As in the case of separable extensions above, any endomorphism of a covering is an automorphism, so it follows that the unary operations can be identified with the Galois group of automorphisms of as a cover. This gives a functor
Fundamental theorem of Galois theory for covering spaces: The above functor is an equivalence of categories.
On the other hand, if is a continuous path in , then path lifting induces a natural transformation depending only on the homotopy class of , and moreover we can show using the universal cover that every natural transformation has this form for a unique homotopy class of paths. Hence:
Theorem: The full subcategory of on fiber functors is equivalent to the fundamental groupoid of .
In particular, , which recovers the usual form of Galois theory for covering spaces, but here we don’t require that be path-connected (in addition to not requiring that covers be path-connected). The full subcategory on the generates a multisorted Lawvere theory (a category with finite products such that each object is the product of objects from some set of objects, not necessarily the product of copies of one object) and together give a functor
which is again an equivalence of categories, even without the hypothesis that is path-connected.
Returning again to the case that is path-connected, by restricting our attention to finite covering spaces we get an equivalence of categories with the category of finite continuous -sets. This strongly suggests that the topological and field-theoretic stories above ought to have something of a common generalization, and they do: Grothendieck defined the correct notion of finite covering space of a suitable class of schemes (which reduces to the notion of finite étale algebra when specialized to ). Relative to this notion of covering space there are usually no universal covers, so the corresponding fiber functors are not representable. However, they are pro-representable, leading to the construction of a profinite group attached to any such scheme called the étale fundamental group. When specialized to this recovers the absolute Galois group, and when specialized to nice schemes over , known comparison results imply that this recovers the profinite completion of the fundamental group of the corresponding topological space, with the analytic topology.
Note that by defining a notion of covering space in this setting, Grothendieck was able to define a notion of fundamental group without defining a notion of path!
For some applications of this formalism see, for example, the relevant section of Milne’s Lectures on Étale Cohomology. Among other things, if is a suitably nice variety over it is possible to use the étale fundamental group to define an action of the absolute Galois group on the profinite completion of by outer automorphisms. A case that has received a lot of attention in particular is ; some keywords here are Belyi’s theorem, dessin d’enfant, and Grothendieck-Teichmüller theory.