Groups are in particular sets equipped with two operations: a binary operation (the group operation) and a unary operation (inverse) . Using these two operations, we can build up many other operations, such as the ternary operation , and the axioms governing groups become rules for deciding when two expressions describe the same operation (see, for example, this previous post).
When we think of groups as objects of the category , where do these operations go? They’re certainly not morphisms in the corresponding categories: instead, the morphisms are supposed to preserve these operations. But can we recover the operations themselves?
It turns out that the answer is yes. The rest of this post will describe a general categorical definition of -ary operation and meander through some interesting examples. After discussing the general notion of a Lawvere theory, we will then prove a reconstruction theorem and then make a few additional comments.
The general definition
Since operations are functions involving underlying sets, the appropriate categorical setting here is not a bare category but a concrete category, namely a category equipped with a faithful functor , which we will think of as the underlying set (forgetful) functor.
What should “operations on the objects in a concrete category” mean, then? Well, an -ary operation is in particular a family of functions
for all objects . Morphisms ought to preserve operations, hence if is a morphism then we ought to have
But this is precisely the condition that defines a natural transformation between the functors and ! Hence we have our definition: an -ary operation on the objects of a concrete category is a natural transformation .
The following straightforward theorem will determine what the -ary operations are in many familiar concrete categories.
Theorem: Let be a concrete category with finite coproducts such that is representable by an object . Then the -ary operations on objects of can be naturally identified with morphisms (the coproduct of copies of ), or equivalently with elements of the underlying set .
Corollary: In particular, the above conclusion holds if has a left adjoint.
Proof. Under the hypotheses, it follows that is represented by , and then the conclusion follows from the Yoneda lemma.
Proof of corollary. If has a left adjoint , then , from which it follows that represents . Note that in this case represents , where denotes an -element set, so we get that -ary operations can be identified with elements of the underlying set .
Examples. Let . In each of these familiar algebraic examples the usual forgetful functor has a left adjoint, from which it follows that the -ary operations can be identified with the underlying set of the free object on elements in each case, namely . In each case this recovers the -ary operations in the usual sense (namely all of the operations one can write down starting from the operations given by the axioms). For example, associated to any element is the operation
on abelian groups, and associated to any polynomial is the operation
on commutative rings.
Example. Let be the category of -sets for a monoid. Again the usual forgetful functor has a left adjoint, so the -ary operations can be identified with the underlying set of the free object on elements, namely (where appears times). Unlike the above cases, in this case the only interesting operations are the unary operations given by multiplication by elements of : all the -ary operations are given by a projection map followed by a unary operation.
Example. Let be the category of -modules for a ring. Again the usual forgetful functor has a left adjoint, so the -ary operations can be identified with the underlying set of the free object on elements, namely . An element gives rise to an operation
in a natural generalization of the case . There would not be much to learn from this case except that there are various ways to define a category equivalent to for a ring without directly specifying . For example, the category of representations of a group over a field is equivalent to , and similarly the category of representations of a Lie algebra over a field is equivalent to . In both cases thinking about -ary operations reveals the presence of a ring whose elements give us additional operations we may not otherwise have noticed: in a representation of we can multiply by any sum of elements of rather than just by elements of , and similarly in a representation of . In the case of finite groups this lets us define useful operations like multiplication by , and in the case of Lie algebras this lets us define useful operations like multiplication by Casimir elements.
Reflecting on the above examples, they should strike you as unsurprising in hindsight: “free object on elements” should mean nothing other than “anything I can get by applying an -ary operation to elements.”
This answers a question which used to bug me, namely “in the standard commutative algebra description of how polynomials work, where did polynomial composition go?” The answer is that to recover composition of, say, polynomials in one variable it suffices to identify with natural transformations where is the forgetful functor. Similarly to recover composition of polynomials in more than one variable.
A quick remark
Studying arbitrary -ary operations is the basic starting point of universal algebra. When we look at -ary operations on an arbitrary concrete category, we are asking in some sense how much of the structure of objects in that category is algebraic structure. In the above cases, where we started with algebraic objects, the answer is more or less “all of it.” In the examples below this will no longer be the case, but it may still be surprising how much algebraic structure we can find.
Less algebraic examples
Example. First, a boring example. Let be the category of topological spaces. The forgetful functor has left adjoint given by the functor which equips a set with the discrete topology. Hence there are no interesting -ary operations: all we get are projections
Even if we look for infinitary operations , where is an infinite set, we only find projection operations. So the structure of topological spaces is not well captured by operations; in some sense, is very far from being algebraic.
Example. Let be the category of compact Hausdorff spaces. The forgetful functor has left adjoint given by the Stone-Čech compactification on discrete spaces, which is just the functor which associates to a set the set of ultrafilters on it (see, for example, this previous post). It follows that the ultrafilters on a set give us the -ary operations on compact Hausdorff spaces.
These operations are precisely given by taking limits with respect to the given ultrafilter. For finite they are uninteresting as in the case of topological spaces, since all ultrafilters on a finite set are principal. For infinite we get an interesting operation for every non-principal ultrafilter on . Taking limits with respect to ultrafilters in this manner completely recovers the topology on a compact Hausdorff space , and in fact it is possible to define the category of compact Hausdorff spaces in this manner (it is precisely the category of algebras over the ultrafilter monad).
Example. Let be the category of complex Banach spaces and weak contractions (see, for example, this previous post). The representable functor sends a Banach space to its unit ball; we will take this as the forgetful functor . Then has a left adjoint , so the unit ball of gives us the -ary operations on unit balls of Banach spaces (we can consider natural transformations for arbitrary ).
The finitary operations are given by combinations of addition and scalar multiplication (subject to the constraint that we stay in the unit ball), but we get interesting infinitary operations that invoke completeness: for example, for any sequence in the unit ball of (so ), we get an infinitary operation
on elements of the unit ball of some Banach space.
Example. Let be the opposite of the category of smooth manifolds. Remembering the slogan that algebra is dual to geometry, this category ought to have something to do with algebras of some kind. Accordingly, for the forgetful functor we will choose the functor sending a smooth manifold to the algebra of smooth real-valued functions on it. This functor is represented by , and has finite coproducts, so the -ary operations on are given by the underlying set of the coproduct of copies of (in the opposite category) or, in other words, the smooth functions on . If is such a function, the corresponding operation is given by
Example. Let be the category of commutative Banach algebras. The holomorphic functional calculus guarantees that every holomorphic function gives an -ary operation . These are precisely all of the -ary operations; see this MO question for a sketch of an argument, although we will give a more general argument in a later post. Note that is not representable; roughly speaking, there is no free commutative Banach algebra on an element because that element must have some fixed norm and hence cannot have arbitrarily large complex numbers in its spectrum.
Example. Similarly, let be the category of commutative C*-algebras. The continuous functional calculus guarantees that every continuous function gives an -ary operation . These are precisely all of the -ary operations by an argument similar to the above argument. Note again that is not representable.
When is a concrete category as above, it’s natural to think of operations more generally as morphisms between the objects in (the category of functors ), where denotes the functor which sends every object to . By the universal property of products, a morphism is precisely an -tuple of morphisms , so classifying such morphisms reduces to the -ary case we’ve already considered, but allowing operations to have multiple outputs as well as inputs highlights the fact that we’re really looking at a certain full subcategory of . If is representable by an object , this is the opposite category of the full subcategory of on the objects (where denotes an initial object), and if has a left adjoint, this is the opposite category of the full subcategory of on the free objects . This category is a Lawvere theory, namely a category with finite products together with a distinguished object (in this case ) such that each object is some power of it.
(This formalism doesn’t capture infinitary operations as in the case of compact Hausdorff spaces above; there we need to consider infinitary Lawvere theories. Here we consider only the finitary case for simplicity.)
Thinking of a Lawvere theory as a specification of a collection of operations and rules for how to compose them, the natural thing to do with a Lawvere theory is to write down its category of models. What should this mean? If is a Lawvere theory with distinguished object , a model of the theory should be a set such that we assign to each morphism a function in a way which is compatible with composition and products.
In short, a model of a Lawvere theory should be a product-preserving functor . The category of models of (in ) is therefore the category of such functors. It naturally comes equipped with a faithful forgetful functor to given by evaluating a functor on the object , making it a concrete category.
Example. If is any of the algebraic examples we encountered earlier, the categories of models of the corresponding Lawvere theories are precisely the original categories respectively.
Example. If with the unit ball forgetful functor, the category of models of the corresponding infinitary Lawvere theory (where we include the countable product and require models to be functors preserving countable products) is the category of totally convex spaces. This is in some sense the closest algebraic theory approximating Banach spaces.
Example. If with the forgetful functor, the category of models of the corresponding Lawvere theory is the category of smooth algebras. The algebras are natural examples of smooth algebras, but smooth algebras can also contain nilpotents like . The object in the opposite of the category of smooth algebras, which behaves like a category of generalized smooth manifolds, represents the tangent bundle functor.
Lawvere theories give us a natural way to construct functors between categories of algebraic objects. First, note that Lawvere theories themselves form a category: a morphism between Lawvere theories should be a product-preserving functor sending the fixed object of to the fixed object of . Next observe that a morphism of Lawvere theories induces a functor between the categories of models in the other direction.
Example. There is a natural inclusion of Lawvere theories given by mapping the addition operations on abelian groups to the addition operations on underlying groups of rings. It induces the underlying group forgetful functor .
Example. There is a natural quotient of Lawvere theories given by abelianization (via the natural map ). It induces the forgetful functor .
Remark. One reason to favor working with Lawvere theories is that we can consider models in categories other than : if is any category with finite products, we can define a model of a Lawvere theory to be a product-preserving functor . For example, a topological group is a model of the Lawvere theory of groups in the category of topological spaces; similarly, a Lie group is a model .
With this generalized notion of model, every Lawvere theory has an initial model, namely the identity functor , which exhibits the distinguished object of as the universal model of . This generalizes the observation made in a previous post about the distinguished object in being the universal group. Moreover, with this notion of model, a morphism of Lawvere theories is just a description of the distinguished object of one Lawvere theory as a model for another Lawvere theory. The two examples we gave above can be summarized by the slogans “the universal ring is an abelian group” and “the universal commutative ring is a ring.”
The category of Lawvere theories gives us a natural interpretation of the fact that the category of models of a Lawvere theory is concrete. Namely, there is an initial Lawvere theory: it has an object and finite products and is the free category on this data. The only morphisms one can write down from this data are the unique morphism , the projection maps , and products and compositions of these. This gives a category equivalent to . We encountered this Lawvere theory when considering with the usual forgetful functor. The category of models of this Lawvere theory in is naturally equivalent to .
By the universal property, every Lawvere theory admits a unique morphism from the initial Lawvere theory, and this induces a functor on categories of models in the other direction which is precisely the usual forgetful functor.
A reconstruction theorem
(Below, “model” means “model in .”)
Above we saw that in some examples, starting from a concrete category , forming the corresponding Lawvere theory , and considering its category of models , we naturally reconstruct the original category. In fact the following general result holds.
Theorem: Let be a concrete category and let be the Lawvere theory generated by . Then there is a naturally defined functor from to the category of models of . This functor is a morphism of concrete categories, and it is initial among all morphisms of concrete categories from to the category of models of a Lawvere theory.
The theorem tells us that the category of models of is in some sense the closest approximation to by the category of models of a Lawvere theory.
Corollary: If is itself the category of models of a Lawvere theory, then the category of models of is naturally equivalent to as a concrete category.
Proof. The idea of the proof is that when we define we’ve already described all -ary operations on the underlying sets preserved by the morphisms of . Any other morphism of concrete categories into the category of models of a Lawvere theory can only supply us with -ary operations we already have.
First, we want to show that there is a natural functor from to the category of models of . This should be clear by definition, but it is good to be explicit. In general, for any categories satisfying mild size conditions, there is a natural evaluation functor
where as usual is the category of functors . We can restrict this functor to various subcategories in either factor. In particular, if is a concrete category, restricting the natural evaluation functor
to the Lawvere theory (regarded as a subcategory of ) generated by gives a functor
and, since functor categories are exponential objects, a functor
Explicitly, this functor sends every object to the functor which assigns an object the object ; in particular, it’s a functor to the category of models of .
We want to prove that this functor is initial among functors to categories of models of Lawvere theories, so let be a morphism of concrete categories (a functor such that ) such that is the category of models of a Lawvere theory . Then defines a functor
which gives a functor
which in turn gives a functor
Letting be the distinguished object, this functor sends to and to (by the assumption that ) and sends morphisms to natural transformations , hence gives a morphism of Lawvere theories inducing a functor on categories of models in the other direction through which factors by inspection. The conclusion follows.
Revisiting topological spaces
This post can be thought of as addressing the following natural question: concrete categories usually arise as categories of structured sets and morphisms preserving that structure. Given a concrete category, can we reconstruct the structure that its morphisms preserve in some appropriate sense? The post shows that the answer is yes for categories of models of a Lawvere theory, where the structure is -ary operations, but it is emphatically no in general, since not all structure is captured by -ary operations. Our worst failure was on the category , where we only managed to get the initial Lawvere theory.
Following the general slogan that algebra is dual to topology, as well as taking a cue from the example of , we might decide to look at instead. A natural forgetful functor to choose here is the functor sending a topological space to its open subsets, which is represented by the Sierpinski space (with the open sets ); here we identify an open subset with its indicator function as an element of . (We will ignore the fact that this functor is not faithful, hence strictly speaking does not define a concretization.)
Now we have lots of operations, but on open subsets rather than points of topological spaces. The obvious guess is that the finitary operations should be generated by finite intersection and union, since these are all preserved by the action of continuous maps on open sets . To verify this, since is represented by , the -ary operations are given by morphisms , or equivalently the open subsets of . Any such open subset is the union of sets of the form where is one of . If any of the are empty then so is the product, in which case the corresponding operation is the one sending any -tuple of open subsets to the empty set. Otherwise, the corresponding operation is the one sending an -tuple of open subsets to an intersection of some of them (the ones at every index such that ). Taking unions of open subsets of corresponds to taking the pointwise union of the corresponding operations, and we recover the desired result.
The Lawvere theory we obtain in this way is the Lawvere theory of (edit:) distributive lattices (posets with finite products and coproducts which distribute over each other). Since open subsets are closed under arbitrary unions but only finite intersection, we get an interesting infinitary Lawvere theory if we look at infinitary operations, namely the Lawvere theory of frames (posets with all coproducts and finite products such that the former distribute over the latter). The category is therefore in some sense the closest approximation to by the category of models of an infinitary Lawvere theory, and consequently its opposite category, the category of locales, approximates itself. Substituting topological spaces with locales leads to pointless topology, which classically is nearly identical to ordinary point-set topology. However, pointless topology behave better with respect to weaker foundations, and in particular behaves better when internalized to, for example, a topos. It also captures precisely the part of topology which is captured by sheaves.