Often in mathematics we define constructions outputting objects which a priori have a certain amount of structure but which end up having more structure than is immediately obvious. For example:
- Given a Lie group , its tangent space at the identity is a priori a vector space, but it ends up having the structure of a Lie algebra.
- Given a space , its cohomology is a priori a graded abelian group, but it ends up having the structure of a graded ring.
- Given a space , its cohomology over is a priori a graded abelian group (or a graded ring, once you make the above discovery), but it ends up having the structure of a module over the mod- Steenrod algebra.
The following question suggests itself: given a construction which we believe to output objects having a certain amount of structure, can we show that in some sense there is no extra structure to be found? For example, can we rule out the possibility that the tangent space to the identity of a Lie group has some mysterious natural trilinear operation that cannot be built out of the Lie bracket?
In this post we will answer this question for the homotopy groups of a space: that is, we will show that, in a suitable sense, each individual homotopy group is “only a group” and does not carry any additional structure. (This is not true about the collection of homotopy groups considered together: there are additional operations here like the Whitehead product.)
Extra structure on a functor
The setting in which we will work is the following. Suppose we have some functor which a priori takes values in a category . To what extent can we lift to a functor taking values in a “more structured” category equipped with a forgetful functor such that the obvious diagram commutes? As phrased, this question is incredibly general, so we will restrict ourselves to lifts which are described by taking into account structure coming from -ary operations, as follows.
Suppose has finite products. Then we can consider natural transformations to be -ary operations (as in this previous post on Lawvere theories) on the outputs of the functor which equip the objects with extra structure. More precisely, the full subcategory of the functor category on the objects is a Lawvere theory, the endomorphism Lawvere theory of (named in analogy with the endomorphism operad). Note that equipping an object in a category with finite products with the structure of a model of a Lawvere theory is equivalent to giving a morphism of Lawvere theories; in particular, itself is tautologically a model of , and this model structure passes to . This lets us lift to a functor taking values in the category of -valued models of , or more precisely the category of product-preserving functors .
If , is representable by some object , and also has finite coproducts, then we can identify natural transformations with morphisms by the Yoneda lemma. Consequently, we can identify with , where is regarded as an object in the opposite category . There is a corresponding story where is a contravariant representable functor; here we just have .
It may be hard to compute the entire endomorphism Lawvere theory of a functor, but any natural transformations that we can find may already provide extra structure that wasn’t there before. More generally it is often possible to identify Lawvere theories and morphisms of Lawvere theories, which allow us to lift to the category of -valued models of . These kinds of observations are already enough to reproduce many familiar examples of extra structure, and generalize the observation that is acted on from the left by the monoid of endomorphisms and from the right by the monoid of endomorphisms .
Example. If is a group object in a category with finite products, then the group operation gives a morphism from the Lawvere theory of groups to . Hence naturally acquires the structure of a group. (Conversely, by the Yoneda lemma, if naturally has the structure of a group then is a group object.)
Example. Dually, if is a cogroup object in a category with finite coproducts, then the cogroup operation gives a morphism from the Lawvere theory of groups to . Hence naturally acquires the structure of a group. (Again, conversely, by the Yoneda lemma, if naturally has the structure of a group then is a cogroup object.)
Example. In the category of schemes over a base ring , the endomorphism Lawvere theory of the affine line is the Lawvere theory of polynomials over , or equivalently the Lawvere theory of commutative -algebras. Hence naturally acquires the structure of a commutative -algebra. (We previously discussed the case for affine schemes in this blog post.)
Example. In the category of topological spaces, the space admits addition and multiplication operations in addition to scalar multiplication operations , and these generate the Lawvere theory of polynomials over . Hence naturally acquires the structure of a commutative -algebra.
Example. A distributive category is a category with finite products and coproducts such that the former naturally distribute over the latter; the standard example is , although and more generally any cartesian closed category also qualify, and and (the category of schemes) are important examples which are not cartesian closed.
In any distributive category, the endomorphism Lawvere theory of the object canonically admits a morphism from the Lawvere theory of Boolean algebras, or equivalently the Lawvere theory of Boolean rings, or equivalently the category of Boolean functions (the full subcategory of on finite sets of size ). Hence naturally acquires the structure of a Boolean algebra, or equivalently a Boolean ring. In this reproduces the lattice of clopen subsets of a topological space. In general I think it should be interpreted as something like the “lattice of decidable properties.”
Example. If is an abelian group, then the group operation is itself a morphism in , giving a morphism from the Lawvere theory of abelian groups to . Hence naturally acquires the structure of an abelian group. (We discussed a more general setting in which such an abelian group structure exists in this previous post on semiadditive categories.)
The homotopy groups are groups
Recall that the pointed homotopy category is the category whose objects are pointed topological spaces and whose morphisms are homotopy classes of pointed continuous maps preserving the base point. Recall also that the homotopy groups are a sequence of functors naturally defined on this category and represented by the spheres with some choice of base point, which we will usually omit in our notation. That the homotopy groups are groups is equivalent to the statement that the spaces , as objects of the pointed homotopy category, are all cogroup objects.
The basic idea is to observe that a pointed map from to a pointed space is the same thing as a map from the -cube to such that the boundary is sent to . In general, morphisms from the -cube can be glued together along any pair of -dimensional faces provided that the images of those faces match. There are distinguished such gluings coming from gluing together each of the copies of in the product in the usual way that one glues two intervals together. These gluing operations are natural, associative, and have inverses up to homotopy. They give compatible group operations on which, when , make it an abelian group by the Eckmann-Hilton argument.
The appearance of maps out of and multiple composition operations suggests a higher-category-theoretic perspective on the situation where we can think of as a suitable automorphism group. More precisely, for any we can associate to an unpointed topological space its fundamental -groupoid , which is the -category whose
- objects are the points of ,
- morphisms are the paths between points of ,
- -morphisms are the homotopies between paths,
- -morphisms are the homotopies between homotopies,
- -morphisms are the homotopy classes of homotopies between homotopies between…
Note that a -morphism can be thought of as a map , with its source and its target determined by its restriction to a suitable choice of two copies of in it. -morphisms have notions of composition given by gluing along the coordinate directions, generalizing horizontal and vertical composition of -morphisms in -categories (in particular, of functors).
The homotopy group of a pointed space can then be interpreted as the group of -automorphisms of the identity -endomorphism of the identity -endomorphism of… of the identity endomorphism of in the fundamental -groupoid.
The homotopy groups are only groups
We would like to show that the homotopy groups are only groups in the sense that the endomorphism Lawvere theories of the functors are generated by the Lawvere theory of groups. In fact we will be able to say slightly more than this.
Theorem: The endomorphism Lawvere theory of is precisely the Lawvere theory of groups.
Proof. By the Yoneda lemma, this means we want to show that the full subcategory of on the finite wedge sums of is equivalent, as a category with finite coproducts, to the full subcategory of on the finitely generated free groups. To show this it more or less suffices to show that the fundamental group of a wedge of circles is the free group generated by each circle (strictly speaking we should show that this identification can be made compatible with partial composition, but we already know this because we already know that the fundamental group is a group), but this follows from Seifert-van Kampen.
In the context of a more general result, not only has fundamental group but is an Eilenberg-MacLane space , since its universal cover is a tree, and the subcategory of on Eilenberg-MacLane spaces (suitably pointed) is known to be equivalent to , with the equivalence given by .
Theorem: The endomorphism Lawvere theory of is precisely the Lawvere theory of abelian groups.
Proof. By the Yoneda lemma, this means we want to show that the full subcategory of on the finite wedge sums is equivalent, as a category with finite coproducts, to the subcategory of on the finitely generated free abelian groups. To show this it more or less suffices to show that is the free abelian group generated by each inclusion of into the wedge (where there are spheres in the wedge) (and, again, strictly speaking we should show compatibility with partial composition, but we already know this).
Since admits a CW-structure with a single -cell and no -cells, , it is -connected by cellular approximation. By the Hurewicz theorem, it follows that the Hurewicz map is an isomorphism, so to compute the former it suffices to compute the latter. But now by Mayer-Vietoris.