Here’s what seems like a silly question: what’s the universal group? That is, what’s the universal example of a set together with maps
satisfying the identities
A moment’s reflection shows that there isn’t such a group; the existence of the groups , where is an arbitrary set, shows that there exist groups of arbitrarily large cardinality, so no particular group can be universal.
However, it still seems like we can say things about the universal group, even though it doesn’t exist. That is, given a first-order statement in the language of groups, we can still give a sensible definition of what it means for that statement to be true in the universal group: it merely has to be true of all groups! Equivalently, by the completeness theorem for first-order logic, it merely has to be deducible from the group theory axioms. For example, it’s true in any group that
so this statement must be true in the universal group.
In fact, the universal group does exist: it’s just not a set! Namely, we can write down all of the objects together with all of the maps implied to exist by the existence of the structure maps , such as the map
and together with all of the equalities between these maps implied by the group theory axioms. This data describes a category , in fact a monoidal category, that can be succinctly described as the free monoidal category on a group object. The object is a group object which deserves the name “universal group” in the sense that, given any other monoidal category , the category of monoidal functors and monoidal natural transformations is equivalent to the category of group objects in .
The category is the Lawvere theory describing the theory of groups. It can be described more concretely as the opposite of the category whose objects are the free groups and whose morphisms are all group homomorphisms between these; indeed, the components of the universal maps are just -tuples of elements of , which are the same thing as homomorphisms , and it’s not hard to see that this identification respects composition.
Describing groups in this way has, to my mind, one major conceptual benefit over the standard definition: it emphasizes that the choice to present the theory of groups using a particular set of maps and axioms is just as arbitrary as the choice to present a group using a particular set of generators and relations. As long as one gets the same Lawvere theory in the end, one is still studying groups. For example, instead of using identity, multiplication, and inversion, we can use identity and the ternary map together with the heap axioms.