Group functor


In mathematics, a group functor is a group-valued functor on the category of commutative rings. Although it is typically viewed as a generalization of a group scheme, the notion itself involves no scheme theory. Because of this feature, some authors, notably Waterhouse and Milne, develop the theory of group schemes based on the notion of group functor instead of scheme theory.
A formal group is usually defined as a particular kind of a group functor.

Group functor as a generalization of a group scheme

A scheme may be thought of as a contravariant functor from the category of S-schemes to the category of sets satisfying the gluing axiom; the perspective known as the functor of points. Under this perspective, a group scheme is a contravariant functor from to the category of groups that is a Zariski sheaf.
For example, if Γ is a finite group, then consider the functor that sends Spec to the set of locally constant functions on it. For example, the group scheme
can be described as the functor
If we take a ring, for example,, then

Group sheaf

It is useful to consider a group functor that respects a topology of the underlying category; namely, one that is a sheaf and a group functor that is a sheaf is called a group sheaf. The notion appears in particular in the discussion of a torsor.
For example, a p-divisible group is an example of a fppf group sheaf.