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 (who followed Waterhouse),^[1] 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 ${\displaystyle {\mathsf {Sch))_{S))$ 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 ${\displaystyle {\mathsf {Sch))_{S))$ to the category of groups that is a Zariski sheaf (i.e., satisfying the gluing axiom for the Zariski topology).

For example, if Γ is a finite group, then consider the functor that sends Spec(R) to the set of locally constant functions on it.^{[clarification needed]} For example, the group scheme

${\displaystyle SL_{2}=\operatorname {Spec} \left({\frac {\mathbb {Z} [a,b,c,d]}{(ad-bc-1)))\right)}$

can be described as the functor

${\displaystyle \operatorname {Hom} _{\textbf {CRing))\left({\frac {\mathbb {Z} [a,b,c,d]}{(ad-bc-1))),-\right)}$

If we take a ring, for example, ${\displaystyle \mathbb {C} }$, then

${\displaystyle {\begin{aligned}SL_{2}(\mathbb {C} )&=\operatorname {Hom} _{\textbf {CRing))\left({\frac {\mathbb {Z} [a,b,c,d]}{(ad-bc-1))),\mathbb {C} \right)\\&\cong \left\((\begin{bmatrix}a&b\\c&d\end{bmatrix))\in M_{2}(\mathbb {C} ):ad-bc=1\right\}\end{aligned))}$

Group sheaf

It is useful to consider a group functor that respects a topology (if any) 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 (where a choice of topology is an important matter).

For example, a p-divisible group is an example of a fppf group sheaf (a group sheaf with respect to the fppf topology).^[2]