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 (who followed Waterhouse), 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.
01Group functor as a generalization of a group scheme
A scheme may be thought of as a contravariant functor from the category 
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. For example, the group scheme
![SL_{2}=\operatorname {Spec} \left({\frac {\mathbb {Z} [a,b,c,d]}{(ad-bc-1)}}\right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/2133f55e11d6e3335112d71493af251cb944841a)
can be described as the functor
![\operatorname {Hom} _{\textbf {CRing}}\left({\frac {\mathbb {Z} [a,b,c,d]}{(ad-bc-1)}},-\right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f401777adb2676cbcbda33c2a61379fc02cd7b1f)
If we take a ring, for example, 
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcb813177f76a1fd8f023179985906de666c2950)
02Group 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).
Sources and credits
This article is adapted from the Wikipedia article “Group functor”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.
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