A scheme may be thought of as a contravariant functor from the category \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 \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. For example, the group scheme :SL_2 = \operatorname{Spec}\left( \frac{\mathbb{Z}[a,b,c,d]}{(ad - bc - 1)} \right) can be described as the functor :\operatorname{Hom}_{\textbf{CRing}}\left(\frac{\mathbb{Z}[a,b,c,d]}{(ad - bc - 1)}, -\right) If we take a ring, for example, \mathbb{C}, then : \begin{align} 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{align} == Group sheaf ==