functor of points

The concept of a **functor of points** as introduced and defined by Grothendieck is either the image of some geometric space by Yoneda embedding into the category of presheaves on some ambient category of geometric spaces or, better, its restriction to some subcategory of local models or nice spaces. In other words, the functor of points corresponding to a space is its corresponding representable presheaf, but the point of the concept is that nonrepresentable functors of points can be studied, as well as their relation to representables (for example, being prorepresentable).

In this approach the spaces are sheaves of sets in some *subcanonical* Grothendieck topology on the category of local models $Aff$. Not only spaces, but also additional structures on spaces (like group structure, equivariance, tangent bundle) are represented as presheaves of sets, of groups, of $O$-modules etc. on $Aff$.

The functor from commutative rings to sets which sends a ring, $R$, to the set of simultaneous solutions in $R^n$ of a set of polynomials, $f_1, \ldots, f_k$ in $\mathbb{Z}[t_1, \ldots,t_n]$ corresponds to the affine scheme $X = Spec(\mathbb{Z}[t_1, \ldots,t_n]/(f_1, \ldots,f_k))$. These $R$-points are then equivalently the hom-space

$Hom_{schemes}(Spec(R), X).$

The functor which sends $R$ to the points of the projective space $\mathbb{P}^n_R$ corresponds to a non-affine scheme.

Revised on August 1, 2014 04:48:26
by David Corfield
(46.208.114.209)