higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
In algebraic geometry, there are two equivalent ways of looking at a scheme: it may be viewed
as a petit topos with a structure sheaf of commutative rings, hence as a locally ringed space,
as an object of the gros topos of sheaves on the site of commutative rings (with étale topology or Zariski topology) satisfying the condition that it is covered by representables via open maps.
In other words, a scheme may be identified with the sheaf it represents; this sheaf is often called the functor of points of the scheme.
To see this, note that by the Yoneda lemma a scheme may be identified with the sheaf it represents on the gros Zariski site of schemes; and since any scheme admits an affine open cover, the comparison lemma says that sheaves on the site of all schemes may be identified with sheaves on the site of affine schemes.
The functor of points approach has the advantage of making certain constructions much simpler (e.g. the fibered product in the category of schemes), and eliminating the need for certain constructions like the Zariski spectrum. In his famous 1973 Buffalo Colloquium talk, Alexander Grothendieck urged that his earlier definition of scheme via locally ringed spaces should be abandoned in favour of the functorial point of view.
Of course, the above discussion generalizes to other types of geometry and even higher geometry. For more see at space.
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
The functor which sends $R$ to the points of the projective space $\mathbb{P}^n_R$ corresponds to a non-affine scheme.
Alexander Grothendieck, Introduction au langage fonctoriel, course in Algiers in November 1965, lecture notes by Max Karoubi, pdf scan.
Alexander Grothendieck, Introduction to functorial algebraic geometry, part 1: affine algebraic geometry, summer school in Buffalo, 1973, lecture notes by Federico Gaeta, pdf scan.
William Lawvere, Grothendieck’s 1973 Buffalo Colloquium, posting to the mailing list [email protected], gmane archive.
Michel Demazure, Pierre Gabriel, Introduction to algebraic geometry and algebraic groups, North-Holland Mathematics Studies Volume 39 (1980).
Zhen Lin Low, Categories of spaces built from local models, doctoral thesis (2016) (web,doi.org/10.17863/CAM.384)