Category of affine schemes ($\mathbf{AffSch}\simeq {\mathbf{CRing}}^{\mathrm{op}}$)

The category of affine schemes has affine schemes as objects and morphisms of schemes as arrows. The contravariant functor $A\mapsto \mathrm{Spec}A$ gives an equivalencewith the opposite of the category of commutative rings. Precisely, every ring homomorphism $A\to B$ corresponds to a unique map $\mathrm{Spec}B\to \mathrm{Spec}A$, and every morphism of schemes between affine schemes arises this way.^[ref]

For example, $k[t]\to k[x]$, $t\mapsto x^2$, induces the squaring map of affine lines. A map from $\mathrm{Spec}k$ to $\mathrm{Spec}A$ instead corresponds to a $k$-algebra map $A\to k$, expressing a rational point. A Cartesian product of affine $k$-schemes corresponds to a tensor product of commutative algebras: $\mathrm{Spec}A{\times }_{\mathrm{Spec}k}\mathrm{Spec}B\cong \mathrm{Spec}(A{\otimes }_{k}B)$. Reversing arrows explains why geometric products become algebraic coproducts.