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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.

The category of commutative rings, often written $\mathbf{CRing}$, has commutative rings with identity as objects and identity-preserving ring homomorphisms as morphisms. Composition and identity morphisms are the usual composition and identity maps. Including the zero ring makes it a terminal object; the integers form an initial object because a unique ring homomorphism sends $n$ to $n{1}_R$.

For example, a ring homomorphism $\mathbb{Z}[x]\to R$ is completely determined by the image of $x$, which can be any element of $R$. In algebraic geometry, applying the spectrum of a ring reverses arrows: a ring homomorphism $R\to S$ induces $\mathrm{Spec}S\to \mathrm{Spec}R$. Thus the category of affine schemes is equivalent to the opposite of the category of commutative rings.^[ref]

A quotient object of an object of a category $X$ is represented by an epimorphism $q:X\to Q$. Two such arrows represent the same quotient if an isomorphism $u:Q\to Q^{\prime }$ satisfies $q^{\prime }=u\circ q$. A quotient is therefore an arrow out of $X$, considered up to this equivalence, rather than an arbitrary smaller-looking object.

In the Category of sets, this recovers surjective maps and quotient sets by equivalence relations. In the category of groups, quotient groups $G/N$ arise from normal subgroups. Quotients can be ordered by factorization: $q_2=h\circ q_1$ means that $q_2$ identifies at least as much as $q_1$.

General epimorphisms need not be surjective maps of underlying sets. In the category of commutative rings, $\mathbb{Z}\hookrightarrow \mathbb{Q}$ is an epimorphism: two unital ring maps out of $\mathbb{Q}$ agreeing on $\mathbb{Z}$ agree on inverses and hence on every rational. Some uses therefore restrict quotient objects to regular epimorphisms, namely arrows realized as coequalizers; the intended convention should be stated.