  We introduce and investigate the category of factorization of a
multiplicative, commutative, cancellative, pre-ordered monoid $A$, which we
denote $\mathcal{F}(A)$. The objects of $\mathcal{F}(A)$ are factorizations of
elements of $A$, and the morphisms in $\mathcal{F}(A)$ encode combinatorial
similarities and differences between the factorizations. We pay particular
attention to the divisibility pre-order and to the monoid $A=D\setminus\{0\}$
where $D$ is an integral domain.
  Among other results, we show that $\mathcal{F}(A)$ is a symmetric and strict
monoidal category with weak equivalences and compute the associated category of
fractions obtained by inverting the weak equivalences. Also, we use this
construction to characterize various factorization properties of integral
domains: atomicity, unique factorization, and so on.
