The Category of Atomic Monoids: Universal Constructions and Arithmetic Properties

We introduce and investigate the category $\mathsf{A}\mathsf{t}\mathsf{o}\mathsf{M}\mathsf{o}\mathsf{n}$ of atomic monoids and atom-preserving monoid homomorphisms, which is a (non-full) subcategory of the usual category of monoids. In particular, we compute all limits and colimits, showing that $\mathsf{A}\mathsf{t}\mathsf{o}\mathsf{M}\mathsf{o}\mathsf{n}$ is a complete and cocomplete category. We also address certain arithmetic properties of products and coproducts, providing explicit formulas for some fundamental invariants associated with factorization lengths in atomic monoids.