{"paper":{"title":"The Category of Factorization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.AC","authors_text":"Brandon Goodell, Sean K. Sather-Wagstaff","submitted_at":"2018-02-18T03:49:39Z","abstract_excerpt":"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.\n  Among other results, we show that $\\mathcal{F}(A)$ is a symmetric and strict monoidal category with weak equivalences"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.06330","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}