{"paper":{"title":"Categorical Algebra of Atomic Monoids: Presentability, Regularity, and Pretorsion Theories","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.CT","authors_text":"Federico Campanini, Laura Cossu","submitted_at":"2026-07-25T10:51:26Z","abstract_excerpt":"We study the category $\\mathsf{AtoMon}$ of atomic monoids and atom-preserving homomorphisms. We prove that $\\mathsf{AtoMon}$ is locally finitely presentable by exhibiting a strong generator consisting of compact objects. We show that $\\mathsf{AtoMon}$ admits (regular epi, mono)-factorizations but that it is not a regular category: we construct a regular epimorphism which is not pullback-stable. We also establish adjunctions for the group of units and explicitly construct the ``cofree atomic monoid'' over an arbitrary monoid. Finally, we exhibit a way to lift torsion theories of $\\mathsf{Grp}$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23144","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.23144/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}