{"paper":{"title":"Monads and Distributive Laws in Substructural Contexts (Extended Version)","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A canonical construction produces a distributive law ST to TS for monads on sets when S is W-operadic and T is W-commutative with respect to a verbal category W.","cross_cats":["math.CT"],"primary_cat":"cs.LO","authors_text":"Ichiro Hasuo, Soichiro Fujii, Yo\\`av Montacute, Yun Chen Tsai","submitted_at":"2026-05-13T13:46:35Z","abstract_excerpt":"We present a categorical theory of monads and distributive laws in substructural contexts. In the study of distributive laws, the roles of (the absence of) structural rules for variable contexts have been recognized; our theory formalizes these substructural situations using Tronin's verbal categories $\\mathbf W$, in a uniform and presentation-independent manner. We introduce the classes of $\\mathbf W$-operadic monads (those defined via the structural rules in $\\mathbf W$) and of $\\mathbf W$-commutative monads (those invariant under the structural rules in $\\mathbf W$). We give a canonical con"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We give a canonical construction of a distributive law ST→TS of monads on Set; it is applicable when S is W-operadic and T is W-commutative (under mild conditions). This accounts for many known and new distributive laws.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That Tronin's verbal categories W provide a uniform and presentation-independent formalization of substructural situations, and that the mild conditions for the canonical construction hold in the intended applications.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A uniform theory of W-operadic and W-commutative monads on Set yields a canonical distributive law ST to TS when S respects the structural rules in W and T is invariant under them.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A canonical construction produces a distributive law ST to TS for monads on sets when S is W-operadic and T is W-commutative with respect to a verbal category W.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ad0f6b52cf2785f50011632f48e4c59b7fafbf8c8778c8352f585adf85f6e9eb"},"source":{"id":"2605.13533","kind":"arxiv","version":1},"verdict":{"id":"b5190bed-480d-44ef-812d-7c83e90f08b1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:18:14.123808Z","strongest_claim":"We give a canonical construction of a distributive law ST→TS of monads on Set; it is applicable when S is W-operadic and T is W-commutative (under mild conditions). This accounts for many known and new distributive laws.","one_line_summary":"A uniform theory of W-operadic and W-commutative monads on Set yields a canonical distributive law ST to TS when S respects the structural rules in W and T is invariant under them.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That Tronin's verbal categories W provide a uniform and presentation-independent formalization of substructural situations, and that the mild conditions for the canonical construction hold in the intended applications.","pith_extraction_headline":"A canonical construction produces a distributive law ST to TS for monads on sets when S is W-operadic and T is W-commutative with respect to a verbal category W."},"references":{"count":72,"sample":[{"doi":"","year":2025,"title":"42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025) , pages =","work_id":"18a80887-db95-49a7-92d3-32ebec68b657","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"On Generalised Coinduction and Probabilistic Specification Formats: Distributive laws in coalgebraic modelling","work_id":"0fc16f81-4f8a-4318-a48a-acd0bc6efbe7","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1969,"title":"Seminar on Triples and Categorical Homology Theory","work_id":"e15d62de-19c7-464d-a03a-cabba41d0664","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"Logical Methods in Computer Science , volume=","work_id":"2b8a63ed-50c2-4708-95f0-a0f1a7fcfaec","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1978,"title":"Journal of Pure and Applied Algebra , volume =","work_id":"4720dd4a-59d3-4041-97a5-17d18db8641a","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":72,"snapshot_sha256":"5e7007ad4c23262805d53d6f479801a2a864006598dd0d0f12e6ea435b58bf02","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"}