{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:MA7C4H6SUVKG2Z6OAKE6TCL4K3","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0765dc9cc51fc0fb400d53ba643e3023463c9e7e6804b8d20595eabc0cec2ec1","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-02-18T03:49:39Z","title_canon_sha256":"82bc017247f04e7ff885b696a11cc74d4d59530a90f1f759e06b263a7dda80ae"},"schema_version":"1.0","source":{"id":"1802.06330","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.06330","created_at":"2026-05-17T23:56:04Z"},{"alias_kind":"arxiv_version","alias_value":"1802.06330v2","created_at":"2026-05-17T23:56:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.06330","created_at":"2026-05-17T23:56:04Z"},{"alias_kind":"pith_short_12","alias_value":"MA7C4H6SUVKG","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"MA7C4H6SUVKG2Z6O","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"MA7C4H6S","created_at":"2026-05-18T12:32:37Z"}],"graph_snapshots":[{"event_id":"sha256:52f7ea34799ce10e8b3164eafb137851c721b244834d789e953503a5d53f06fe","target":"graph","created_at":"2026-05-17T23:56:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Brandon Goodell, Sean K. Sather-Wagstaff","cross_cats":["math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-02-18T03:49:39Z","title":"The Category of Factorization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.06330","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:e4a3c4dd7830b77c7418804f037005f7596494279b859095dcef435a10d3f869","target":"record","created_at":"2026-05-17T23:56:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0765dc9cc51fc0fb400d53ba643e3023463c9e7e6804b8d20595eabc0cec2ec1","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-02-18T03:49:39Z","title_canon_sha256":"82bc017247f04e7ff885b696a11cc74d4d59530a90f1f759e06b263a7dda80ae"},"schema_version":"1.0","source":{"id":"1802.06330","kind":"arxiv","version":2}},"canonical_sha256":"603e2e1fd2a5546d67ce0289e9897c56d2a02420e5022d5946eb70fff43e9188","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"603e2e1fd2a5546d67ce0289e9897c56d2a02420e5022d5946eb70fff43e9188","first_computed_at":"2026-05-17T23:56:04.056821Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:56:04.056821Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"11RRLZrOnwG1L/+KCnbDD+LNb7TJIpCaCKtvMNRA8bHSYmTsXvJsOlr279ZAvhqPM6MK+bV2n4NIjkTD4xsDDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:56:04.057441Z","signed_message":"canonical_sha256_bytes"},"source_id":"1802.06330","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e4a3c4dd7830b77c7418804f037005f7596494279b859095dcef435a10d3f869","sha256:52f7ea34799ce10e8b3164eafb137851c721b244834d789e953503a5d53f06fe"],"state_sha256":"90ad6559681c13ff829e2d292b267e0dbfa3a1b65e70894b2aeff5226d4994e8"}