{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:XPLPMJTTZIV7G6NOPSKICRKKKO","short_pith_number":"pith:XPLPMJTT","schema_version":"1.0","canonical_sha256":"bbd6f62673ca2bf379ae7c9481454a53a2447082d2ee4057d84c194ed3691bb5","source":{"kind":"arxiv","id":"2207.04731","version":1},"attestation_state":"computed","paper":{"title":"Skew category algebras and modules on ringed finite sites","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.RT","authors_text":"Fei Xu, Mawei Wu","submitted_at":"2022-07-11T09:26:03Z","abstract_excerpt":"Let $\\mathcal{C}$ be a small category. We investigate ringed sites $(\\mathbf{C},\\mathfrak{R})$ on $\\mathcal{C}$ and the resulting module categories $\\mathfrak{M}{\\rm od}\\text{-}\\mathfrak{R}$. When $\\mathcal{C}$ is finite, based on Grothendieck and Verdier's classification of finite topoi, we prove that each $\\mathfrak{M}{\\rm od}\\text{-}\\mathfrak{R}$ is equivalent to ${\\rm Mod}\\text{-}\\mathfrak{R}|_{\\mathcal{D}}[\\mathcal{D}]$, where $\\mathfrak{R}|_{\\mathcal{D}}[\\mathcal{D}]$ is the skew category algebra, canonically defined on $(\\mathbf{C},\\mathfrak{R})$, for a uniquely determined full subcateg"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2207.04731","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2022-07-11T09:26:03Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"099680f4235982c39bc83144732f75f2cc9aefdf6d49f7c50ac27ec485676acb","abstract_canon_sha256":"a9a425a928bd20430ebd32efa3e292e64f74e076c29a9487840f7a18353ca1f1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:07:31.332166Z","signature_b64":"kSzaapbkLonauskILLFIqGJzkEplEzTD5Cmm80q59+zkeJLRZWhoOs8SqSkjNpT/FFj3jw9TH+xXN5rB0kUvCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bbd6f62673ca2bf379ae7c9481454a53a2447082d2ee4057d84c194ed3691bb5","last_reissued_at":"2026-07-05T06:07:31.331703Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:07:31.331703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Skew category algebras and modules on ringed finite sites","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.RT","authors_text":"Fei Xu, Mawei Wu","submitted_at":"2022-07-11T09:26:03Z","abstract_excerpt":"Let $\\mathcal{C}$ be a small category. We investigate ringed sites $(\\mathbf{C},\\mathfrak{R})$ on $\\mathcal{C}$ and the resulting module categories $\\mathfrak{M}{\\rm od}\\text{-}\\mathfrak{R}$. When $\\mathcal{C}$ is finite, based on Grothendieck and Verdier's classification of finite topoi, we prove that each $\\mathfrak{M}{\\rm od}\\text{-}\\mathfrak{R}$ is equivalent to ${\\rm Mod}\\text{-}\\mathfrak{R}|_{\\mathcal{D}}[\\mathcal{D}]$, where $\\mathfrak{R}|_{\\mathcal{D}}[\\mathcal{D}]$ is the skew category algebra, canonically defined on $(\\mathbf{C},\\mathfrak{R})$, for a uniquely determined full subcateg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.04731","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/2207.04731/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2207.04731","created_at":"2026-07-05T06:07:31.331768+00:00"},{"alias_kind":"arxiv_version","alias_value":"2207.04731v1","created_at":"2026-07-05T06:07:31.331768+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2207.04731","created_at":"2026-07-05T06:07:31.331768+00:00"},{"alias_kind":"pith_short_12","alias_value":"XPLPMJTTZIV7","created_at":"2026-07-05T06:07:31.331768+00:00"},{"alias_kind":"pith_short_16","alias_value":"XPLPMJTTZIV7G6NO","created_at":"2026-07-05T06:07:31.331768+00:00"},{"alias_kind":"pith_short_8","alias_value":"XPLPMJTT","created_at":"2026-07-05T06:07:31.331768+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.08685","citing_title":"A torsion theoretic interpretation for sheaves of modules and Grothendieck topologies on directed categories","ref_index":31,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XPLPMJTTZIV7G6NOPSKICRKKKO","json":"https://pith.science/pith/XPLPMJTTZIV7G6NOPSKICRKKKO.json","graph_json":"https://pith.science/api/pith-number/XPLPMJTTZIV7G6NOPSKICRKKKO/graph.json","events_json":"https://pith.science/api/pith-number/XPLPMJTTZIV7G6NOPSKICRKKKO/events.json","paper":"https://pith.science/paper/XPLPMJTT"},"agent_actions":{"view_html":"https://pith.science/pith/XPLPMJTTZIV7G6NOPSKICRKKKO","download_json":"https://pith.science/pith/XPLPMJTTZIV7G6NOPSKICRKKKO.json","view_paper":"https://pith.science/paper/XPLPMJTT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2207.04731&json=true","fetch_graph":"https://pith.science/api/pith-number/XPLPMJTTZIV7G6NOPSKICRKKKO/graph.json","fetch_events":"https://pith.science/api/pith-number/XPLPMJTTZIV7G6NOPSKICRKKKO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XPLPMJTTZIV7G6NOPSKICRKKKO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XPLPMJTTZIV7G6NOPSKICRKKKO/action/storage_attestation","attest_author":"https://pith.science/pith/XPLPMJTTZIV7G6NOPSKICRKKKO/action/author_attestation","sign_citation":"https://pith.science/pith/XPLPMJTTZIV7G6NOPSKICRKKKO/action/citation_signature","submit_replication":"https://pith.science/pith/XPLPMJTTZIV7G6NOPSKICRKKKO/action/replication_record"}},"created_at":"2026-07-05T06:07:31.331768+00:00","updated_at":"2026-07-05T06:07:31.331768+00:00"}