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Suppose that $\\varphi : X \\rightarrow X$ is a morphism and $(\\varphi_1, Z, \\varphi_2)$ is an (epic, monic) factorization of $\\varphi$ through $Z$, then $\\varphi$ is core invertible if and only if $(\\varphi^{\\ast})^2\\varphi_1$ and $\\varphi_2\\varphi_1$ are both left invertible if and only if $((\\varphi^{\\ast})^2\\varphi_1, Z, \\varphi_2)$, $(\\varphi_2^{\\ast}, Z, \\varphi_1^{\\ast}\\varphi^{\\ast}\\varphi)$ and $(\\varphi^{\\ast}\\varphi_2^{\\ast}, Z, \\varphi_1^{\\ast}\\varphi)$ are all essentially unique (epic, monic) factorizations of $(\\varphi^{\\as"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.08817","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":""},"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":"1804.08817","created_at":"2026-05-18T00:17:37.248305+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.08817v1","created_at":"2026-05-18T00:17:37.248305+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.08817","created_at":"2026-05-18T00:17:37.248305+00:00"},{"alias_kind":"pith_short_12","alias_value":"MZQGGVKU6ML3","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_16","alias_value":"MZQGGVKU6ML3HZHA","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_8","alias_value":"MZQGGVKU","created_at":"2026-05-18T12:32:40.477152+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MZQGGVKU6ML3HZHAKNBMFT3NAO","json":"https://pith.science/pith/MZQGGVKU6ML3HZHAKNBMFT3NAO.json","graph_json":"https://pith.science/api/pith-number/MZQGGVKU6ML3HZHAKNBMFT3NAO/graph.json","events_json":"https://pith.science/api/pith-number/MZQGGVKU6ML3HZHAKNBMFT3NAO/events.json","paper":"https://pith.science/paper/MZQGGVKU"},"agent_actions":{"view_html":"https://pith.science/pith/MZQGGVKU6ML3HZHAKNBMFT3NAO","download_json":"https://pith.science/pith/MZQGGVKU6ML3HZHAKNBMFT3NAO.json","view_paper":"https://pith.science/paper/MZQGGVKU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.08817&json=true","fetch_graph":"https://pith.science/api/pith-number/MZQGGVKU6ML3HZHAKNBMFT3NAO/graph.json","fetch_events":"https://pith.science/api/pith-number/MZQGGVKU6ML3HZHAKNBMFT3NAO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MZQGGVKU6ML3HZHAKNBMFT3NAO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MZQGGVKU6ML3HZHAKNBMFT3NAO/action/storage_attestation","attest_author":"https://pith.science/pith/MZQGGVKU6ML3HZHAKNBMFT3NAO/action/author_attestation","sign_citation":"https://pith.science/pith/MZQGGVKU6ML3HZHAKNBMFT3NAO/action/citation_signature","submit_replication":"https://pith.science/pith/MZQGGVKU6ML3HZHAKNBMFT3NAO/action/replication_record"}},"created_at":"2026-05-18T00:17:37.248305+00:00","updated_at":"2026-05-18T00:17:37.248305+00:00"}