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Then the gauge action by exponents of the zero degree component $\\Bbbk(\\g_1,\\g_2)^0$ on $MC\\subset\\Bbbk(\\g_1,\\g_2)^1$ gives an explicit \"homotopy relation\" between two $L_\\infty$ morphisms. We prove that the quotient category by this relation (that is, the category whose objects are $L_\\infty$ algebras and morphisms are $L_\\infty$ morphisms modulo the gauge rela"},"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":"0706.1333","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.KT","submitted_at":"2007-06-09T23:55:50Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"6291a8e0dd5c3801b14ac3adcc123dd3c976bf6d82ebe8f57e73746f612b284f","abstract_canon_sha256":"4ebe17d21c97a60094d2ef6486bd1f085f56d39251e1b066fcc1f2e0a6a90381"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:01:44.817294Z","signature_b64":"gzgYERnlVeIrjjp5ddTqHOu0HSNP0uhFABbKDW6JCKsN96dw0XqnX6DsvZPQkm9RYArPHr19XEKFMr3+A0IzDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3812c5299ed96f4a8c8673a51c2035c14af6a0a94e7d75283a400bdfca3dbed5","last_reissued_at":"2026-07-04T15:01:44.816902Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:01:44.816902Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An explicit construction of the Quillen homotopical category of dg Lie algebras","license":"","headline":"","cross_cats":["math.QA"],"primary_cat":"math.KT","authors_text":"Boris Shoikhet","submitted_at":"2007-06-09T23:55:50Z","abstract_excerpt":"Let $\\g_1$ and $\\g_2$ be two dg Lie algebras, then it is well-known that the $L_\\infty$ morphisms from $\\g_1$ to $\\g_2$ are in 1-1 correspondence to the solutions of the Maurer-Cartan equation in some dg Lie algebra $\\Bbbk(\\g_1,\\g_2)$. 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