{"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 $\\Bbbk(\\g_1,\\g_2)$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0706.1333","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/0706.1333/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"}