{"paper":{"title":"Tensor Products of $A_\\infty$-algebras with Homotopy Inner Products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Ronald Umble, Thomas Tradler","submitted_at":"2011-02-01T00:39:24Z","abstract_excerpt":"We show that the tensor product of two cyclic $A_\\infty$-algebras is, in general, not a cyclic $A_\\infty$-algebra, but an $A_\\infty$-algebra with homotopy inner product. More precisely, we construct an explicit combinatorial diagonal on the pairahedra, which are contractible polytopes controlling the combinatorial structure of an $A_\\infty$-algebra with homotopy inner products, and use it to define a categorically closed tensor product. A cyclic $A_\\infty$-algebra can be thought of as an $A_\\infty$-algebra with homotopy inner products whose higher inner products are trivial. However, the highe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.0047","kind":"arxiv","version":3},"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"}