{"paper":{"title":"Twisting structures and strongly homotopy morphisms","arxiv_id":"1007.1934","primary_cat":"math.AT","submitted_at":"2010-07-12T16:25:16+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13-20260917 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.96,"id":"s0","own":true,"role":"contribution","text":"In an application of the notion of twisting structures introduced by Hess and Lack, we define twisted composition products of symmetric sequences of chain complexes that are degreewise projective and finitely generated.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.04,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.96,"interpretation":0.0}},{"i":1,"p":0.54,"id":"s1","own":false,"role":"background","text":"Let Q be a cooperad and let BP be the bar construction on the operad P.","hedge":{"p":0.65,"label":"asserted"},"own_p":0.22,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.0,"method":0.33,"result":0.0,"background":0.54,"limitation":0.0,"future_work":0.0,"contribution":0.13,"interpretation":0.0}},{"i":2,"p":0.95,"id":"s2","own":true,"role":"contribution","text":"To each morphism of cooperads g from Q to BP is associated a P-co-ring, K(g), which generalizes the two-sided Koszul and bar constructions.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.82,"main_claim_p":0.08,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.03,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.95,"interpretation":0.0}},{"i":3,"p":0.98,"id":"s3","own":true,"role":"result","text":"When the co-unit from K(g) to P is a quasi-isomorphism, we show that the Kleisli category for K(g) is isomorphic to the category of P-algebras and of their morphisms up to strong homotopy, and we give the classifying morphisms for both strict and homotopy P-algebras.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.68,"quantitative":false,"quantitative_p":0.23,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.98,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}},{"i":4,"p":0.98,"id":"s4","own":true,"role":"contribution","text":"Parametrized morphisms of (co)associative chain (co)algebras up to strong homotopy are also introduced and studied, and a general existence theorem is proved.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.08,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.02,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.98,"interpretation":0.0}},{"i":5,"p":0.98,"id":"s5","own":true,"role":"contribution","text":"In the appendix, we study the particular case of the two-sided Koszul resolution of the 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