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(1) Describe the nonsymmetric dg operad A ∞: = ΩAs ¡. (2) Show that A∞-algebras are the algebras over the nonsymmetric dg operad A∞. Exercise 2 (∞-morphisms for A∞-algebras). (1) Make explicit the definition of an A∞-algebra using the four equivalent definitions of the Rosetta Stone. (2) Make explicit the notion of ∞-morphism bewteen A∞-algebras. (This notion is also known as A∞-morphism.) Exercise 3 (Lie algebra up to homotopy, alias L∞-algebra). (1) Make explicit the definition of an algebra over the operad L ∞: = ΩLie ¡ using the four equivalent definitions of the Rosetta Stone. (2) Make explicit the notion of ∞-morphism between L∞-algebras. (This notion is also known as L∞-morphism.) Exercise 4 (Homotopy mixed complexes, alias multicompexes). (1) Make explicit the definition of an algebra over the operad D ∞: = ΩD ¡ , where D: = K[ε]/(ε 2) is the algebra of dual numbers, using the four equivalent definitions of the Rosetta Stone. (When |ε | = 1, a D-algebra is called a mixed complex, in the literature, and a D∞-algebra is called a multicomplex.) (2) Make explicit the notion of ∞-morphism between D∞-algebras. (The notion of ∞-quasi-isomorphism allows one to compare the various definitions of cyclic homology, for instance.) Exercise 5 (Homotopy transfer theorem). (1) Make the formulae for the HTT explicit in the cases of the operads D: = K[ε]/(ε 2), As, and Lie. (2) In the case of the algebra of dual numbers D, explain the relationship between this formula and the construction of the spectral sequence associated to a bicomplex by diagram chasing. Exercise 6 (Commutative algebra up to homotopy, alias C∞-algebra). (1) Describe the morphisms of operads which correspond to the functors Lie → Ass → Com, Lie algebras ← associative algebras ← commutative algebras. (2) Show that this induces morphisms of dg cooperad

Year: 2013

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