KOSZUL DUALITY FOR OPERADS MASTERCLASS UNIVERSITY OF COPENHAGUEN

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KOSZUL DUALITY FOR OPERADS MASTERCLASS, UNIVERSITY OF COPENHAGUEN SHEET I LECTURE 1. OPERADS Exercise 1 (Operadic ideal). Let (P, ?, ?) be an operad, that is here a monoid in the monoidal category (S-Mod, ?, I) of S-modules. An ideal of the operad P is a sub-S-module I ? P such that ?(µ, ?1, . . . , ?k) ? I when at least one of the µ, ?1, . . . , ?k lives in I. (1) Show that the quotient S-moduleP/I is endowed with a canonical operad structure which satisfies the classical property of quotients. (2) Prove that the free ns operad T ( ? ? ) on one binary generator is given by the space spanned by planar binary trees, with the operadic composition given by the grafting of trees. (3) Make explicit the ideal generated by ? ? ? ? ? ? ? ? ? ? in the free nonsymmetric operad T ( ? ? ). (4) Recover the nonsymmetric operad As. Exercise 2 (Diassociative algebras). By definition, a dimonoid is a set D equipped with two maps a : D ?D ? D and : D ?D ? D , called the left operation and the right operation respectively, satisfying the following five relations ? ????? ????? (x a