Kaj Börjeson: On the Operad of 2-Gerstenhaber Algebras

The purpose of this thesis is to provide an introduction to the theory of operads with a focus on homological algebra and to analyze the homological properties of the operad of 2-Gerstenhaber algebras. With this view we present among other things the fundamentals of category theory, monoidal categories and operads with a focus on topological and algebraic operads. We define the cobar construction of an operad and the concept of Koszul operad. Koszulity of an operad provides a straightforward method of constructing a minimal resolu- tion via the cobar construction. We analyze the operad 2-Gerst of 2-Gerstenhaber algebras that was introduced by Etingof et al. in [EHKR] as the cohomology of the moduli space of stable genus zero real algebraic curves. Our main new result is a proof that this operad is a Koszul operad. This is done via the use of a distributive law as introduced by Markl in [Mar].