Six operations formalism for generalized operads

Abstract

This paper shows that generalizations of operads equipped with their respective bar/cobar dualities are related by a six operations formalism analogous to that of classical contexts in algebraic geometry. As a consequence of our constructions, we prove intertwining theorems which govern derived Koszul duality of push-forwards and pull-backs. Contents 1. Introduction. 1 2. Categorical six operations formalism. 6 3. Quillen adjunctions from morphisms of Feynman categories. 7 4. The functor L ! . 10 5. The functor R ! . 16 6. Quadratic intertwining and Manin products. 18 7. Six operations formalisms. 22 8. Verdier duality is Koszul duality 25 9. Examples. 27 Appendix A. Feynman categories reference guide. 35 References 38