Operads, homotopy algebras and all that jazz…

Vasily Dolgushev

The goal of this series of lectures is to give an answer to the question "What do homotopy algebras form?". My presentation will be based on recent papers arXiv:1406.1744 and arXiv:1406.1751. In the first lecture, I will recall the notions of operad, cooperad and cobar construction. I will define homotopy algebras, give several examples and formulate the homotopy transfer theorem. In the second lecture, I will introduce L-infinity algebras, the Deligne-Getzler-Hinich(DGH) infinity groupoid and talk about categories enriched "over L-infinity algebras". In the third lecture, I will show that homotopy algebras of a fixed type naturally form a category enriched "over L-infinity algebras". Finally, I will explain in what sense this enriched category stands behind the homotopy category of homotopy algebras of a fixed type.

http://scgp.stonybrook.edu/video_portal/video.php?id=1368

Simons- Workshop: Homological Methods in Quantum Field Theory