$P$-functors

$P^n$ objects are a class of objects in derived categories of algebraic varieties first studied by Huybrechts and Thomas. They were shown to give rise, naturally, to derived autoequivalences. It was also shown that they could sometimes be produced out of spherical objects by taking a hyperplane section of the ambient variety.
In this talk we'll explain how to generalise the above to the notion of $P$-functors between (enhanced) triangulated categories. We'll also discuss a closely related notion of a non-commutative line bundle over such category. This is based on work in progress with Rina Anno and Ciaran Meachan.