Résume  The class of NSOP_1 theories contains the simple theories and many interesting nonsimple theories, such as the omegafree PAC fields or generic vector spaces with a nondegenerate bilinear form. With Itay Kaplan, we introduced Kimindependence which agrees with nonforking independence within the simple theories and shares many of its nice properties within the simple NSOP_1 context. One very basic roadblock in lifting simplicity theory to the NSOP_1 setting, however, was transitivity: a free extension of a free extension should still be a free extension. This is almost immediate for nonforking extensions in a simple theory, but becomes more involved for free extensions in the sense of Kimindependence. We will describe and motivate the basic theory, and then discuss our recent proof of transitivity. This is joint with Itay Kaplan.
