A directed version of homotopy colimits

In my talk I will discuss a variant of lax colimit and lax limit for diagrams of weak (infinity, infinity)-categories thought of as directed homotopy types. I will demonstrate that this notion is appropriate to perform directed analogues of classical constructions of homotopy theory like suspensions, loop spaces and homotopy fibers, and therefore may be thought of as a directed version of homotopy colimits and homotopy limits. This is joint work with David Gepner.