Barwick’s model is described in Lurie’s survey of the cobordism hypothesis, and also in Barwick-Schommer-Pries. You should describe it as a Bousfield localization.
We described Segal n-categories last semester.
How do these compare?
Why iterated complete Segal spaces fail to be Cartesian.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/4676