Work of Joyal, Lurie and many other contributors can be summarized by saying that ordinary 1-category theory extends to (∞,1)-category theory: that is, there exist homotopical/derived analogs of 1-categorical results. As “brave new algebra” grows in influence, many areas of mathematics now require homotopical/derived analogs of 2-categorical results and this work largely remains to be done in a rigorous fashion. In my talks, I will give an overview of the development of (∞, 1)-category theory in the quasi-categorical model and describe the main idea behind the proof that this theory is “model independent.” I’ll then suggest some models of (∞, 2)- categories that might prove fertile for studying extensions of 2-category theory and sketch a possible strategy to demonstrate model independence. Reading List: • J. Lurie “(∞, 2)-categories and the Goodwillie Calculus I”, October 8, 2009. 
Available from http://www.math.harvard.edu/∼lurie/papers/GoodwillieI.pdf. 
 • D. Gaitsgory and N. Rozenblyum , Appendix A of A study in derived algebraic geometry, Mathematical Surveys and Monographs, Vol. 221 (2017), pp. 419 - 524.
Available from http://www.math.harvard.edu/∼gaitsgde/GL/. 
 • G. M. Kelly “Elementary observations on 2-categorical limits”, Bull. Austral. Math. Soc., Vol. 39 (1989), pp. 301-317. 
Potential participants should skim bits of the first two, but need not read either in full.