[iheartmemcache]

(To contextualize [hopefully without misrepresenting their positions], Lurie and Vlad both think/thought that axiomatic set theory is not a proper mathematical foundation. Their disagreement lays in how to construct the reformalization of mathematics (Univalent Foundations vs Higher Topos Theory).

A (very rough) analogy would be the general consensus of climate change scientists agreeing on global warming being the result of the rapid re-release of fossil fuels into the ecosystem but disagreeing on whether the cause is from shipping container barges or the rapid industrialization of the BRIC nations.)

Here's what's regarded as the seminal resource[1] on Univalent Foundations. "Homotopy Type Theory: Univalent Foundations of Mathematics" which Vlad was working on at IAS. Not only mathematicians but logicians and computer scientists have made large contributions to this work. Names like Awodey and Robert Harper will certainly sound familiar to the C.S. crowd here.

[1]https://homotopytypetheory.org/book/