Question

What are the prerequisites for Jacob Lurie's Higher Algebra?

Answer 1

IMO the prerequisites are to have a problem that uses infinity category (whether it be in your research or a paper you are reading) and follow the relevant sections of higher algebra along side it.

Answer 2

You should know homological algebra, e.g. the first half of Weibel's book. Simplicial methods there is particularly important to know. I'd also suggest the third part of Adams' blue book to get to know spectra more concretely, although ignore his construction of the category of spectra.

Answer 3

Apart from the formal pre requisites already mentioned, make sure you know and are comfortable with plenty of explicit, concrete geometric examples before going into higher categories. For example, become comfortable first with spaces such as Grassmanians, Projective spaces, Spheres, Torus, Klein bottles, fibre bundles of the many sorts, and do many computations with them like fundamental groups, cohomology, etc