I've been told that it's important to know modern physics, Differential Geometry and Algebraic Topology for understanding higher structures. Is there any other prerequisite for understanding Lurie's work? Since the title of the book indicates, I guess Algebraic Geometry is also important. Please tell me if I'm wrong. Moreover how deep should I known on those subjects and others I do not mention?

31$\begingroup$ I don't understand any of Lurie's work, but the advice you've been given is awfully broad. Differential Geometry and Algebraic Topology are very broad subjects, including many subfields. Any useful advice should include at least "...on the level of (insert book here)". And "modern physics"? I hope an understanding on how to build experiments with attosecond lasers (for example) is not necessary to read Lurie's work. $\endgroup$ – Jan Jitse Venselaar Sep 6 '11 at 10:39

2$\begingroup$ I believe his second book is called "Higher Algebra". See math.harvard.edu/~lurie $\endgroup$ – S. Carnahan♦ Sep 6 '11 at 10:39

4$\begingroup$ Heavy use of simplicial language and homotopy theory are the things that limit my ability to understand the stuff :). Modern physics and curvature tensors probably not so much. $\endgroup$ – Daniel Pomerleano Sep 6 '11 at 11:41

6$\begingroup$ Jacob Lurie mentions the prerequisites in the intros of his books. On the other hand it is great that he answers here directly. $\endgroup$ – Martin Brandenburg Sep 15 '11 at 16:44
To read Higher Topos Theory, you'll need familiarity with ordinary category theory and with the homotopy theory of simplicial sets (Peter May's book "Simplicial Objects in Algebraic Topology" is a good place to learn the latter). Other topics (such as classical topos theory) will be helpful for motivation.
To read "Higher Algebra", you'll need the above and familiarity with parts of "Higher Topos Theory". Several other topics (stable homotopy theory, the theory of operads) will be helpful for motivation.
To read the papers "Derived Algebraic Geometry ???", you need all of the above plus familiarity with Grothendieck's theory of schemes, along with some more recent ideas in algebraic geometry (stacks, etcetera).
Since no knowledge of modern physics was required to write any of these books and papers, I can't imagine that you need any such knowledge to read them.

5$\begingroup$ Thank you very much. You have recommended J.P. May's "Simplicial Objects in Algebraic Topology" for the homotopy theory of simplicial sets, would you like to recommend some other specific books to read before reading your books? $\endgroup$ – Chuang Sep 6 '11 at 15:26

25$\begingroup$ "Sheaves in Geometry and Logic" by Moerdijk and MacLane is a pretty good read (as is Uncle John, but I've never seen topos theory in there). $\endgroup$ – Jacob Lurie Sep 7 '11 at 23:35

3$\begingroup$ I'm usually pretty up on my math injokes, but I've never heard of Uncle John's bathroom reader. Can anyone enlighten me? $\endgroup$ – Andy Putman Sep 8 '11 at 5:16

5$\begingroup$ en.wikipedia.org/wiki/Uncle_John%27s_Bathroom_Reader $\endgroup$ – algori Sep 9 '11 at 0:54

8$\begingroup$ Dear prof Lurie I am reading your works and after reading Higher Topoi I would be delighted to start your book on Higher Algebra posted on your website, very interesting for my research focused on HoTT and Set Theory. Can I ask you to post or send a version with expanded font size (12 or better 14) since it is very difficult for me to read it with the actual small font size? Thank you in advance Best regards Giovanni $\endgroup$ – user67032 Feb 12 '15 at 11:29