How to learn motivic homotopy theory？

Question

What prerequisite knowledge do I need to know to learn motivic homotopy theory？ And what materials can I refer to to learn motivic homotopy theory?

Answer 1

For a newcomer, there are a number of references that don't have any real prerequisites. The most famous (and my favorite) is probably Lectures at a Summer School in Nordfjordeid. A quick reference is this handbook. Other good references are A primer for unstable motivic homotopy theory and Notes on Homotopy and $A^1$ homotopy, which are more for geometers. A classic text is the Morel-Voevodsky article (which I am not sure is a good starting point). For the abstract and conceptual approach to the construction of the motivic category I definitely recommend Dugger's amazing texts: https://math.mit.edu/~dspivak/files/cech.pdf and https://arxiv.org/pdf/math/0007070.pdf .

It has to be noted that while almost all the references try to give you model categories, homotopy theory and even algebraic geometry from scratch, I'd rather read about these things in specialized textbooks first, for otherwise it looks hard to digest. For simplicial stuff and model categories, Simplicial Homotopy Theory is the standard reference (the first two chapters plus the one on localizations should be enough). Plus for abstract homotopy theory, Riehl's book is good. For sheaves, Dugger's texts should be enough. Finally, algebraic geometry and algebraic topology literature must have been described elsewhere.