Notes on homotopy limits, singular constructions, fiber sequences (handwritten notes for the reading group talk):
I have made some notes and remarks about motivic homotopy theory while working on my master dissertation during the summer of 2019.
Affine representability results:
For remarks on section 2.3 and Lemma 2.3.2 of  (also quoted as Proposition 5.5 and Proposition 5.6 in ), we take the approach of stack and descent. By relating hyperdescent condition for simplicial presheaves with descent for stacks , we show that how this implies classifying space of a stack satisfies hyperdescent.
After that, we give a different proof and some remarks of Lemma 2.3.2 based on my understanding, see Proposition 0.0.2 and Proposition 0.0.3 in Stack and descent.
Exercises on the construction of motivic homotopy theory:
I also have some informal notes about the exercises in . The following are some of my proofs for the exercises, in the remarks, I also pointed out some typos I found which may be helpful for other readers: Exercises on A Primer.
 Antieau, B., & Elmanto, E. (2017). A primer for unstable motivic homotopy theory. Surveys on recent developments in algebraic geometry, 95, 305-370.
 Asok, A., Hoyois, M., & Wendt, M. (2018). Affine representability results in 𝔸1–homotopy theory, II: Principal bundles and homogeneous spaces. Geometry & Topology, 22(2), 1181-1225.
 Jardine, J. F. (2015). Local homotopy theory. Springer.