Notes on homotopy limits, singular constructions, fiber sequences (handwritten notes for the reading group talk):

# Category: Motivic homotopy

# Notes and Remarks on Unstable Motivic Homotopy Theory

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 [2] (also quoted as Proposition 5.5 and Proposition 5.6 in [1]), we take the approach of stack and descent. By relating hyperdescent condition for simplicial presheaves with descent for stacks [3], 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 [1]. 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.

[1] Antieau, B., & Elmanto, E. (2017). A primer for unstable motivic homotopy theory. *Surveys on recent developments in algebraic geometry*, *95*, 305-370.

[2] 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.

[3] Jardine, J. F. (2015). *Local homotopy theory*. Springer.