Introduction In the last post we studied homology theories as abstract functors from stable $\infty$-categories to abelian categories. We showed that for every stable $\infty$-category $\mathcal{C}$ there is a universal homology theory which all others factor through, namely the Yoneda embedding into the Freyd envelope,
$$ y\colon \mathcal{C}\longrightarrow A(\mathcal{C}). $$
The fact that this is universal means that for any homology theory $H\colon \mathcal{C}\longrightarrow \mathcal{A}$ there is an essentially unique factorization...

Schemes

The first two posts ([1],[2]) I ever did on this blog - now over a year ago - were posts about algebraic geometry. In particular we explored the geometric implications of some of the algebraic results I was learning in my commutative algebra class. Last summer I also wrote a post about sheaves, and left it off by claiming to soon write about schemes. If you scroll through the blog we have covered a bunch of different topics, but the blog post on schemes, seems to have fallen through the cracks....

Sheaves

This summer I’m participating in Ravi Vakils pseudocourse on algebraic geometry, AGITTOC. Hence this summer serves as a wonderful opportunity to learn and write about cool mathematics. For long I have wanted to dive deeper into this abstract topic after just dipping my toes in during my bachelor thesis, and now it is time. Ravi though us in the first lecture that we shouldn’t study abstract objects without a cause, i....