学术报告（李冠宇 6.7）

Abstract:

Homological Algebra explores derived functors. They come from left/right exact functors (between abelian categories) that are not exact. Their constructions, which involve projective/injective resolutions,can be challenging to grasp, but they hold significance across various mathematical disciplines. We will use the example of $\mathrm{Tor}$ functor, to explain why the  derived functor are special, in the sense that it is the functor closest to the original one among all the functors preserving quasi-isomorphisms. This property allows us to extend the notion of derived functors to a wide range of settings, including non-abelian categories. For instance, the singular homology functor defined on the category of topological spaces is a derived functor. With
such an enhancement, we can establish the foundation of derived algebraic geometry.