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Jan, 2021
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Homological Algebra

When people try to learn algebraic topology, they are forced to get familiar with chain complexes and hence basic homological algebra.

Then it will not be hard to learn homological algebra in more general settings based on Abelian categories. Or Abelian categories are defined by extracting essential properties of categories of modules.

People have been trying to extend homological algebra.

We sometimes need generalizations and variations of Abelian categories.

We need exact categories for algebraic K -theory .

From a homotopy theoretic point ov view, homological algebra is a part of homotopical algebra.

Derived categories and triangulated categories are now popular tools in various fields in algebra and geometry. And we can reformulate them in the language of model categories . For example, it has been recognized that the notion of triangulated categories is not sufficient in many algebraic and geometric problems. People start to think about dg categories and spectral categories without taking their derived categories. The language of model categories is essential in such cases. As is stated in [ ? ] by Ben-Zvi, Francis, and Nadler, such structures are now called enhanced triangulated categories, after Bondal adn Kapranov [ ? ] .

The surge of notions of spectral categories and stable -categories required us to unify the two origins of triangulated categories, stable homotopy theory and homological algebra.