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 BenZvi, 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.
