As the name suggests, algebraic tools are
indispensable in this field. For example, one of the
motivations of
homological algebra
was to develope tools to study homology groups in
algebraic topology. The notion of
Hopf algebras
was introduced and developed in algebraic topology.
We use standard algebraic structures, such as
groups
and
rings
. We also need algebraic structures that only satisfy
a part of conditions for groups or rings.
Conversely, tools from algebraic topology have been
used in algebra. For example,
cohomology operations
are used in the study of the cohomology of groups,
for the cohomology of a group
G
be regarded as the cohomology of the
classifying space
BG
. Furthermore, in the modern treatment of homological
algebra, homotopy theoretic viewpoints, such as theory
of
model categories
, are playing the central role.
