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Algebraic Tools

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.