Algebraic topology

Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces.

Setting in category theory

In general, all constructions of algebraic topology are functorial: the notions of category, functor and natural transformation originated here. Fundamental groups, homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups; a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings.

Related Topics:
Functorial - Category - Functor - Natural transformation - Homeomorphic

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