What is homological algebra used for?

Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other ‘tangible’ mathematical objects. A powerful tool for doing this is provided by spectral sequences.

What is homology in algebra?

homology, in mathematics, a basic notion of algebraic topology. Intuitively, two curves in a plane or other two-dimensional surface are homologous if together they bound a region—thereby distinguishing between an inside and an outside.

Who invented homological algebra?

Homological algebra had its origins in the 19th century, via the work of Riemann (1857) and Betti (1871) on “homology numbers,” and the rigorous development of the notion of homology numbers by Poincaré in 1895.

What is homology group used for?

When the underlying object has a geometric interpretation as topological spaces do, the nth homology group represents behavior in dimension n. Most homology groups or modules may be formulated as derived functors on appropriate abelian categories, measuring the failure of a functor to be exact.

What is an example of homology?

An example of homologous structures are the limbs of humans, cats, whales, and bats. Regardless of whether it is an arm, leg, flipper or wing, these structures are built upon the same bone structure. Homologies are the result of divergent evolution.

Is geometric algebra useful?

Geometric algebra has been advocated, most notably by David Hestenes and Chris Doran, as the preferred mathematical framework for physics. Proponents claim that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity.

How can I understand Yoneda lemma?

Roughly speaking, the Yoneda lemma says that one can recover an object X up to isomorphism from knowledge of the hom-sets Hom(X,Y) for all other objects Y. Equivalently, one can recover an object X up to isomorphism from knowledge of the hom-sets Hom(Y,X).