Homotopy Category Of Chain Complexes
In homological algebra in mathematics, the homotopy category ''K(A)'' of chain complexes in an additive category ''A'' is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes ''Kom(A)'' of ''A'' and the derived category ''D(A)'' of ''A'' when ''A'' is abelian; unlike the former it is a triangulated category, and unlike the latter its formation does not require that ''A'' is abelian. Philosophically, while ''D(A)'' turns into isomorphisms any maps of complexes that are quasi-isomorphisms in ''Kom(A)'', ''K(A)'' does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence. Thus, ''K(A)'' is more understandable than ''D(A)''. Definitions Let ''A'' be an additive category. The homotopy category ''K(A)'' is based on the following definition: if we have complexes ''A'', ''B'' and maps ''f'', ''g'' from ''A'' to ''B'', a chain homoto ...
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Homological Algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of module (mathematics), modules and Syzygy (mathematics), syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariant (mathematics), invariants of ring (mathematics), rings, modules, topological spaces, and other 'tan ...
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