On Topological Methods in Homological Algebra

Homological Algebra

Notes on Basic Homological Algebra

Categories and Homological Algebra

The aim of these Notes is to introduce the reader to the language of categories with emphazis on homological algebra. We treat with some details basic homological algebra, that is, categories of complexes in additive and abelian categories and construct with some care the derived functors. We also introduce the reader to the more sophisticated concepts of triangulated and derived categories. Ou...

Some homological algebra

1.1 Adjoint functors Let C and D be categories and let f∗ : C → D and f∗ : D → C be functors. Then f∗ is a right adjoint to f∗ and f∗ is a left adjoint to f∗ if, for each D ∈ D and C ∈ C there is a natural vector space isomorphism HomC(fD,C) Φ −→HomD(D, f∗C). For C ∈ C and D ∈ D define τC = Φ(idf∗C) ∈ HomC(ff∗C,C) and φD = Φ(idf∗D) ∈ HomD(D, f∗fD). In general τC and φD are neither injective nor...

Homological Algebra and Data

These lectures are a quick primer on the basics of applied algebraic topology with emphasis on applications to data. In particular, the perspectives of (elementary) homological algebra, in the form of complexes and co/homological invariants are sketched. Beginning with simplicial and cell complexes as a means of enriching graphs to higher-order structures, we define simple algebraic topological...