Abstract |
: |
Historically, homological algebra grew out of algebraic topology, where one attempts to simplify and solve problems involving geometric structures by associating to them algebraic ones. Homological algebra then gives the calculus for extracting the required information from the algebraic objects. Today, the subject also has many applications in other areas of mathematics apart from algebraic topology, and so one may want to begin studying homological algebra without necessarily going through its geometric background. This is precisely what we will do in these notes, which attempt to give a quick introduction to basic concepts and constructions of homological algebra. Our approach here is inspired by a recent work of Marco Grandis, which bases the foundation of homological algebra on an interplay between Galois connections between lattices of ``substructures'', and ``structure preserving maps'' between the ``structures''. In fact, we slightly generalise the theory of Marco Grandis to cover further examples where homological algebra has been applied/developed. In this sense, our approach is new. This aspect of these notes is part of an ongoing joint research project with Zurab Janelidze. |