  In these self-contained low prerequisite introductory notes we first present
(in part 1) basic concepts of set theory and algebra without explicit category
theory. We then present (in part 2) basic category theory involving a somewhat
detailed discussion of system limits and the exact imbedding of abelian
categories. This is followed (in part 3) by a discussion of localization,
homological algebra, and generalizations of additive and abelian categories
such as triangulated and derived categories. Based on the concepts of category
theory from parts 2 and 3, (in part 4) we provide hints for constructive
discussions on familiar mathematics such as representation theory and
topological geometry/analysis (i.e., topology-based geometry/analysis). If
events permit, the notes will be revised/updated regularly.
