Linear Algebra

Expository Study Notes on Linear Algbera

Abstract

These notes follow a textbook-style roadmap through category theory, moving from concrete definitions to abstract machinery and onward to cross-domain applications. I open with the core anatomy of a category: objects, hom-sets, associative composition and identity arrows; emphasis is placed on locally-small categories and the dual-category trick that instantly doubles one’s supply of examples. Functors enter next as structure-preserving translations between categories, preparing the ground for natural transformations—“maps of functors” whose commutative-diagram discipline is illustrated with explicit calculations. The Yoneda Lemma is then derived step-by-step, showing how every fobject lives vicariously through its representable functor and why “proof by probing” becomes a legitimate strategy. With this embedding tool in hand I explore limits and colimits (products, equalizers, pull-/push-outs) and spell out their universal properties diagram-by-diagram; adjoint functors appear soon after as “optimal solution providers”, exemplified by free groups and the Stone–Čech compactification. A final chapter bridges to computer science: algebraic data types as algebras for endofunctors, concurrent processes as coalgebras, and how monadic composition captures side effects—culminating in the Beck-style monadicity theorem. Throughout, definitions, theorems and exercises alternate with motivational remarks, mirroring the rhythm of a lecture course and keeping the abstract landscape navigable.

Keywords

Natural Transformation, Functor, Adjoint Functor, Limits

1 Category Theory

1.1 Code Examples