🗺️ Map of Content - Category Theory
Basic structures
In the forties and fifties (mostly in the works of Cartan, Eilenberg, MacLane, and Steenrod), it was realized that there was a systematic way of developing certain relations of linear algebra, depending only on fairly general constructions which were mostly arrow-theoretic, and were affectionately called abstract nonsense by Steenrod.
My source: Riehl
..."category" has been defined in order to be able to define "functor" and "functor" has been defined in order to be able to define "natural transformation."
My source: Mac Lane
Categories
Functors
Natural transformations
Functor categories
Special morphisms
Universal properties
In the forties and fifties (mostly in the works of Cartan, Eilenberg, MacLane, and Steenrod), it was realized that there was a systematic way of developing certain relations of linear algebra, depending only on fairly general constructions which were mostly arrow-theoretic, and were affectionately called abstract nonsense by Steenrod.
My source: Riehl
Universal Properties I - Inspiring Examples
Universal Properties II - Commutative diagrams, cones and limits
Universal arrows and elements
Yoneda's Lemma
Examples of universal properties - Revisited
Adjoints
... good general theory does not search for the maximum generality, but for the right generality.
My source: Mac Lane
Adjoints
Examples of adjoints
Properties of adjoints
Abelian categories
... the subject languished until ... the discovery by Grothendieck that categories of sheaves (of abelian groups) over a topological space were abelian categories but not categories of modules, and that homological algebra in these categories was needed for a complete treatment of sheaf cohomology. With this impetus, abelian categories joined the establishment.
My source: Mac Lane
Basic structures
Ab-categories
Additive categories
Abelian categories
Chain complexes
Chain complexes
Exact sequences
Double complexes and mural maps
The Salamander Lemma
Diagram lemmas
Diagram chases without elements