🗺️ Map of Content - Category Theory

Basic structures

Serge Lang

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

Mac Lane

..."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

Serge Lang

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

Mac Lane

... 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

Mac Lane

... 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