Comments or suggestions:

Categorical mathematics

Set theory?

Category theory

Homotopical algebra

Higher category theory

Theory of quasi-categories

Higher quasi-categories?

Geometry?

Elementary geometry?

Differential geometry?

Lie theory?

Algebraic geometry?

Homotopical algebraic geometry?

Number theory?

Elementary number theory?

Algebraic number theory?

Algebra

Universal algebra?

Group theory?

Rings and modules?

Commutative algebras?

Lie algebras?

Representation theory?

Operads?

Homological algebra

Logic?

Boolean algebra?

First order theory?

Model theory?

Categorical logic?

Topology?

General topology

Algebraic topology?

Homotopy theory?

Theory of locales?

Topos theory?

Higher topos theory?

Combinatorics?

Combinatorial geometry?

Enumerative combinatorics?

Algebraic combinatorics?

Contributors

References

Introduction

Basic category theory

Weak factorisation systems

Factorisation systems

Distributors and barrels

Model structures on Cat

Homotopy factorisation systems in Cat?

Accessible categories?

Locally presentable categories?

Algebraic theories and varieties of algebras?