Mathematics / advanced
Categories, functors, and universal constructions
Organize mathematical structures with universal properties, natural transformations, adjunctions, and categorical limits.
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Inside this module
Module 01 advanced
Categories, functors, and universal constructions
Organize mathematical structures with universal properties, natural transformations, adjunctions, and categorical limits.
- 01
Categories, morphisms, and composition
Build categories from typed arrows and distinguish smallness, composition, and isomorphism.
34 min - 02
Functors and natural transformations
Preserve categorical structure and test compatibility across every arrow of a diagram.
34 min - 03
Universal properties and representable functors
Describe an object by a natural family of mapping problems and a universal element.
34 min - 04
The Yoneda lemma and fully faithful functors
Recover natural transformations from identity arrows and track the variance of represented presheaves.
34 min - 05
Products, coproducts, and equalizers
Build finite universal constructions and verify their mapping directions explicitly.
34 min - 06
Limits, colimits, and comma categories
Organize compatible diagrams with universal cones and compute finite pullbacks and pushouts.
34 min - 07
Adjoint functors and the unit–counit equations
Calculate a natural Hom bijection and verify the triangle identities in a finite currying example.
34 min - 08
Monads and algebras for a monad
Flatten nested finite subsets and verify the unit and algebra laws by explicit calculation.
34 min - 09
Equivalences, localizations, and inverting morphisms
Compare categorical presentations and enforce inversion by a universal mapping property.
34 min - 10
Enriched and monoidal categories
Replace Hom sets by structured objects and interpret directed shortest paths as enriched composition.
34 min