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Category Theory
Explore objects and maps; universal constructions; representability and algebra. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Function composition
Composition applies one function after another where the output fits the next domain.
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Objects and maps
Categories
A category has objects, morphisms, associative composition and identity morphisms.
Video explanations to be addedFunctors
A functor maps objects and morphisms while preserving identities and composition.
Video explanations to be addedNatural transformations
Compatible morphisms between functor values commute with every source-category morphism.
Video explanations to be addedCategorical isomorphisms
A morphism is an isomorphism when it has a two-sided compositional inverse.
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Universal constructions
Universal properties
A universal property characterizes an object by unique compatible maps.
Video explanations to be addedProducts and coproducts
Products receive universal compatible maps; coproducts reverse the direction.
Video explanations to be addedLimits and colimits
Limits and colimits generalize universal constructions over diagrams of objects and arrows.
Video explanations to be addedAdjunctions
An adjunction relates morphism sets naturally between two functors.
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Representability and algebra
Yoneda lemma
Maps from a representable functor into a functor correspond naturally to an element at its representing object.
Video explanations to be addedEquivalence of categories
Functors inverse up to natural isomorphism express equivalence without literal equality of categories.
Video explanations to be addedMonoidal categories
A tensor-like product and unit object obey coherent associativity and unit constraints.
Video explanations to be addedAbelian categories
An abelian category supports additive structure, kernels, cokernels and exactness with compatible axioms.
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