Index: Foundations

Foundations

Routing Summary

This folder covers the foundational language of category theory: categories, the maps between them (functors), the maps between those maps (natural transformations), and the categories formed by functors. It also includes size issues needed for the Yoneda lemma.

• Need the definition of a category or examples? → Categories
• Need the definition of a functor, full/faithful, equivalence? → Functors
• Need the naturality condition or natural isomorphisms? → Natural Transformations
• Need presheaf categories or the hom bifunctor? → Functor Categories

Concept Map

Concept	Note	Type	Depends On	Key Result
Category definition	Categories	definition	—	Objects + morphisms + composition + identities
Isomorphism	Categories	definition	—	Invertible morphism
Opposite category	Categories	definition	—	Reverse all arrows for duality
Small/locally small	Categories	definition	—	Size distinctions for set-theoretic safety
Functor definition	Functors	definition	Categories	Preserves composition and identities
Full/faithful	Functors	definition	Functors	Bijective/injective/surjective on hom-sets
Equivalence of categories	Functors	definition	Functors	Fully faithful + essentially surjective
Diagonal functor	Functors	definition	Functors	Used in limits: $\Delta :\mathcal{A}\to [\mathcal{I},\mathcal{A}]$
Natural transformation	Natural Transformations	definition	Functors	Commuting squares for every map
Natural isomorphism	Natural Transformations	definition	Natural Transformations	Components are all isomorphisms
Vertical/horizontal composition	Natural Transformations	definition	Natural Transformations	Two ways to compose nat. trans.
Functor category $[\mathcal{A},\mathcal{B}]$	Functor Categories	definition	Natural Transformations	Objects = functors, morphisms = nat. trans.
Presheaf category $\hat{\mathcal{A}}$	Functor Categories	definition	Functor Categories	$[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$

Notes

• Categories — CONTAINS: category definition, examples (Set/Grp/Top/monoid/preorder), isomorphisms, opposite category, size distinctions (small/locally small/large)
• Functors — CONTAINS: functor definition, core examples (forgetful/free/power set), contravariant functors, full/faithful, equivalence of categories, diagonal functor
• Natural Transformations — CONTAINS: natural transformation definition, natural isomorphisms, double-dual example, vertical and horizontal composition, interchange law
• Functor Categories — CONTAINS: functor category definition, presheaf category definition, examples (diagrams as functor categories, sheaves), isomorphisms in functor categories, hom bifunctor

Sources

• 1612.09375v2.pdf — Basic Category Theory, Tom Leinster, Ch. 1.1–1.4 and Ch. 3

See Also

• Adjunctions — Adjoint functors, built on functors and natural transformations
• Representables — Yoneda lemma, uses functor categories heavily