Category theory MOC

Natural transformation

A natural transformation is a morphism in a so-called functor category, that is it is a morphism between two functors, or a 2-morphism in Category of small categories. If 𝐹,𝐺 :𝖢 →𝖣, then a natural transformation 𝜂 :𝐹 ⇒𝐺 :𝖢 →𝖣 consists of a morphism 𝜂𝑋 :𝐹𝑋 →𝐹𝑌 for every 𝑋 ∈𝖢 such that the following diagram commutes: cat

i.e. 𝜂𝑌 𝐹𝑓 =𝐺𝑓 𝜂𝑋 for every 𝑋,𝑌 ∈𝖢.¹

If 𝜂𝑋 :𝐹𝑋 →𝐺𝑋 is an isomorphism for every 𝑋 ∈𝖢, then it is called a Natural isomorphism and we say 𝐹 ≅𝐺.

A slight generalization is an (Extra)natural transformation.

Properties

• The most fundamental result in category theory: the Yoneda lemma
• Identity natural transformation

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Footnotes

1. 2020, Topology: A categorical approach, pp. 11–12 (Definition 0.9) ↩