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Small categories, functors, and natural transformations form the strict 2-category
Statement
Small categories, functors, and natural transformations form a strict 2-category .
Facts & Assumptions
Given: Small categories .
A strict 2-category has hom-categories, strictly associative and unital horizontal composition, and interchange (Strict 2-category).
Functors and natural transformations form functor categories (Functor category ), these hom-categories are small for small endpoints (If is small and is locally small then is locally small; if both are small it is small), and interchange holds (Horizontal and vertical composition of natural transformations satisfy the interchange law).
Proof
Take small categories as objects and as each hom-category; its objects are functors and its morphisms are natural transformations.
Functor composition gives horizontal composition, whiskering gives its action on natural transformations, and ordinary composition of functions makes associativity and units literal equalities.
The interchange theorem makes horizontal composition functorial with respect to vertical composition; the class convention of Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why is not formed treats the object collection schematically and never forms , so all strict 2-category axioms hold for .
Depends on
- Strict 2-category
- Functor category $[\mathcal C,\mathcal D]$
- Horizontal and vertical composition of natural transformations satisfy the interchange law
- If $\mathcal C$ is small and $\mathcal D$ is locally small then $[\mathcal C,\mathcal D]$ is locally small; if both are small it is small
- Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $\mathbf{CAT}$ is not formed
Used by
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Sources
- Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)