Alphabeta MathHow statement and proof provenance work
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Strict 2-category
Definition
A strict 2-category consists of a class of objects and, for every ordered pair , a hom-category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection). Objects of a hom-category are 1-morphisms and its morphisms are 2-morphisms.
There are identity 1-morphisms and horizontal-composition functors
where the product and functor notions are those of Product category and its projection functors and Covariant functor, identity functor, composite functor, and contravariant functor. They are associative and unital as literal equalities. Because horizontal composition is functorial, it satisfies the interchange law with the vertical composition inside each hom-category. The adjective strict refers to these equalities rather than coherent isomorphisms.
Depends on
Used by
- A Cat-enriched category is exactly a strict 2-category with a set of objects and small hom-categories Theorem
- Set-object enriched categories, enriched functors, and enriched natural transformations form a strict 2-category Theorem
- Small categories, functors, and natural transformations form the strict 2-category Cat Theorem
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)