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

Depends on

• Functor category $[\mathcal C,\mathcal D]$
• Small, locally small, and large categories

Used by

• A category satisfying the explicit SAFT intersection hypotheses is cocomplete Corollary
• Global Kan extensions as adjoints to restriction Definition
• Set-weighted limits and colimits Definition
• The arrow category of an abelian category Definition
• The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding Definition
• FALSE: A monad is a monoid object in the endofunctor category for every category False statement
• FALSE: the free-cocompletion theorem holds for an arbitrary large locally small source category with no change in meaning False statement
• A weighted limit of a set-valued diagram is the set of natural transformations from the weight Proposition
• Additive functors and natural transformations form a preadditive category Proposition
• An adjunction induces postcomposition and precomposition adjunctions on legitimate functor categories Proposition
• Local smallness does not make every natural-transformation collection a set, but the Yoneda construction proves sethood in the representable-source case Remark
• The Kleisli and Eilenberg–Moore universal properties are schematic Remark
• The monoid description of a monad requires an endofunctor category Remark
• A presheaf category on a small category is cartesian closed Theorem
• Additive functors from a small preadditive category to an abelian category form an abelian category Theorem
• Density theorem for a small category Theorem
• For a small source category, the set of natural transformations is an end of the hom-bifunctor of the values Theorem
• For small source and index categories, chosen target limits and colimits compute the corresponding functor-category limits and colimits pointwise Theorem
• Small categories, functors, and natural transformations form the strict 2-category Cat Theorem
• The category of small categories is cartesian closed Theorem
• The endofunctor category of a small category is strict monoidal under composition Theorem
• The presheaf category on a small category is the free cocompletion Theorem
• The Yoneda embedding is its own pointwise left Kan extension Theorem

Dependency tree · two levels

7 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)