The Yoneda lemma states that for and there is a natural bijectionIf is an epimorphism in the functor category, it is pointwise surjective, so is surjective. Yoneda identifies this map withThus every representable functor is a projective object in a category.
The colimit form of the Special adjoint functor theorem says that a colimit-preserving functor from a locally small, cocomplete, well-copowered category with a small generating family into a locally small category has a right adjoint functor. For small , the category is locally small and has colimits pointwise. Quotients of are represented by compatible equivalence relations on the sets , so they form a set; hence the category is well-copowered. The set of representables generates it by the Yoneda lemma. The theorem therefore gives a right adjoint to every small-colimit-preserving functorIn particular, product with a fixed functor is computed pointwise, and preserves colimits in the Category of sets. Hence preserves all small colimits and has a right adjoint . Thus is a cartesian closed category.
Now work in and write . If has binary products, thenThus exponentiation by is precomposition with . Precomposition between functor categories has a right adjoint given by Right Kan extension, so is a tiny object.
Conversely, suppose has a terminal object and is tiny. The representable is the terminal presheaf, and the exponential adjunction plus Yoneda givesSince is tiny, is a left adjoint and preserves all colimits; evaluation at also preserves pointwise colimits. Therefore the hom functor preserves coproducts and epimorphisms. Preservation of epimorphisms makes projective, while preservation of coproducts makes it indecomposable.
Articles by others on the same topic
There are currently no matching articles.