Graduated locally finitely presentable categories are introduced, examples include categories of sets, vector spaces, posets, presheaves and Boolean algebras. A finitary functor between graduated locally finitely presentable categories is proved to be a right adjoint if and only if it preserves countable limits. For endofunctors on vector spaces or pointed sets even countable products are sufficient. Surprisingly, for set functors there is a single exception of a (trivial) finitary functor preserving countable products but not countable limits.
Keywords: Locally finitely presentable categories, finitary functors
2020 MSC: 18A22, 18A35, 18A40, 18B05
Theory and Applications of Categories, Vol. 41, 2024, No. 53, pp 1919-1936.
Published 2024-11-19.
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