We describe all left continuous triangular norms for which the category [0,1]-Cat of real-enriched categories and functors is cartesian closed. We furthermore show that the cartesian closedness of [0,1]-Cat is equivalent to the cartesian closedness of either (and thus all) of the following subcategories: the full subcategory of Cauchy complete [0,1]-categories; the subcategory of Yoneda complete [0,1]-categories and Yoneda continuous [0,1]-functors; the full subcategory of Smyth complete [0,1]-categories; and the full subcategory of finite [0,1]-categories.
Keywords: enriched categories, cartesian closed, triangular norm, Cauchy complete, Yoneda complete, Smyth complete
2020 MSC: 18B35, 18D20, 06D10, 06F07
Theory and Applications of Categories, Vol. 45, 2026, No. 8, pp 289-306.
Published 2026-02-05.
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