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Real-enriched categories are categories with real numbers as enrichment. Precisely, a real-enriched category is a category enriched over the commutative and unital quantale composed of the unit interval and a continuous t-norm. These notes present a brief introduction to real-enriched categories, focusing on the presheaf monad and its submonads in the category of such categories.
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Dexue Zhang (2024) studied this question.
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