There’s also work by Orchard et al. here on category-graded monads:
]]>Our approach can be summarised as the horizontal categorification of monads.
A monad in a bicategory is a lax functor . This definition admits a natural generalisation to a set equipped with a lax functor from the chaotic bicategory on (i.e. having a single 1-cell between each pair of elements of ), which is exactly a category enriched in . Thus categories enriched in a bicategory are many-object generalisations of monads. They were called “polyads” by Bénabou. In particular, your -enriched categories are bicategory-enriched categories where every object has the same extent (namely, ). The paper Enriched categories as a free cocompletion by Garner and Shulman takes this perspective on enriched categories, for instance.
]]>Hi. There’s an MO question What are the internal categories in an endofunctor category which may be of interest.
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