We investigate the properties of the Kleisli category Kl T of a monad T on a category E and in particular the existence of (some kind of) pullbacks. This culminates when the monad is cartesian. In this case, we show that any T-category in E in the sense of A. Burroni coincides with a special kind of internal category in Kl T. Thus, it is the case in particular for T-operads and T-multicategories. More unexpectedly, this, in turn, sheds new conceptual lights on internal categories and n-categories.
Keywords: Internal categories and groupoids, Kleisli categories, T-categories, internal n-categories and n-groupoids, operads, multicategories.
2020 MSC: 18C20, 18D40, 18M60, 18M65, 18N50
Theory and Applications of Categories, Vol. 41, 2024, No. 34, pp 1108-1159.
Published 2024-09-10.
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