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I added more to idempotent monad, in particular fixing a mistake that had been on there a long time (on the associated idempotent monad). I had wanted to give an example that addresses Mike’s query box at the bottom, but before going further, I wanted to track down the reference of Joyal-Tierney, or perhaps have someone like Zoran fill in some material on classical descent theory for commutative algebras (he wrote an MO answer about this once) to illustrate the associated idempotent monad.
Some of this (condition 2 in the proposition in the section on algebras) was written as a preparatory step for a to-be-written nLab article on Day’s reflection theorem for symmetric monoidal closed categories, which came up in email with Harry and Ross Street.
Thanks! I fixed a bit of formatting, and changed some appearances of $R L$ to $M$ which it seemed like they wanted to be.
I have made more explicit the statement that the Eilenberg-Moore category of an idempotent monad induced by a reflection reproduces the underlying reflective subcategory; at idempotent monad, at reflective subcategory and at Eilenberg-Moore category, pointing also to Borceux vol 2.
Added a reference to idempotent adjunction.
added this pointer:
To the list of equivalent characterizations I have added (here) the one saying that
$\array{ Hom_{\mathcal{C}}\big( T(c) ,\, U(a) \big) & \overset{\sim}{\longrightarrow} & Hom_{\mathcal{C}}\big( c ,\, U(a) \big) \\ \big( T(c) \overset{f}{\longrightarrow} U(a) \big) &\mapsto& \big( c \overset{\eta}{\longrightarrow} T(c) \overset{f}{\longrightarrow} U(a) \big) }$Added an original result formalised in Agda: the monoidal functor induced by a strong idempotent monad is idempotent. I plan to create the page idempotent monoidal functor soon; the definition is quite natural and can be found in Programming contextual computations by Dominic Orchard.
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