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Jan 31 at 0:22 comment added David White Was this ever resolved? Alexander said he'd ask the experts back in 2015. Would be nice to know what they said!
May 13, 2015 at 11:18 comment added Zhen Lin I wonder if there's an actual proof in the literature somewhere? Anyway, Ignacio López Franco suggested to me an argument for reducing the monadicity of $\mathfrak{M}$ to the monadicity of categories with finite (co)limits, so I can more-or-less believe that there is a 2-monad of the desired form. But it would be nice to have some intuition for when a 2-category of categories-with-structure is 2-monadic.
May 12, 2015 at 9:26 comment added Alexander Campbell The final paragraph of section 6.4 of Blackwell-Kelly-Power suggests that there is, but I will ask the experts tomorrow.
May 12, 2015 at 8:24 comment added Zhen Lin But is there such a finitary 2-monad? I'm not even sure the underlying ordinary adjunction here is monadic.
May 12, 2015 at 8:14 history answered Alexander Campbell CC BY-SA 3.0