Dear Mike, For T = Ind = free cocompletion under filtered colimits such a string w -| a -| e : A --> TA of adjunctions defines what Peter Johnstone and Andre Joyal call continuous categories, see P.Johnstone, A.Joyal, Continuous categories and exponentiable toposes, JPAA 25 (1982), 225--296 The left adjoint w then gives (the generalisation of) the way-below relation from domain theory. Since lax-idempotent T axiomatises a ``colimit formation'', perhaps T-continuous algebras might be a good name for general algebras (A,a) having an additional adjoint w -| a. All the best, Jirka On Fri, 15 Nov 2013, Michael Shulman wrote:
Let T be a lax-idempotent (i.e. Kock-Zoberlein) 2-monad (or pseudomonad). Then to give a pseudo T-algebra structure on an object A is to give a left adjoint a : TA -> A to the unit e : A -> TA. Has anyone studied and/or named the class of T-algebras for which the algebra structure map admits a further left adjoint? In examples, this seems to be a sort of "super-exactness" condition.
Mike
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