Yes, these "super-exact algebras" have been extensively studied in particular cases (e.g. the "continuous categories" studied by Andre Joyal and myself in JPAA 25 (1982)). What you can say about them in general is contained in Corollary B1.1.15 of the Elephant (page 254): they are exactly the retracts of free algebras (provided idempotent 2-cells split in the underlying 2-category), and they all occur as coadjoint retracts of free algebras. Peter Johnstone 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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