adjoints to lax-idempotent algebra structures
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
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Jiri Velebil -
Michael Shulman -
Prof. Peter Johnstone