Re: adjoints to lax-idempotent algebra structures

[Michael Shulman <shulman@sandiego.edu>]

Thanks again to everyone who replied with examples and comments.  I've
created an nLab page which hopefully includes everything I learned:
http://ncatlab.org/nlab/show/continuous+algebra

On Sun, Nov 17, 2013 at 9:14 AM, Michael Shulman <shulman@sandiego.edu> wrote:
> On Sun, Nov 17, 2013 at 6:53 AM, Prof. Peter Johnstone
> <P.T.Johnstone@dpmms.cam.ac.uk> wrote:
>> 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.
>
> That's exactly the sort of thing I was looking for; thanks!
>
> Continuous categories are one of the examples I had in mind.  Another
> interesting almost-example is totally distributive categories.  And
> when T is a monad for coproducts, such a left adjoint seems to
> decompose every object into a coproduct of connected ones (although I
> have not analyzed this case carefully).
>
> T-continuous is a reasonable name, but it would also be nice for a
> name to suggest B1.1.15.  Is there a general name for algebras that
> are retracts of free ones?  In particular cases they are "projective"
> or "cofibrant", but it seems doubtful that either of those terms
> applies literally here.
>
> Mike


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