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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