# References requestion : Pretopos are algebras for a composed monad?

Unless I'm mistaken the "Free completion under finite limits monad" $C \mapsto C^{lex}$ and the "free co-completion monad" $C \mapsto \widehat{C}$ (the categories of small presheaves) satisfies a distributivity law, and the algebra for the composed monad $C \mapsto \widehat{C^{lex}}$ are the infinitary pretopos.

Has this been written out somewhere ? I have a vague memories of seeing something like that some time ago (or maybe it was for completion under co-products and extensive categories), but I have not been able to find it anywhere...

Is there a similar statement for finitary pretoposes ?

I assume $C \mapsto \widehat{C}$ has to be replaced by completion under finite co-products and some co-equalizer, but that seems to be a little bit more tricky. I guess in technical terms what I want to know is: "is it possible to write the free pre-topos monad as the composite of a co-KZ-monad and a KZ-monad satisfying a distributivity law ?"

• This might be implicitly present in the Garner-Lack paper "Lex colimits"? Mar 9 '18 at 19:14
• Proposition 4.3 and Remark 6.6 of Day and Lack, Limits of small functors at least show that the distributive law between finite limits and small colimits exists (and is unique because it's between a lax-idempotent and a colax-idempotent 2-monad). The last section of A classification of accessible categories treats a similar family of distributive laws, and at least says something about the algebras for the composite 2-monads. Mar 10 '18 at 19:51