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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 ?"

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  • $\begingroup$ This might be implicitly present in the Garner-Lack paper "Lex colimits"? $\endgroup$
    – Mike Shulman
    Commented Mar 9, 2018 at 19:14
  • $\begingroup$ 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. $\endgroup$
    – Tim Campion
    Commented Mar 10, 2018 at 19:51

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I think Mike is right: in Garner and Lack, Lex colimits, Proposition 2.3 and the discussion following it show that the free cocompletion 2-monad lifts to the 2-category of finitely complete categories, which is equivalent to the existence of a distributive law between these two 2-monads. Since the 2-monad for finite limits is colax-idempotent and the 2-monad for small colimits is lax-idempotent, one doesn't need to appeal to the theory of fully weak distributive laws between 2-monads for this -- the KZ theory will suffice.

Then Proposition 2.5 shows that the algebras for the composite 2-monad are precisely the infinitary pretoposes.

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  • $\begingroup$ Yes. Thanks to both of you for pointing out that paper ! $\endgroup$
    – Simon Henry
    Commented Mar 10, 2018 at 23:53
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    $\begingroup$ I think that this also follows from the proof of Theorem 4.6 in my paper with Adámek and Vitale, On algebraically exact categories and essential localizations of varieties, J. Alg. 244 (2001), 450-477. There we take all limits and get "complete pretoposes" where products also distribute over coproducts. $\endgroup$
    – Jiří Rosický
    Commented Mar 11, 2018 at 9:30

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