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In the non-relative case, we have a theorem, that an Eilenberg-Moore category of algebras of a Monad $T$ on a topos is itself a topos if the monad in question has a right adjoint.

Now how does this generalize for the relative case? Let $C$, $D$ be topoi, let functor $T : C \rightarrow D$ be a monad relative to $J : C \to D$ and let $EM(T)$ be its Eilenberg-Moore category as defined in "Monads need not be endofunctors". Do we have any conditions for when this category is a topos? Such as if $T$ has a right adjoint or more generally a suitable relative right adjoint.

Notably normally the proof for Eilenberg-Moore category goes through the co-Eilenberg-Moore category of left exact comonad (the right adjoint). But in the relative case the theory bifurcates into left-adjoint/monad and right-adjoint/comonad theory. Can we still translate a relative monad structure into a relative comonad structure along a relative adjunction under some conditions? Or perhaps replay the topos proof but unwind it?

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  • $\begingroup$ Do you have an example when the Eilenberg-Moore category is a topos? Or isn't? $\endgroup$ Mar 10 at 22:19
  • $\begingroup$ One very simple observation is that the category of algebras admits limits when $D$ admits limits. So the problem reduces to finding conditions for which the category of algebras admits power objects. $\endgroup$
    – varkor
    Mar 11 at 22:53
  • $\begingroup$ @AndrejBauer no counter-examples as of yet and only the obvious example when it reduces to the non-relative case. $\endgroup$
    – Ilk
    Mar 12 at 9:10
  • $\begingroup$ @varkor is this proven in the relative case? $\endgroup$
    – Ilk
    Mar 12 at 9:11
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    $\begingroup$ Yes, it's Proposition 2.5 of Relative monadicity. $\endgroup$
    – varkor
    Mar 12 at 9:23

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