# When is the Eilenberg-Moore category of a relative monad between two topoi a topos?

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?

• Do you have an example when the Eilenberg-Moore category is a topos? Or isn't? Mar 10 at 22:19
• 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. Mar 11 at 22:53
• @AndrejBauer no counter-examples as of yet and only the obvious example when it reduces to the non-relative case.
– Ilk
Mar 12 at 9:10
• @varkor is this proven in the relative case?
– Ilk
Mar 12 at 9:11
• Yes, it's Proposition 2.5 of Relative monadicity. Mar 12 at 9:23