# Does the Eilenberg Moore Construction Preserve fibrations?

Say we have a Grothendieck fibration $$p : E \to B$$ and a monad $$T$$ on $$B$$ and a lift $$T'$$ of $$T$$ to $$E$$, i.e. a monad on $$E$$ such that $$pT' = Tp$$ and $$p$$ preserves $$\eta, \mu$$.

Then because the Eilenberg–Moore construction is functorial, we have a morphism $$EM(p)$$ from $$T'\text{-}Alg$$ to $$T\text{-}Alg$$. Is $$EM(p)$$ generally a fibration? If not, under what conditions is $$EM(p)$$ a fibration?

Let $$C$$ be a 2-category and $$Mnd(C)$$ the 2-category of monads in $$C$$. As explained by Street in the formal theory of monads, the Eilenberg-Moore construction is right 2-adjoint to the inclusion 2-functor $$C \to Mnd(C)$$ sending an object to the identity monad on it. Therefore it preserves 2-limits.
Thus it's sufficient for $$p$$ to be a fibration object in the 2-category $$Mnd(C)$$.
• Fibrations can be detected by homming in, so one answer will probably be a tautological one: $p: E \to B$ is a fibration if $Mnd(C)(X,E) \to Mnd(C)(X,B)$ is a fibration for all monads $X$, which will probably reduce in the case $C = Cat$ to the induced map of Eilenberg-Moore objects being a fibration. – Tim Campion Dec 30 '18 at 15:00
• I think it would probably be easier to understand fibrations in Mnd(C) using the characterization of fibrations in terms of limits. Since the forgetful functor $\mathit{Mnd}(C)\to C$ also preserves limits (when C has Kleisli objects it has a left adjoint), being a fibration in Mnd(C) just means being a fibration in C together with the extra structure that the universal lift-assigning functor and transformation lift to Mnd(C). Then just work out what that means explicitly. – Mike Shulman Dec 30 '18 at 15:42
• When you say the forgetful functor $Mnd(C) \to C$ do you mean the inclusion $C \to Mnd(C)$ that picks the identity monad instead? That's what it looks like it says in Street. – Max New Jan 2 at 20:46