Does the Eilenberg Moore Construction Preserve fibrations?












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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 $Ttext{-}Alg$. Is $EM(p)$ generally a fibration? If not, under what conditions is $EM(p)$ a fibration?






ct.category-theory monads fibration







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    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 $Ttext{-}Alg$. Is $EM(p)$ generally a fibration? If not, under what conditions is $EM(p)$ a fibration?






    ct.category-theory monads fibration







    share|cite|improve this question



























      4












      4








      4







      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 $Ttext{-}Alg$. Is $EM(p)$ generally a fibration? If not, under what conditions is $EM(p)$ a fibration?






      ct.category-theory monads fibration







      share|cite|improve this question















      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 $Ttext{-}Alg$. Is $EM(p)$ generally a fibration? If not, under what conditions is $EM(p)$ a fibration?






      ct.category-theory monads fibration




      ct.category-theory monads fibration






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      Max New

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          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 adjoint to the forgetful functor $Mnd(C) to C$. Therefore it preserves 2-limits.



          A fibration in a 2-category may be defined in terms of certain 2-limits (namely slice categories) and adjoints, and so is preserved by any 2-functor preserving 2-limits, and in particular by the Eilenberg-Moore construction.



          Thus it's sufficient for $p$ to be a fibration object in the 2-category $Mnd(C)$.






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            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 adjoint to the forgetful functor $Mnd(C) to C$. Therefore it preserves 2-limits.



            A fibration in a 2-category may be defined in terms of certain 2-limits (namely slice categories) and adjoints, and so is preserved by any 2-functor preserving 2-limits, and in particular by the Eilenberg-Moore construction.



            Thus it's sufficient for $p$ to be a fibration object in the 2-category $Mnd(C)$.






            share|cite|improve this answer


























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              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 adjoint to the forgetful functor $Mnd(C) to C$. Therefore it preserves 2-limits.



              A fibration in a 2-category may be defined in terms of certain 2-limits (namely slice categories) and adjoints, and so is preserved by any 2-functor preserving 2-limits, and in particular by the Eilenberg-Moore construction.



              Thus it's sufficient for $p$ to be a fibration object in the 2-category $Mnd(C)$.






              share|cite|improve this answer
























                2












                2








                2






                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 adjoint to the forgetful functor $Mnd(C) to C$. Therefore it preserves 2-limits.



                A fibration in a 2-category may be defined in terms of certain 2-limits (namely slice categories) and adjoints, and so is preserved by any 2-functor preserving 2-limits, and in particular by the Eilenberg-Moore construction.



                Thus it's sufficient for $p$ to be a fibration object in the 2-category $Mnd(C)$.






                share|cite|improve this answer












                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 adjoint to the forgetful functor $Mnd(C) to C$. Therefore it preserves 2-limits.



                A fibration in a 2-category may be defined in terms of certain 2-limits (namely slice categories) and adjoints, and so is preserved by any 2-functor preserving 2-limits, and in particular by the Eilenberg-Moore construction.



                Thus it's sufficient for $p$ to be a fibration object in the 2-category $Mnd(C)$.







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                answered 1 hour ago









                Tim Campion

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