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 $Ttext{-}Alg$. Is $EM(p)$ generally a fibration? If not, under what conditions is $EM(p)$ a fibration?
ct.category-theory monads fibration
asked 3 hours ago
Max New
30518
add a comment |
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
asked 3 hours ago
Max New
30518
add a comment |
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
asked 3 hours ago
Max New
30518
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
asked 3 hours ago
Max New
30518
asked 3 hours ago
Max New
30518
edited 1 hour ago
asked 3 hours ago
Max New
30518
asked 3 hours ago
Max New
30518
asked 3 hours ago
Max New
30518
30518
add a comment |
add a comment |
1 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 $Mnd(C)$.
answered 1 hour ago
Tim Campion
13.3k354122
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1 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 $Mnd(C)$.
answered 1 hour ago
Tim Campion
13.3k354122
add a comment |
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 $Mnd(C)$.
answered 1 hour ago
Tim Campion
13.3k354122
add a comment |
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 $Mnd(C)$.
answered 1 hour ago
Tim Campion
13.3k354122
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 $Mnd(C)$.
answered 1 hour ago
Tim Campion
13.3k354122
answered 1 hour ago
Tim Campion
13.3k354122
answered 1 hour ago
Tim Campion
13.3k354122
answered 1 hour ago
Tim Campion
13.3k354122
13.3k354122
add a comment |
add a comment |
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