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1. • CommentRowNumber1.
   • CommentAuthorZhen Lin
   • CommentTimeAug 11th 2012
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   Author: Zhen Lin
   Format: MarkdownItexThe Eilenberg–Moore construction is essentially a 2-functor from the "2-category of monads on locally small categories" to the "2-category of locally small categories", modulo certain size issues. In the reverse direction, it is known that any functor $\Phi : \mathcal{C}^\mathbb{S} \to \mathcal{D}^\mathbb{T}$ such that $U^\mathbb{T} \Phi = F U^\mathbb{S}$ for some functor $F : \mathcal{C} \to \mathcal{D}$, where $U^\mathbb{S} : \mathcal{C}^\mathbb{S} \to \mathcal{C}$ and $U^\mathbb{T} : \mathcal{D}^\mathbb{T} \to \mathcal{D}$ are the respective forgetful functors, must come from a unique morphism of monads $\mathbb{S} \to \mathbb{T}$. (This is basically a stronger version of Theorem 6.3 in _Toposes, triples and theories_.) It is also not hard to come up with functors $\mathcal{C}^\mathbb{S} \to \mathcal{D}^\mathbb{T}$ that do _not_ arise in this fashion. But what about the 2-cells? Is every natural transformation between functors induced from a morphism of monads also induced by a 2-cell between the monad morphisms? Clearly, the answer is no – any natural transformation that doesn't factor through the forgetful functor $U^\mathbb{S}$ can't be induced by a 2-cell between monad morphisms. But I haven't been able to come up with a proof or counterexample when I assume that the natural transformation _does_ factor through $U^\mathbb{S}$.

   The Eilenberg–Moore construction is essentially a 2-functor from the “2-category of monads on locally small categories” to the “2-category of locally small categories”, modulo certain size issues. In the reverse direction, it is known that any functor such that for some functor , where and are the respective forgetful functors, must come from a unique morphism of monads . (This is basically a stronger version of Theorem 6.3 in Toposes, triples and theories.) It is also not hard to come up with functors that do not arise in this fashion.

   But what about the 2-cells? Is every natural transformation between functors induced from a morphism of monads also induced by a 2-cell between the monad morphisms? Clearly, the answer is no – any natural transformation that doesn’t factor through the forgetful functor can’t be induced by a 2-cell between monad morphisms. But I haven’t been able to come up with a proof or counterexample when I assume that the natural transformation does factor through .

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 11th 2012
   • (edited Aug 11th 2012)
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   Author: Todd_Trimble
   Format: MarkdownItex> Is every natural transformation between functors induced from a morphism of monads also induced by a 2-cell between the monad morphisms? Clearly, the answer is no – any natural transformation that doesn't factor through the forgetful functor $U^\mathbb{S}$ can't be induced by a 2-cell between monad morphisms. But I haven't been able to come up with a proof or counterexample when I assume that the natural transformation _does_ factor through $U^\mathbb{S}$. Good question -- the short answer is 'yes'. Just to lay all the cards on the table, I assume that by a monad morphism from a monad $(C, S, m_S: S S \to S, u_S: 1_C \to S)$ to a monad $(D, T, m_T: T T \to T, u_T: 1_D \to T)$, you mean a pair $(F: C \to D, \phi_F: T F \to F S)$ where the natural transformation $\phi_F$ satisfies an obvious compatibility with the monad multiplications and the monad units (taking the shape of a pentagon and a unit, much as in the case of a distributive law, which is actually a special case). (I point this out because if you don't think about it too hard, one could guess the opposite direction for $\phi_T$!) By a 2-cell from a monad morphism $(F, \phi_F)$ to a monad morphism $(G, \phi_G)$, you must mean a transformation $\eta: F \to G$ satisfying an obvious compatibility with the $\phi$'s. Okay, suppose functors $F, G: C \to D$ have lifts to functors $\Phi, \Psi: C^S \to D^T$; then, as you say, $F$ and $G$ become endowed with appropriate transformations $\phi_F$, $\phi_G$, making them monad morphisms. Now suppose we have a transformation $\theta: \Phi \to \Psi$ which "descends" to a transformation $\eta: F \to G$, in the sense that $\eta U_S = U_T \theta$, where $U_S$, $U_T$ are the forgetful functors from Eilenberg-Moore categories. We want to show $\eta$ is a transformation between the monad morphisms. Here is the critical diagram you need to stare at: $$\array{ T F & \stackrel{T u_T F}{\to} & T T F & \stackrel{T\phi_F}{\to} & T F S & \stackrel{\phi_F S}{\to} & F S S & \stackrel{F m_S}{\to} & F S \\ ^\mathllap{T \eta} \downarrow & & & & \downarrow^\mathrlap{T\eta S} & & & & \downarrow^\mathrlap{\eta S} \\ T G & \underset{T u_T G}{\to} & T T G & \underset{T\phi_G}{\to} & T G S & \underset{\phi_G S}{\to} & G S S & \underset{G m_S}{\to} & G S }$$ It's not too hard to see, using the fact that $(F, \phi_F)$ is a monad morphism and one of the unit laws for a monad, that the long top horizontal composite is $\phi_F$; similarly, the long bottom horizontal composite is $\phi_G$. So we're done if we show the entire diagram commutes. Now the right-hand rectangle commutes essentially because $\eta$ lifts to a transformation $\theta: \Phi \to \Psi$. The left rectangle commutes because it's $T$ applied to a naturality square for $\eta$ in disguise -- one has to rewrite $\phi_F \circ u_T F = F u_S$ and $\phi_G \circ u_T G = G u_S$, again using the fact that $(F, \phi_F)$ and $(G, \phi_G)$ are monad morphisms, to remove the disguise.

   Is every natural transformation between functors induced from a morphism of monads also induced by a 2-cell between the monad morphisms? Clearly, the answer is no – any natural transformation that doesn’t factor through the forgetful functor can’t be induced by a 2-cell between monad morphisms. But I haven’t been able to come up with a proof or counterexample when I assume that the natural transformation does factor through .

   Good question – the short answer is ’yes’.

   Just to lay all the cards on the table, I assume that by a monad morphism from a monad to a monad , you mean a pair where the natural transformation satisfies an obvious compatibility with the monad multiplications and the monad units (taking the shape of a pentagon and a unit, much as in the case of a distributive law, which is actually a special case). (I point this out because if you don’t think about it too hard, one could guess the opposite direction for !)

   By a 2-cell from a monad morphism to a monad morphism , you must mean a transformation satisfying an obvious compatibility with the ’s.

   Okay, suppose functors have lifts to functors ; then, as you say, and become endowed with appropriate transformations , , making them monad morphisms. Now suppose we have a transformation which “descends” to a transformation , in the sense that , where , are the forgetful functors from Eilenberg-Moore categories. We want to show is a transformation between the monad morphisms.

   Here is the critical diagram you need to stare at:

   It’s not too hard to see, using the fact that is a monad morphism and one of the unit laws for a monad, that the long top horizontal composite is ; similarly, the long bottom horizontal composite is . So we’re done if we show the entire diagram commutes. Now the right-hand rectangle commutes essentially because lifts to a transformation . The left rectangle commutes because it’s applied to a naturality square for in disguise – one has to rewrite and , again using the fact that and are monad morphisms, to remove the disguise.

3. • CommentRowNumber3.
   • CommentAuthorZhen Lin
   • CommentTimeAug 12th 2012
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   Author: Zhen Lin
   Format: MarkdownItexPerfect, thanks! I got stuck because I subdivided that diagram too much...

   Perfect, thanks! I got stuck because I subdivided that diagram too much…

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 12th 2012
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   Author: Todd_Trimble
   Format: MarkdownItexYup! The same thing happened to me :-)

   Yup! The same thing happened to me :-)