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Stream: learning: questions

Topic: Hint for an exercise on monad morphisms

Adrián Doña Mateo (Feb 20 2025 at 15:03):

Exercise 4.8.7 in Vol. 2 of Borceux's Handbook, asks the following:
Let $T$ and $S$ be monads on $\mathcal{C}$, and $\lambda : S \to T$ be a morphism of monads, which induces a functor $Q : \mathcal{C}^T \to \mathcal{C}^S$ over $\mathcal{C}$. Show that $Q$ is full and faithful iff $\lambda_C$ is epic for all $C$ in $\mathcal{C}$.

That $\lambda$ being objectwise epic implies that $Q$ is full and faithful follows from a simple diagram chase. I am completely stumped by the other direction though. Does anyone have any hints or ideas?

I'll say briefly what I've tried. Since $Q$ is always faithful, the hypothesis says that if $(C,\alpha)$ and $(D,\beta)$ are $T$-algebras, and $f : C \to D$ is an $S$-algebra morphism $Q(C,\alpha) \to Q(D,\beta)$, then $f$ is also an a $T$-algebra morphism $(C,\alpha) \to (D,\beta)$. To apply this hypothesis, it seems like I need to get my hands on two $T$-algebras, and the only $T$-algebras that are guaranteed to exist are the free ones. But such an argument seems like it will only be able to say something about the components of $\lambda T$, and not those of $\lambda$.

Nathanael Arkor (Feb 20 2025 at 18:43):

I believe the statement in Borceux is incorrect: being pointwise epic is sufficient, but not necessary. The question of the induced functor being fully faithful is the topic of Karazeris–Velebil's Dense morphisms of monads. See Remark 3.5 there in particular.