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Stream: learning: questions

Topic: Hint for an exercise on monad morphisms


view this post on Zulip Adrián Doña Mateo (Feb 20 2025 at 15:03):

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

That λ\lambda being objectwise epic implies that QQ 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 QQ is always faithful, the hypothesis says that if (C,α)(C,\alpha) and (D,β)(D,\beta) are TT-algebras, and f:CDf : C \to D is an SS-algebra morphism Q(C,α)Q(D,β)Q(C,\alpha) \to Q(D,\beta), then ff is also an a TT-algebra morphism (C,α)(D,β)(C,\alpha) \to (D,\beta). To apply this hypothesis, it seems like I need to get my hands on two TT-algebras, and the only TT-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 λT\lambda T, and not those of λ\lambda.

view this post on Zulip 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.