If I read the nLab page correctly, the TW V-monoids turn out to be the monads T on the category of V-algebras such that the underlying functor T: V-Alg -> V-Alg preserves colimits.

You could say that monads on V-Alg the things that act on it; these are just particularly nice actions.

Edited to address Peter's comment: The nLab page does not say what I said. But what I think happens is that the underlying functor T: V-Alg -> V-Alg has a right adjoint C. It is formal that C is a comonad. I believe that in this algebraic setting, T-algebras turn out to be the same thing as C-coalgebras.

The functor C: V-Alg -> V-Alg is the gadget that is "corepresentable"; that is, there is a co-V-object P in V-Alg such that

CA = Hom(P, A)

for any V-algebra A.

This is certainly what happens in the case I understand best: plethories (as mentioned in Greg's answer).

If I read the nLab page correctly, the TW V-monoids turn out to be the monads T on the category of V-algebras such that the underlying functor T: V-Alg -> V-Alg preserves colimits.

You could say that monads on V-Alg the things that act on it; these are just particularly nice actions.

Edited to address Peter's comment: The nLab page does not say what I said. But what I think happens is that the underlying functor T: V-Alg -> V-Alg has a right adjoint C. It is formal that C is a comonad. I believe that in this algebraic setting (edit: no this happens always), T-algebras turn out to be the same thing as C-coalgebras.

The functor C: V-Alg -> V-Alg is the gadget that is "corepresentable"; that is, there is a co-V-object P in V-Alg such that

CA = Hom(P, A)

for any V-algebra A.

This is certainly what happens in the case I understand best: plethories (as mentioned in Greg's answer).

If I read the nLab page correctly, the TW V-monoids turn out to be the monads T on the category of V-algebras such that the underlying functor T: V-Alg -> V-Alg preserves colimits.

You could say that monads on V-Alg the things that act on it; these are just particularly nice actions.

If I read the nLab page correctly, the TW V-monoids turn out to be the monads T on the category of V-algebras such that the underlying functor T: V-Alg -> V-Alg preserves colimits.

You could say that monads on V-Alg the things that act on it; these are just particularly nice actions.

Edited to address Peter's comment: The nLab page does not say what I said. But what I think happens is that the underlying functor T: V-Alg -> V-Alg has a right adjoint C. It is formal that C is a comonad. I believe that in this algebraic setting, T-algebras turn out to be the same thing as C-coalgebras.

The functor C: V-Alg -> V-Alg is the gadget that is "corepresentable"; that is, there is a co-V-object P in V-Alg such that

CA = Hom(P, A)

for any V-algebra A.

This is certainly what happens in the case I understand best: plethories (as mentioned in Greg's answer).