# Why are morphisms of monads lax and not oplax natural transformations

Monads in a bicategory $\mathscr B$ correspond to lax functors $* → \mathscr B$, so one expects morphisms of monads should correspond to nautral transformations between them.

A natural transformation between two lax functors $F, G : \mathscr B → \mathscr B'$ is given by a family $φ_B : FB → GB$ of 1-morphisms in $\mathscr B'$ and a 2-cell $φ_f$ replacing the usual commutative square. Requiring $φ_f$'s to be invertible is obviously too restrictive, but then one must choose a direction: $φ_f : Gf ∘ φ_A ⇒ φ_B ∘ Ff$ (what nlab calls lax natural transformation) or $φ_f : φ_B ∘ Ff ⇒ Gf ∘ φ_A$ (oplax).

Spelling this all out for the familiar case of monads $(\mathscr C, T)$ and $(\mathscr D, S)$ in $\mathrm{Cat}$, we have a functor $F = φ_* : \mathscr C → \mathscr D$ and a natural transformation $φ = φ_\mathrm{id} : SF ⇒ FT$ in the lax, or $FT ⇒ SF$ in the oplax case, and everyone seems to agree that lax is the way to go. So I guess my question is: is the oplax case uninteresting and is there a reason it's uninteresting, or am I misunderstanding something? (My understanding of 2-categories is very basic and superficial so I'll admit that the second option is a strong contender)

Note that when $F = \mathrm{Id}$, which is the case I'm somewhat familiar with for $\mathscr C = \mathrm{Ab}$ (where cocontinuous monads correspond to rings) and $\mathscr C = \mathrm{Set}$ (finitary monads = algebraic theories), it doesn't matter which direction we choose because they're dual (although the oplax one which collapses to $\operatorname{Hom}(T, S) ⊆ \operatorname{Nat}(T, S)$ certainly looks like the more sensible choice, and it's exactly what you get if you take monads on $\mathscr C$ to be monoids in $\mathrm{End}(\mathscr C)$.

Finally, for $\mathscr B = \mathrm{Rel}$ monads are preorders, and unless I miscalculated somewhere, the oplax case is the one that gives the expected notion of a morphism and this is actually what made me think about the direction of that 2-morphism.

• Good question! Maybe the nlab convention should be vice versa? – Martin Brandenburg May 21 '15 at 13:01
• @MartinBrandenburg: You mean for what a morphism of monads is? I don't think it should be changed, I'm asuming it's useful for something. Instead, I'm wondering why the alternative isn't mentioned. In Cat at least, you can check that monad morphisms as defined on nlab induce a functor between E-M categories, while oplax transformations would give rise to a functor between Kleisli categories instead, which doesn't sound as useful. But what about other bicategories? – user54748 May 21 '15 at 14:35

The 2-category $\mathrm{Mnd}K$ of monads in a 2-category $K$ consists of lax morphisms as its 1-cells, and the one $\mathrm{Mnd}^* K$ with oplax morphisms is (defined to be) $\mathrm{Mnd}^\mathrm{op} K^\mathrm{op}$. Both categories are interesting in the sense that one yields EM construction and the other Kleisli construction. Given a left adjoint $J$ to $F$, every lax morphism $(F, \varphi\colon SF \to FT)$ corresponds uniquely to an oplax morphism $(J: X \to Y, \varphi^\sharp\colon JS \to TJ)$ (via mate correspondence) as observed in [Street, 1972].
The oplax morphism of monads gives rise to a lifting $\hat J$ of $J$ along the left adjoints $X \to \mathrm{Kl}(S)$ and $Y\to \mathrm{Kl}(T)$. Kleisli maps are used to model "side effects" in theoretical computer science, so the above lifting can be thought as a way of making a type construction (for $X = Y$) effectful.