A *unidetermined contramonad* is a 2-monad $T : {\cal C}\to \cal C$ such that

- $T$ is contravariant, i.e. a contravariant endofunctor;
- the multiplication $\mu_A : TTA \to TA$ is determined as $T\eta_A = T(A\to TA)$.

Q(-1): Is this even a thing? Does this definition already exist under another name?

An example of such monad is the presheaf construction $P : A\mapsto [A°,Set]$; it has the Yoneda embeddings as units, and it is in fact possible to show that $T\eta_A$ acts as a multiplication, in view of the general fact that $$ [[PA°,Set]°,Set] \underset{PP\eta}{\overset{P\eta P}\leftrightarrows} [PA°,Set] $$ is an adjunction ($PP\eta\dashv P\eta P$). (notation: whenever $X$ is a large category, $[X°,Set]$ is the category of small functors).

Q0: Or is it (coming from the adjunction $P\eta\dashv \eta P$)? I'm able to find two natural transformations:

- $\alpha : F \Rightarrow P\eta(\eta P(F))$, induced by the cowedge $$ Fa\times A(-,a)\to F$$ (or, rather, induced by the action of $F$ on arrows): $\alpha$ mates to $$ \tilde\alpha : Fa \to Set(A(-,a),F) $$ (and $\tilde\alpha$ is invertible, for that matter: Yoneda lemma).
- $\beta : \Theta \Rightarrow \eta P(P\eta(\Theta))$, induced by the action of $\Theta$ on morphisms: its components are $$ PA(\hom(-,a),F)\to PA(\Theta F,\Theta(\hom(-a,)))$$ who mate to a family of maps $$ \tilde\beta : \Theta(F) \to PA(F, \Theta(\hom(-,a)))$$
Now... who's the unit? Who's the counit?

$P$, being a free cocompletion, is a KZ-monad. As soon as one wants to write down explicitly what this structure is, however, they have to face a few slight inaccuracies in how $P$ was defined;

first of all it is not only contravariant, but also partially defined; it is a so-called relative monad, like a monad but not an endofunctor. In this particular case, $P : cat^\text{coop}\to Cat$ is a monad relative to $i^\text{coop} : cat^\text{coop} \to Cat^\text{coop}$, the inclusion of small into locally small categories.

**Q1**: Am I wrong if I define a contravariant (total) monad to be a contravariant endofunctor $T : {\cal C}\to \cal C$ which is relative to $1^\text{op} : {\cal C}^\text{op} \to {\cal C}^\text{op}$?second, it seems to satisfy mixed properties of a lax and a colax idempotent 2-monad: in particular, it seems to me that every $P$-algebra $a :PA \to A$ is a left adjoint to the unit (so lax), and yet $\mu \dashv \eta P$ (so colax).

**Q2**: Am I committing a mistake? If not, does these mixed properties have to do with the fact that $P$ is contravariant?third it is "unidetermined", i.e. $\mu$ is determined by $\eta$.

**Q3**: How does this affect the equations defining a KZ-monad, if at all?

lax-idempotent monad. $\endgroup$ – fosco Oct 20 '18 at 10:485more comments