By now hopefully we suspect that the categorical concept of monad is important for probing possible definitions of observable. A monad $T$ naturally defines $T$-algebras. Let’s look at an example from Mac Lane’s classic Categories for the Working Mathematician (p 138, 1st edition).

Define a functor $P$ on Set as follows. On sets, $P$ sends $X$ to the set of all subsets of $X$. A function $f$ gets sent to $Pf$, which sends $S$ to the direct image of $S$ under $f$, as a subset of $X$. There is a natural transformation whose components are arrows from $X$ to $PX$ which take elements of $X$ to one point sets, and yet another natural transformation with arrows from $PPX$ to $PX$ which takes sets of sets to a union of sets. This data makes $P$ a monad, called the power set monad.

Recall that a complete semi-lattice $C$ satisfies that every subset $S$ has a least upper bound in $C$. A $P$-algebra is a complete semi-lattice with $x \leq y$ given by $h \{ x,y \} = y$ where $h$ is part of the data for a $P$-algebra, and it also gives the least upper bound for $S$. So the category of $P$-algebras is the category of all complete semi-lattices along with the appropriate arrows.

This has been mentioned a number of times before, so I hope I’m not boring you to death. Alas, I must run again.