M Theory Lesson 8

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.

1 Response so far »

  1. 1

    L. Riofrio said,

    Happy New Year to you, too! YOur posts aren’t boring at all, they are much more fun than NEW. Recent events show that these maths can be applied to reality.

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