Monads and algebras for a monad

A monad on $\mathcal C$ consists of an endofunctor $T$, a natural transformation $\eta:\mathrm{Id}\Rightarrow T$, and a natural transformation $\mu:T^2\Rightarrow T$. Both ways of inserting a trivial layer into an already structured object must flatten back to that object. Flattening three layers in either order must agree. The subscripts specify which layer is being manipulated, so the two unit expressions are different maps even when their composites with multiplication coincide.

An adjunction $F\dashv G$ produces a monad $T=GF$ with the adjunction's unit and with multiplication $G\varepsilon F$. The triangle identities imply the unit laws, while naturality of the counit supplies associativity. This explains one source of monads without claiming that every useful calculation needs the adjunction written down. In the concrete example below, singleton insertion and union can be verified directly, including their compatibility with maps.

On FinSet, let $T(X)$ be the set of all subsets of $X$. For a function $f$, define $T(f)$ by direct image of subsets. This is covariant: direct images preserve identities and compose in the same order as functions. Set $\eta_X(x)=\{x\}$, and let $\mu_X$ take the union of a family of subsets. The empty subset and the empty family are included. Direct image commutes with union and singleton formation, proving naturality of both transformations.

The two unit laws say that unioning a singleton family containing S or unioning the family of singleton subsets of S both recover S. Associativity says membership in the final union is equivalent to membership in some subset belonging to some family, regardless of which union is performed first. Since X is finite, all iterated powersets remain finite; for a three-point X their first two sizes are eight and 256. Inverse image would have the wrong variance for this endofunctor.

An algebra for a monad is an object X with a map $\alpha:TX\to X$ satisfying $\alpha\eta_X=\mathrm{id}_X$ and $\alpha\mu_X=\alpha T(\alpha)$. The first says interpreting a trivial structure returns its original element. The second says interpreting after flattening agrees with interpreting each inner structure and then interpreting the resulting outer structure. These equations involve the algebra map, which has a different target from the monad multiplication.

For the finite powerset monad, a finite chain with a least element becomes an algebra by taking the maximum of a nonempty subset and interpreting the empty subset as the least element. Singleton interpretation is immediate. For a family, the largest element appearing anywhere is also the largest of the inner maxima, with the empty case handled by the least element. More generally these algebras express finite joins with a bottom element; the empty-set convention is part of the laws, not a disposable special case.