Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 23 / 7

A monad on $\mathcal{C}$ consists of an endofunctor $T:\mathcal{C}\to \mathcal{C}$ and natural transformations $\eta :1_{\mathcal{C}}\Rightarrow T$ and $\mu :T^2\Rightarrow T$ satisfying, at every object $A$,These are the unit and associativity laws for the unit and multiplication of a monad.

An algebra for a monad is a pair $(A,a)$ with $a:TA\to A$ satisfyingA monad algebra morphism $f:(A,a)\to (B,b)$ is a morphism $f:A\to B$ with $fa=bTf$. Identities satisfy this condition, and composing two such maps preserves it by functoriality of $T$. The resulting Eilenberg-Moore category is denoted ${\mathcal{C}}^T$. Its forgetful functor $U:{\mathcal{C}}^T\to \mathcal{C}$ sends $(A,a)$ to $A$ and leaves the underlying arrows unchanged.

Write the adjunction unit as $\eta :1_{\mathcal{C}}\Rightarrow GF$ and the adjunction counit as $\epsilon :FG\Rightarrow 1_{\mathcal{D}}$. Define the monad induced by an adjunction bywith unit $\eta$. Naturality of $\epsilon$ makes $\mu$ a natural transformation.

The triangle identities for an adjunction give the two unit laws:For associativity, apply naturality of $\epsilon$ to the arrow ${\epsilon }_{FA}:FGFA\to FA$ in $\mathcal{D}$:Applying $G$ yieldsThus the induced data satisfy all the monad laws, rather than merely supplying an endofunctor and two transformations.

Define the free algebra functor byThe monad laws say precisely that ${\mu }_X$ defines an monad algebra structure on $TX$, while naturality of $\mu$ makes $Tf$ an monad algebra morphism. We construct the adjunction explicitly.

For an monad algebra $(A,a)$ and a morphism $f:X\to A$, put $\overline{f}=aTf:TX\to A$. It is an monad algebra morphism becauseConversely, from an monad algebra morphism $h:(TX,{\mu }_X)\to (A,a)$ obtain $h{\eta }_X:X\to A$. These operations are inverse: naturality of $\eta$ and the monad algebra unit law giveand the algebra-morphism equation together with the monad unit law givesThe formulas commute with precomposition in $X$ and composition with monad algebra morphisms in $(A,a)$, so the bijections are natural. Therefore the free-forgetful Eilenberg-Moore adjunction isIts unit is ${\eta }_X$ and its counit at $(A,a)$ is the monad algebra action $a$.