Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 18 / 8 / i / Solution

A monad on $\mathcal{C}$ consists of an endofunctor $T$ and natural transformations $\eta :1\Rightarrow T$, $\mu :T^2\Rightarrow T$, satisfyingAn algebra for a monad is $(A,a)$ with $a:TA\to A$, $a{\eta }_A=1_A$ and $a{\mu }_A=aT(a)$. A morphism $(A,a)\to (B,b)$ is an arrow $h:A\to B$ with $ha=bT(h)$. These form the Eilenberg-Moore category ${\mathcal{C}}^T$.

A functor $G:\mathcal{D}\to \mathcal{C}$ is monadic when it has a left adjoint $F$ and its comparison $K:\mathcal{D}\to {\mathcal{C}}^{GF}$ is an equivalence; this is the monadic adjunction convention. Some strict formulations require $K$ to be an isomorphism over the base. The distinction affects the meaning of literal limit creation, discussed in part (iii), but not the algebra construction.