Algebra for a monad ($(A,a)$)

An algebra for a monad $(T,\eta ,\mu )$ is an object $A$ with a morphism $a:TA\to A$ satisfying $a{\eta }_A=1_A$ and $aT(a)=a{\mu }_A$.

Every algebra for a monad $(A,a)$ is the coequalizer, in the Eilenberg-Moore category, of $F(TA)\rightrightarrows F(A)$ with underlying arrows ${\mu }_A$ and $Ta$, followed by $a:F(A)\to (A,a)$. The pair has common section $F{\eta }_A$. If an algebra morphism $h:F(A)\to (B,b)$ equalizes the pair, the unique induced algebra morphism is $h{\eta }_A:(A,a)\to (B,b)$. This explicit proof does not require general colimits of algebras.

For an algebra for a monad $(X,a)$, its action $a:TX\to X$ satisfies $a{\eta }_X=1_X$.

For a monad $(T,\eta ,\mu )$, the free algebra functor $F:\mathcal{C}\to {\mathcal{C}}^T$ sends $X$ to $(TX,{\mu }_X)$ and $f$ to $Tf$. It is a left adjoint to the forgetful functor $U$, via $h\mapsto h{\eta }_X$ and $u\mapsto \gamma Tu$ between ${\mathcal{C}}^T(FX,(C,\gamma ))$ and $\mathcal{C}(X,C)$.

A morphism $f:(A,a)\to (B,b)$ of algebras for a monad satisfies $fa=bT(f)$. Such morphisms are the morphisms of the Eilenberg-Moore category.