Eilenberg-Moore algebra

Moore–Eilenberg algebra

Given a monad (or triple) $T$ in a category $\mathcal{C}$, a $T$-algebra is a pair $( A , \alpha )$, $\alpha : T A \rightarrow A$, $A \in \mathcal{C}$, such that the diagram \begin{equation} \begin{array}{crccc} A & \stackrel{\eta_A}{\rightarrow} & T(A) & & T(T(A)) \\ & {}_{\mathrm{id}_A}\nwarrow & \downarrow{}_\alpha & \stackrel{\mu_A}{\leftarrow} & \downarrow{}_{T(\alpha)} \\ & & A & & T(A) \end{array} \end{equation} commutes. Such a $T$-algebra is also called an Eilenberg–Moore algebra. The forgetful functor from the category of Eilenberg–Moore algebras $\mathcal{C} ^ { T }$ to $\mathcal{C}$ has a left adjoint, exhibiting the monad $T$ as coming from a pair of adjoint functors (the Eilenberg–Moore construction).

See also Adjoint functor.

References

[a1]	F. Borceux, "Handbook of categorical algebra: Categories and structures" , 2 , Cambridge Univ. Press (1994) pp. Chap. 4