View source for Eilenberg-Moore algebra

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''Moore–Eilenberg algebra''

Given a monad (or [[triple]]) $T$ in a [[Category|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====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  F. Borceux,   "Handbook of categorical algebra: Categories and structures" , '''2''' , Cambridge Univ. Press  (1994)  pp. Chap. 4</td></tr>
</table>