Difference between revisions of "Eilenberg-Moore algebra"
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| + | ''Moore–Eilenberg algebra'' | ||
| − | commutes. Such a | + | 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 [[ | + | See also [[Adjoint functor]]. |
====References==== | ====References==== | ||
| − | <table>< | + | <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> | ||
Latest revision as of 09:49, 19 July 2020
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 |
How to Cite This Entry:
Eilenberg-Moore algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Eilenberg-Moore_algebra&oldid=22373
Eilenberg-Moore algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Eilenberg-Moore_algebra&oldid=22373
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article