Eilenberg-Moore algebra

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Moore–Eilenberg algebra

Given a monad (or triple) in a category , a -algebra is a pair , , , such that the diagram

commutes. Such a -algebra is also called an Eilenberg–Moore algebra. The forgetful functor from the category of Eilenberg–Moore algebras to has a left adjoint, exhibiting the monad as coming from a pair of adjoint functors (the Eilenberg–Moore construction).

See also Adjoint functor.


[a1] F. Borceux, "Handbook of categorical algebra: Categories and structures" , 2 , Cambridge Univ. Press (1994) pp. Chap. 4
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Eilenberg-Moore algebra. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article