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Eilenberg-Moore algebra

From Encyclopedia of Mathematics
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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

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=50795
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article