Algebra over a Monad

Definition

A $T$-algebra (or an algebra for the monad $T$) is a pair $(A,\alpha )$ consisting of:

• An objects $A:\mathcal{C}$ (the underlying object).
• A morphism $\alpha :TA\to A$ in $\mathcal{C}$ (the structure map or algebra action).

This pair must satisfy the following two conditions:

• Associativity: $\alpha \circ T\alpha =\alpha \circ {\mu }_A$.
• Unit Law: $\alpha \circ {\eta }_A=1_A$.

Homomorphisms

A morphism of $T$-algebras (or $T$-homomorphism) from $(A,\alpha )$ to $(B,\beta )$ is a morphism $f:A\to B$ in $\mathcal{C}$ that commutes with the algebra actions. That is, $f\circ \alpha =\beta \circ Tf$.

References

• riehl2016-category-theory