Category theory; homological algebra Definition

Algebra over a monad

Definition

Let $(T, \eta, \mu)$ be a monad on a category $\mathcal{C}$ . A $T$ -algebra is a pair $(A, a)$ of an object $A$ of $\mathcal{C}$ and a morphism $a \colon T(A) \to A$ with these two properties.

Unit law: $a \circ \eta_A = \id_A$ .
Associativity law: $a \circ T(a) = a \circ \mu_A$ as morphisms $T(T(A)) \to A$ .

A morphism of $T$ -algebras $f \colon (A, a) \to (B, b)$ is a morphism $f \colon A \to B$ of $\mathcal{C}$ with $f \circ a = b \circ T(f)$ .

Remarks

Example: let $T$ be the list monad on sets, so $T(X)$ is the free monoid of finite words in $X$ . For a monoid $M$ , let $a$ send a word $(m_1, \dots, m_n)$ to the product $m_1 m_2 \cdots m_n$ in $M$ , with the empty word sent to the identity. The unit law holds since a one-letter word $m$ evaluates to $m$ . The associativity law holds since multiplying the blocks of a word of words and then multiplying the results gives the product of the concatenated word. Every $T$ -algebra arises in this way, so $T$ -algebras are the same as monoids.

Non-example: for the same monad, let $a \colon T(\N) \to \N$ send a word to its length. Then $a(\eta_{\N}(5)) = 1 \ne 5$ , so the unit law fails.

For every object $X$ , the pair $(T(X), \mu_X)$ is a $T$ -algebra, the free $T$ -algebra on $X$ .

Reading path

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Free monoid concept
Monad concept
Monoid concept
Algebra over a monad definition

Tag	`CZWDNA1HA7`
URI	`m2://CZWDNA1HA7`
Kind	definition
Writing	W0 stub
MSC	18C20, 18C15
Defines	Algebra over a monad
Uses	Monad, Free monoid, Monoid

Algebra over a monad

Definition

Remarks

Reading path

Further reading

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