10.2 Algebras for a monad

Monad algebras provide a way to interpret monadic structures in specific contexts. They consist of a carrier object and a structure map, which must satisfy coherence conditions to ensure consistency with the monad's properties.

These algebras allow us to evaluate monadic computations in various ways. Examples include the identity monad and list monad algebras. Constructing a monad algebra involves defining a structure map and verifying it meets the required conditions.

Algebras for a Monad

Components of monad algebra

• An algebra for a monad $(T, \eta, \mu)$ consists of two components:
  • Carrier object $A$ represents the target object in the category $\mathcal{C}$ (e.g., sets, vector spaces)
  • Structure map $a: TA \to A$ defines a morphism in $\mathcal{C}$ that transforms a monadic value of type $TA$ into a value of type $A$
• The structure map $a$ must satisfy two coherence conditions to ensure consistency with the monadic structure:
  • Interpreting a pure value using $\eta$ should not change the value: $a \circ \eta_A = id_A$
  • Interpreting a monadic computation using $\mu$ and then applying the structure map is equivalent to first applying the structure map to the subcomputations and then interpreting the result: $a \circ Ta = a \circ \mu_A$

Algebras and monadic computations

• Algebras for a monad provide a way to interpret the monadic structure in a specific context determined by the carrier object $A$
• Monadic computations build up a structure using monadic operations ($\eta$ and $\mu$)
• Algebras allow us to evaluate or interpret this structure in a specific way defined by the structure map $a$
• The coherence conditions ensure that the interpretation is consistent with the monadic structure and preserves the properties of the monad

Examples of common monad algebras

• Identity monad $(Id, \eta, \mu)$:
  • Carrier object: any object $A$ in the category
  • Structure map: identity morphism $id_A: A \to A$
• List monad $(List, \eta, \mu)$:
  • Carrier object: any object $A$ in the category
  • Structure map $a: List(A) \to A$ summarizes a list of values into a single value
    • Sum, product, maximum, or minimum of a list of numbers
    • Concatenating a list of strings into a single string

Construction of monad algebras

• To construct an algebra for a given monad $(T, \eta, \mu)$ and a carrier object $A$:
  1. Define a structure map $a: TA \to A$
  2. Check that the structure map satisfies the coherence conditions
• Example: Constructing an algebra for the list monad with carrier object $\mathbb{N}$ (natural numbers)
  1. Define the structure map $a: List(\mathbb{N}) \to \mathbb{N}$ as the sum of the elements in the list
  2. Check the coherence conditions:
     • $a \circ \eta_\mathbb{N} = id_\mathbb{N}$ holds because the sum of a singleton list is the element itself
     • $a \circ Ta = a \circ \mu_\mathbb{N}$ holds because summing a list of lists and then summing the result is equivalent to first summing each sublist and then summing the resulting list