nLab
writer comonad

Idea

In a cartesian closed category/type theory $\mathcal{C}$, given any object/type $W$ there is a comonad

$$W \times (-) \colon \mathcal{C} \to \mathcal{C}$$

given by forming the Cartesian product with $W$

If here $W$ is equipped with the structure of a monoid, then $W \times (-)$ also canonically inherits the structure of a monad.

In the context of monads in computer science this is called the writer monad. This is the case of interest in functional programming languages such as Haskell, where a program for the form $X \longrightarrow W \times Y$ behaves like a program taking input of type $X$ to output of type $Y$ while in addition producing output of type $W$, which is aggregated using the the monoid when the functions are composed, creating a sort of side channel.

Properties

Relation to reader monad and state monad

In a cartesian closed category/type theory $\mathcal{C}$, the writer comonad $W\times (-)$ is left adjoint to the reader monad $[W,-]$.

The composite of writer comonad followed by reader monad is the state monad.

In terms of dependent type theory

If the type system is even a locally Cartesian closed category/dependent type theory then for each type $W$ there is the base change adjoint triple

$$\mathcal{C}_{/W} \stackrel{\overset{\sum_W}{\longrightarrow}}{\stackrel{\overset{W^\ast}{\longleftarrow}}{\underset{\prod_W}{\longrightarrow}}} \mathcal{C}$$

In terms of this then the writer comonad is equivalently the composite

$$\sum_W W^\ast = W\times (-) \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}$$

of context extension followed by dependent sum.

One may also think of this as being the integral transform through the span

$$\ast \leftarrow W \rightarrow \ast$$

(with trivial kernel) or as the polynomial functor associated with the span

$$\ast \leftarrow W \stackrel{id}{\rightarrow} W \rightarrow \ast \,.$$

Related concepts

• writer monad

• maybe monad

• continuation monad

• reader monad, state monad

References

• Philip Wadler, The essence of functional programming, 1992.

• Olivier Verdier, The Reader and Writer Monads and Comonads, 2014