nLab coreader 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 $W$, with the coproduct induced by the diagonal map on $W$ and the counit induced by the terminal map $W \to \ast$.

Notice then if $W$ is furthermore equipped with the structure of a monoid, then $W \times (-)$ also canonically inherits the structure of a monad, allowing aggregation of a program’s $W$ outputs, corresponding to a sort of side channel. Equipped with this monad-structure, the operation $W \times (-)$ is known as the writer monad, see there for more.

On the other hand, the canonical comonad structure on $W \times (-)$ is left adjoint to the reader monad, so that it is known as the reader comonad (eg. in the Haskell documentation for Control.Comonad.Reader) or coreader comonad (e.g. Ahman & Uustalu (2019)). It is also known as the product comonad (e.g. Uustalu & Vene 2008, p. 270).

Properties

Relation to reader monad and state monad

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

The composite of coreader 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 coreader 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

Eg.

• Danel Ahman, Tarmo Uustalu, p. 3 of: Decomposing Comonad Morphisms, CALCO 2019, Leibniz International Proceedings in Informatics (LIPIcs) 139 (2019) [doi:10.4230/LIPIcs.CALCO.2019.14]

• Tarmo Uustalu, p. 13 of: Monads and Interaction Lecture 3, lecture notes for MGS 2021 (2021) [pdf, pdf]

• Tarmo Uustalu, Varmo Vene, Comonadic Notions of Computation, Electronic Notes in Theoretical Computer Science 203 5 (2008) 263-284 [doi:10.1016/j.entcs.2008.05.029]

In Haskell:

• Control.Comonad.Reader

See also:

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

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