comonad.md

Comonad

A comonad is the exact categorical dual of a monad: an endofunctor $W$ equipped with a counit (extract) and a comultiplication (duplicate), satisfying co-unit and co-associativity laws. Where monads model effectful computation flowing into a context, comonads model context-dependent computation flowing out.

Definition

A comonad on $\mathcal{C}$ is a triple $(W, \varepsilon, \delta)$ where:

• $W : \mathcal{C} \to \mathcal{C}$ is an endofunctor
• $\varepsilon : W \Rightarrow \mathrm{Id}$ is the counit (natural transformation; called extract in Haskell)
• $\delta : W \Rightarrow W \circ W$ is the comultiplication / duplicate (natural transformation)

Dually to the monad's Kleisli triple, the comonad has a co-Kleisli triple:

• extract :: W a -> a — observe the focused value
• extend :: (W a -> b) -> W a -> W b — shift the focus and re-observe (cokleisli extension)

The two presentations are related by:

$$\mathrm{extend}; f = W f \circ \delta \qquad \delta = \mathrm{extend};\mathrm{id}$$

Co-Kleisli category

For a comonad $W$, the co-Kleisli category $\mathcal{C}^W$ has:

• The same objects as $\mathcal{C}$
• Morphisms $A \to B$ in $\mathcal{C}^W$ are morphisms $W A \to B$ in $\mathcal{C}$ (co-Kleisli arrows)
• Composition of $f : W A \to B$ and $g : W B \to C$ is $g \circ_W f = g \circ W f \circ \delta_A$
• Identity: $\varepsilon_A : W A \to A$

Duality table

Monad	Comonad
$\eta : \mathrm{Id} \Rightarrow M$ (unit / return)	$\varepsilon : W \Rightarrow \mathrm{Id}$ (counit / extract)
$\mu : M \circ M \Rightarrow M$ (multiplication / join)	$\delta : W \Rightarrow W \circ W$ (comultiplication / duplicate)
Kleisli arrow $A \to M B$	Co-Kleisli arrow $W A \to B$
>>= (bind)	=>> (cobind / extend)
models effects flowing in	models context flowing out

Laws

Counit left:

$$\varepsilon W \circ \delta = \mathrm{id}_W$$

Counit right:

$$W \varepsilon \circ \delta = \mathrm{id}_W$$

Co-associativity:

$$\delta W \circ \delta = W \delta \circ \delta$$

These are exactly the monad laws with all arrows reversed.

FP Analog

CT concept	Haskell
Comonad $(W, \varepsilon, \delta)$	class Functor w => Comonad w
Counit $\varepsilon$	extract :: w a -> a
Comultiplication $\delta$	duplicate :: w a -> w (w a)
Extension $\mathrm{extend}$	extend :: (w a -> b) -> w a -> w b
Co-Kleisli arrow $W A \to B$	w a -> b (context-dependent transform)
Store s a comonad	Lens as a comonad: extract = get, extend = sets
Env e a comonad	Reader as a comonad (reads from fixed environment)
Traced m a comonad	Writer dual: produces values indexed by a monoid

The Store comonad Store s a = (s -> a, s) is the denotational basis of the Lens type: a lens is precisely a co-Kleisli arrow of Store.

→ FP track: 20. Comonad | 27. Lens / Optics (Store comonad as lens) | 19. Monad (the dual)

CTFP Reference

Resource	Link
CTFP blog — Comonads	Part 3 Ch 7
CTFP LaTeX source Ch 3.7	src/content/3.7/

See Also

• Monad — the categorical dual; reversing all arrows gives the comonad laws
• Adjunction — every adjunction gives both a monad and a comonad
• F-Algebra — coalgebras (the dual of algebras) are the formal basis of anamorphisms; comonads arise as terminal coalgebras