eilenberg-moore

Tag. eilenberg-moore

eilenberg-moore

Notes (4)

The co-Eilenberg–Moore category as a displayed category co-eilenberg-moore-displayed

A comonad on $𝒞︀$ is a monad $𝑊$ on ${𝒞︀}^{op}$ . Everything about its coalgebras is then inherited from the Eilenberg–Moore construction, instantiated at the opposite category — nothing is defined twice.

Algebras of $𝑊$ over ${𝒞︀}^{op}$ are coalgebras $𝛾 \in 𝒞︀ (𝑥, 𝑊 𝑥)$ of the underlying endofunctor, and the monad algebra laws, read in ${𝒞︀}^{op}$ , are the comonad coalgebra laws — the unit and multiplication of $𝑊$ , viewed in $𝒞︀$ , are the counit $𝜀$ and comultiplication $𝛿$ :

𝛾 ⋆ 𝜀_{𝑥} = {𝗂𝖽}_{𝑥} 𝛾 ⋆ 𝛿_{𝑥} = 𝛾 ⋆ 𝑊 𝛾 .

The co-Eilenberg–Moore category is the opposite of the total category:

coEM (𝑊) = {(EM (𝑊))}^{op} .

Definition. The comparison functor of an adjunction comparison-functor

An adjunction $𝐹 ⊣ 𝑈$ with $𝐹 : 𝒞︀ \to 𝒟︀$ and $𝑈 : 𝒟︀ \to 𝒞︀$ induces a monad $𝑇 = 𝑈 \circ 𝐹$ on $𝒞︀$ . Write $𝜀 : 𝐹 \circ 𝑈 \Rightarrow 𝖨𝖽$ for the counit of the adjunction. Every object $𝑑$ of $𝒟︀$ then induces a $𝑇$ -algebra carried by the object $𝑈 𝑑$ , witnessed by the map

𝑈 𝜀_{𝑑} : 𝑈 𝐹 𝑈 𝑑 \to 𝑈 𝑑 .

This assignment extends to a functor into the Eilenberg–Moore category,

𝐾 : 𝒟︀ \to EM (𝑇),

the comparison functor of the adjunction.

Dually, an adjunction induces a comonad on the other side and a comparison into the co-Eilenberg–Moore category. When these comparisons are equivalences we say that the adjunction $𝐹 ⊣ 𝑈$ is (co)monadic.

The Eilenberg–Moore category as a displayed category eilenberg-moore-displayed

Fix a monad $(𝑇, 𝜂, 𝜇)$ on $𝒞︀$ . Its Eilenberg–Moore category arises in two displayed layers. The first layer is the displayed category of algebras $AlgStr (𝑇)$ of the underlying endofunctor.

The second layer, $EMStr (𝑇)$ , is displayed over the total category $Alg (𝑇)$ . Over an algebra $(𝑥, 𝛼)$ the displayed objects are the propositions that $𝛼$ satisfies the monad algebra laws:

𝜂_{𝑥} ⋆ 𝛼 = {𝗂𝖽}_{𝑥} 𝜇_{𝑥} ⋆ 𝛼 = 𝑇 𝛼 ⋆ 𝛼 .

The Eilenberg–Moore category is the total category of the tower:

EM (𝑇) = \int EMStr (𝑇) .

Definition. Monadicity and comonadicity monadicity-comonadicity

An adjunction $𝐹 ⊣ 𝑈$ with $𝐹 : 𝒞︀ \to 𝒟︀$ and $𝑈 : 𝒟︀ \to 𝒞︀$ induces a monad $𝑇 = 𝑈 \circ 𝐹$ on $𝒞︀$ , and a comparison functor

𝐾 : 𝒟︀ \to EM (𝑇)

sending each object of $𝒟︀$ to the $𝑇$ -algebra it carries.

The functor $𝑈$ is monadic when $𝐾$ is an equivalence: the adjunction exhibits $𝒟︀$ as objects of $𝒞︀$ equipped with algebraic structure for $𝑇$ , the Eilenberg–Moore category.

Comonadicity is monadicity in the opposite category: a left adjoint $𝐿 : 𝒟︀ \to 𝒞︀$ with right adjoint $𝑅$ induces a comonad $𝑊 = 𝐿 \circ 𝑅$ on $𝒞︀$ , a comparison $𝒟︀ \to coEM (𝑊)$ into the co-Eilenberg–Moore category, and $𝐿$ is comonadic when this comparison is an equivalence.