adjunction

Tag. adjunction

adjunction

Notes (5)

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.

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.

The adjoint triple between presheaves and families presheaf-family-adjoint-triple

A family over $𝒞︀$ is a set $𝐴 (𝑥)$ for each object $𝑥$ , with no action of morphisms. Families form a category $Fam (𝒞︀)$ : a morphism $𝐴 \to 𝐵$ is a function $𝐴 (𝑥) \to 𝐵 (𝑥)$ for each $𝑥$ .

Forgetting the restriction maps of a presheaf gives a functor

𝑈 : 𝖯𝗌𝗁 𝒞︀ \to Fam (𝒞︀) .

It has both a left and a right adjoint,

Free ⊣ 𝑈 ⊣ Cofree .

The two adjoints demonstrate different means of forcing a family to be functorial. The right adjoint universally quantifies over morphisms in,

Cofree (𝐴) (𝑥) = \prod_{𝑦} 𝒞︀ (𝑦, 𝑥) \to 𝐴 (𝑦),

with restriction along $𝑓$ given by precomposition. The left adjoint instead existentially quantifiers over morphisms out:

Free (𝐴) (𝑥) = \sum_{𝑦} 𝒞︀ (𝑥, 𝑦) \times 𝐴 (𝑦),

with restriction acting on the first component. (For $Free$ we ask that $𝒞︀$ have a set of objects, so that this sum is a set and thus $Free$ defines a presheaf.)

Theorem. Presheaves are monadic and comonadic over families presheaves-monadic-comonadic-over-families

The adjoint triple $Free ⊣ 𝑈 ⊣ Cofree$ induces a monad $𝑇 = 𝑈 \circ Free$ and a comonad $𝑊 = 𝑈 \circ Cofree$ on $Fam (𝒞︀)$ .

Both comparison functors are equivalences: presheaves are the Eilenberg–Moore algebras of $𝑇$ and the co-Eilenberg–Moore coalgebras of $𝑊$ ,

𝖯𝗌𝗁 𝒞︀ ≃ EM (𝑇) 𝖯𝗌𝗁 𝒞︀ ≃ coEM (𝑊) .

So presheaves are both monadic and comonadic over families.

Reading the algebra structure concretely: a $𝑇$ -algebra on a family $𝐴$ is a map $\sum_{𝑦} 𝒞︀ (𝑥, 𝑦) \times 𝐴 (𝑦) \to 𝐴 (𝑥)$ for each $𝑥$ , subject to the monad algebra laws — that is, exactly a functorial action of restriction.

The comonadic reading is the same structure seen from the element’s side: a $𝑊$ -coalgebra is a map $𝐴 (𝑥) \to \prod_{𝑦} 𝒞︀ (𝑦, 𝑥) \to 𝐴 (𝑦)$ , giving each value its restriction along every morphism into $𝑥$ . Where the monad says restriction acts on values, the comonad says a value already carries all of its restrictions — and the coalgebra laws say it does so coherently.

Theorem. Day Convolution is Closed day-closed-structure

Let $𝒱︀$ be a symmetric monoidal closed category that is complete and cocomplete. Let $(𝒞︀, \otimes_{𝒞︀}, 𝐼)$ be a small monoidal $𝒱︀$ -enriched category and $𝐴, 𝐵$ be $𝒱︀$ -enriched presheaves on $𝒞︀$ . Define

(𝐴 ⊸ 𝐵) 𝑐 = \int_{𝑢} 𝐴 𝑢 ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑐) .

Then, for Day convolution $\otimes_{Day}$ ,

𝐴 \otimes_{Day} - ⊣ 𝐴 ⊸_{Day} - .

Proof. Proof that Day Convolution is Closed day-closed-structure-proof

For $𝑋, 𝐵 : {𝒞︀}^{op} \to 𝒱︀$ , the enriched hom in $[{𝒞︀}^{op}, 𝒱︀]$ is given by the end:

\begin{aligned} \int_{𝑐} ((𝐴 \otimes_{Day} 𝑋) 𝑐 ⊸_{𝒱︀} 𝐵 𝑐) & ≅ \int_{𝑐} ((\int^{𝑢, 𝑣} 𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 𝑐) \\ ≅ \int_{𝑐} \int_{𝑢, 𝑣} ((𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 𝑐) \\ ≅ \int_{𝑢, 𝑣} ((𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} \int_{𝑐} (𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) ⊸_{𝒱︀} 𝐵 𝑐)) \\ ≅ \int_{𝑢, 𝑣} ((𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑣)) \\ ≅ \int_{𝑣} (𝑋 𝑣 ⊸_{𝒱︀} \int_{𝑢} (𝐴 𝑢 ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑣))) \\ ≅ \int_{𝑣} (𝑋 𝑣 ⊸_{𝒱︀} (𝐴 ⊸ 𝐵) 𝑣) . \end{aligned}

∎

Symmetrically, $(𝐵 ⟜ 𝐴) 𝑐 = \int_{𝑣} 𝐴 𝑣 ⊸_{𝒱︀} 𝐵 (𝑐 \otimes_{𝒞︀} 𝑣)$ and $- \otimes_{Day} 𝐴 ⊣ - ⟜ 𝐴$ , so the enriched presheaf category $[{𝒞︀}^{op}, 𝒱︀]$ is biclosed [1].