ADJOINT

This article introduces the Adjoint Triple, a structure in category theory where three functors between three categories form a chain of two consecutive adjunctions. An adjunction triple consists of functors , , and , where and . This configuration arises in settings like monad-comonad interactions, algebraic geometry, and type theory.

Key applications include: 1) Modeling reflective and coreflective subcategories in tandem; 2) Constructing monads and comonads via composite adjunctions; 3) Providing a framework for Kan extensions and limits. This article is a companion to studies of adjoint functors and monad theory, offering both geometric intuition and formal categorical constructions.

Intuition

Imagine three categories as spaces, with functors as bridges. An adjunction triple is like a triangular pathway where each bridge has a dual, allowing a "round-trip" with specific universal properties. The functor plays a pivotal role, acting as both a left and right adjoint, creating a balanced structure.

Definitions

(Adjoint). An adjunction between categories and consists of functors and , with a natural isomorphism:

We write , with the left adjoint and the right adjoint.

(Adjoint Triple). An adjunction triple consists of three categories , and functors , , , such that:

Equivalently, there are natural isomorphisms:

The diagram for an adjunction triple can be visualized as:

where and .

Properties

Adjoint triples yield rich structures, such as monads and comonads, and preserve certain categorical properties like limits and colimits.

Monad and Comonad

(Monad from Adjoint Triple). Given an adjunction triple , the composite functor forms a monad on , with unit and multiplication derived from the adjunction counits and units.

monad T: B -> B = H ∘ G
  unit: (b: B) -> T b
  multiply: (b: B) -> T (T b) -> T b

Theorem (Comonad from Adjoint Triple). The composite functor forms a comonad on , with counit and comultiplication derived from the adjunction units and counits.

comonad S: C -> C = H ∘ F
  counit: (c: C) -> S c -> c
  comultiply: (c: C) -> S c -> S (S c)

Reflective and Coreflective Subcategories

Definition (Reflective Subcategory). A subcategory is reflective if the inclusion functor has a left adjoint .

Proposition. In an adjunction triple , if is fully faithful, then is a reflective subcategory of , and if is fully faithful, is a coreflective subcategory of .

Examples

Free-Forgetful Adjoint Triple

Consider the categories (sets), (monoids), and (groups). Define: - , the free monoid functor; - , the inclusion of groups into monoids; - , the group completion functor. These form an adjunction triple , where is the pivot, embedding groups as monoids with inverses.

Kleisli Categories

Given a monad on a category , the Kleisli category and the Eilenberg-Moore category form an adjunction triple with . The functors are: - , mapping objects to free -algebras; - , comparing algebras; - , mapping Kleisli arrows to algebra homomorphisms.

kleisli_adj: (B: Cat) (T: Monad B) -> Adjoint B (Kleisli T)
em_adj: (B: Cat) (T: Monad B) -> Adjoint (Kleisli T) (EilenbergMoore T)

Literature

[1]. Saunders Mac Lane, Categories for the Working Mathematician