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‎index.html‎

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‎intro/index.html‎

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‎mathematics/categories/adjoint/index.html‎

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‎mathematics/categories/adjoint/index.pug‎

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include ../../../header
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block title
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title ADJOINT
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block content
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nav
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<a href='https://anders.groupoid.space/'>ANDERS</a>
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<a href='https://anders.groupoid.space/lib/'>LIB</a>
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<a href='#'>ADJOINT</a>
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section
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article.main.list
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section
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h1 ADJOINT
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aside
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time Published: 16 OCT 2017
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+tex.
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This article introduces the Adjoint Triple,
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a structure in category theory where three functors between three
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categories form a chain of two consecutive adjunctions. An adjunction
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triple consists of functors $ F: \mathcal{A} \to \mathcal{B} $,
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$ G: \mathcal{B} \to \mathcal{C} $, and $ H: \mathcal{C} \to \mathcal{B} $,
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where $ F \dashv H $ and $ H \dashv G $. This configuration arises in
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settings like monad-comonad interactions, algebraic geometry, and type theory.
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+tex.
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Key applications include:
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1) Modeling reflective and coreflective subcategories in tandem;
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2) Constructing monads and comonads via composite adjunctions;
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3) Providing a framework for Kan extensions and limits.
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This article is a companion to studies of adjoint functors and monad theory, offering both geometric intuition and formal categorical constructions.
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section
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h1 Intuition
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+tex.
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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 $ H $ plays a pivotal role, acting as both a left and right adjoint, creating a balanced structure.
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section
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h2 Definitions
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+tex.
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$\mathbf{Definition}$ (Adjoint). An adjunction between
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categories $ \mathcal{A} $ and $ \mathcal{B} $ consists
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of functors $ F: \mathcal{A} \to \mathcal{B} $ and
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$ G: \mathcal{B} \to \mathcal{A} $, with a natural isomorphism:
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+tex(true, false).
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$$
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\text{Hom}_{\mathcal{B}}(F(A), B) \cong \text{Hom}_{\mathcal{A}}(A, G(B)).
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$$
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+tex.
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We write $ F \dashv G $, with $ F $ the left adjoint and $ G $ the right adjoint.
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+tex.
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$\mathbf{Definition}$ (Adjoint Triple). An adjunction triple consists of three categories $ \mathcal{A}, \mathcal{B}, \mathcal{C} $, and functors $ F: \mathcal{A} \to \mathcal{B} $, $ H: \mathcal{C} \to \mathcal{B} $, $ G: \mathcal{B} \to \mathcal{C} $, such that:
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+tex(true, false).
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$$
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F \dashv H \quad \text{and} \quad H \dashv G.
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$$
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+tex.
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Equivalently, there are natural isomorphisms:
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+tex(true, false).
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$$
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\begin{array}{c}
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\text{Hom}_{\mathcal{B}}(F(A), B) \cong \text{Hom}_{\mathcal{A}}(A, H(C)), \\
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\text{Hom}_{\mathcal{C}}(H(B), C) \cong \text{Hom}_{\mathcal{B}}(B, G(C)).
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\end{array}
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$$
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+tex.
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The diagram for an adjunction triple can be visualized as:
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+tex(true, false).
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$$
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\begin{array}{ccc}
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\mathcal{A} & \xrightarrow{F} & \mathcal{B} \\
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& \nwarrow H & \downarrow G \\
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& & \mathcal{C}
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\end{array}
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$$
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+tex.
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where $ F \dashv H $ and $ H \dashv G $.
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section
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h1 Properties
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p.
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Adjoint triples yield rich structures, such as monads and comonads, and preserve certain categorical properties like limits and colimits.
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section
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h2 Monad and Comonad
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+tex.
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$\mathbf{Theorem}$ (Monad from Adjoint Triple). Given an adjunction
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triple $ F \dashv H \dashv G $, the composite functor $ T = H G: \mathcal{B} \to \mathcal{B} $
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forms a monad on $ \mathcal{B} $, with unit and multiplication derived from the
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adjunction counits and units.
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+code.
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monad T: B -> B = H ∘ G
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unit: (b: B) -> T b
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multiply: (b: B) -> T (T b) -> T b
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+tex.
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<b>Theorem</b> (Comonad from Adjoint Triple). The composite functor $ S = H F: \mathcal{C} \to \mathcal{C} $ forms a comonad on $ \mathcal{C} $, with counit and comultiplication derived from the adjunction units and counits.
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+code.
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comonad S: C -> C = H ∘ F
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counit: (c: C) -> S c -> c
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comultiply: (c: C) -> S c -> S (S c)
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h2 Reflective and Coreflective Subcategories
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+tex.
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<b>Definition</b> (Reflective Subcategory). A subcategory $ \mathcal{B} \subseteq \mathcal{C} $ is reflective if the inclusion functor $ H: \mathcal{B} \to \mathcal{C} $ has a left adjoint $ G: \mathcal{C} \to \mathcal{B} $.
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+tex.
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<b>Proposition</b>. In an adjunction triple $ F \dashv H \dashv G $, if $ H $ is fully faithful, then $ \mathcal{C} $ is a reflective subcategory of $ \mathcal{B} $, and if $ F $ is fully faithful, $ \mathcal{A} $ is a coreflective subcategory of $ \mathcal{B} $.
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section
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h1 Examples
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section
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h2 Free-Forgetful Adjoint Triple
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+tex.
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Consider the categories $ \mathbf{Set} $ (sets), $ \mathbf{Mon} $ (monoids), and $ \mathbf{Grp} $ (groups). Define:
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- $ F: \mathbf{Set} \to \mathbf{Mon} $, the free monoid functor;
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- $ H: \mathbf{Grp} \to \mathbf{Mon} $, the inclusion of groups into monoids;
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- $ G: \mathbf{Mon} \to \mathbf{Grp} $, the group completion functor.
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These form an adjunction triple $ F \dashv H \dashv G $, where $ H $ is the pivot, embedding groups as monoids with inverses.
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+tex(true, false).
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$$
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\begin{array}{ccc}
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\mathbf{Set} & \xrightarrow{F} & \mathbf{Mon} \\
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& \nwarrow H & \downarrow G \\
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& & \mathbf{Grp}
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\end{array}
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$$
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h2 Kleisli Categories
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+tex.
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Given a monad $ T $ on a category $ \mathcal{B} $, the Kleisli category $ \mathcal{B}_T $ and the Eilenberg-Moore category $ \mathcal{B}^T $ form an adjunction triple with $ \mathcal{B} $. The functors are:
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- $ F: \mathcal{B} \to \mathcal{B}_T $, mapping objects to free $ T $-algebras;
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- $ H: \mathcal{B}^T \to \mathcal{B}_T $, comparing algebras;
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- $ G: \mathcal{B}_T \to \mathcal{B}^T $, mapping Kleisli arrows to algebra homomorphisms.
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+code.
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kleisli_adj: (B: Cat) (T: Monad B) -> Adjoint B (Kleisli T)
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em_adj: (B: Cat) (T: Monad B) -> Adjoint (Kleisli T) (EilenbergMoore T)
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section
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h1 Literature
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section
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p
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| [1]. Saunders Mac Lane,
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a(href='https://www.springer.com/gp/book/9780387984032') Categories for the Working Mathematician
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include ../../../footer

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