Alphabeta MathAdjunctions Units and Counits
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Homomorphisms and the Isomorphism Theorems
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Rings, Subrings, Integral Domains and Fields
Category Theory
Number Theory
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- Topology of ℝ
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
Topology
- Compactness
- Compactness in Metric Spaces
- Convergence: Nets and Filters
- Hausdorff via the Diagonal
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Tychonoff Embedding and the Stone–Čech Compactification
- Topological Spaces and Continuity
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
Adjunction by unit, counit, and the triangle identities
Remark
The unit-counit definition imposes no local-smallness hypothesis
Proposition
An adjoint equivalence is an adjunction whose unit and counit are natural isomorphisms
Definition
Adjuncts and transposition under an adjunction
Lemma
The unit and counit transpose formulas are mutually inverse
Theorem
Under local smallness, transposition gives the natural hom-set bijection, and conversely
Theorem
A square commutes if and only if its transposed square commutes
Theorem
Unit components are initial in comma categories, and counit components are terminal
Theorem
The unit-counit, hom-set, unit-universal, and counit-universal encodings of an adjunction are equivalent
Theorem
A left adjoint exists exactly when chosen initial objects are supplied in every comma category
Theorem
Chosen objectwise universal arrows assemble uniquely into a left adjoint
Theorem
A supplied pointwise right adjoint extends uniquely to a functor
Remark
Objectwise existence does not itself supply a class-sized choice of adjoint values
Theorem
Adjoints are unique up to a unique natural isomorphism compatible with the adjunction data
Theorem
Adjunctions compose with the composite unit and counit formulas
Proposition
An adjunction induces postcomposition and precomposition adjunctions on legitimate functor categories
Theorem
Natural transformations have mates under a pair of adjunctions
Remark
Conjugation preserves invertibility but the general mates correspondence need not
Proposition
An adjunction restricts to an equivalence on the subcategories fixed by its unit and counit
Theorem
Fullness and faithfulness of a right adjoint are detected by its counit
Definition
Adjoint triple L⊣ M⊣ R
Proposition
An adjoint triple induces adjunctions between its associated endofunctors
Definition
Mutually left and mutually right adjoint contravariant functors
Definition
Galois connection between preorders
Corollary
A Galois connection between posets satisfies FGF=F and GFG=G
Proposition
In a poset adjunction the triangle identities are automatic
Theorem
Right adjoints preserve every limit that exists
Corollary
Left adjoints preserve every colimit that exists
Remark
The direct preservation theorem carries no size hypothesis
Theorem
Under local smallness, representable functors give a second proof that right adjoints preserve small limits
Definition
Left exact and right exact functors
Corollary
A right adjoint is left exact and a left adjoint is right exact
Corollary
Right adjoints preserve monomorphisms and left adjoints preserve epimorphisms
Theorem
Chosen limits and colimits are adjoint to the diagonal functor
Theorem
Direct image, preimage, and universal image form an adjoint triple on power sets
Remark
The image-preimage adjunctions explain the set-operation preservation laws
Theorem
The free-group functor is left adjoint to the underlying-set functor
Theorem
The free-module functor is left adjoint to the underlying-set functor
Lemma
Finite words satisfy the free-monoid universal property
Theorem
The free-monoid functor is left adjoint to the underlying-set functor
Theorem
Abelianisation is left adjoint to the inclusion of abelian groups
Theorem
Discrete topology, underlying set, and indiscrete topology form an adjoint triple
Theorem
Under the ultrafilter lemma and dependent choice, Stone-Cech compactification is left adjoint to the compact-Hausdorff inclusion
Proposition
The underlying-set functor on fields has no left adjoint
Lemma
Coextension of scalars Hom_R(S,M) carries its canonical left S-module structure
Theorem
Coextension of scalars is right adjoint to restriction of scalars
Theorem
Currying gives the adjunction -× A⊣(-)^A in Set
False statement
A unit and counit determine an adjunction without the triangle identities
False statement
Left adjoints preserve limits
False statement
Every functor with a left adjoint also has a right adjoint
False statement
The hom-set form of an adjunction needs no size hypothesis
Sources
- Emily Riehl, Category Theory in Context, 2nd ed., Definition 4.2.5
- Tom Leinster, Basic Category Theory, Theorem 2.2.5
- Emily Riehl, Category Theory in Context, 2nd ed., Sections 4.1–4.2
- Tom Leinster, Basic Category Theory, Section 2.2
- Emily Riehl, Category Theory in Context, 2nd ed., Proposition 4.3.5
- Tom Leinster, Basic Category Theory, Section 1.3 and Theorem 2.2.5
- Emily Riehl, Category Theory in Context, 2nd ed., Section 4.2
- Tom Leinster, Basic Category Theory, Lemma 2.2.4
- Tom Leinster, Basic Category Theory, Lemmas 2.2.2 and 2.2.4
- Emily Riehl, Category Theory in Context, 2nd ed., Theorem 4.2.7
- Emily Riehl, Category Theory in Context, 2nd ed., Lemma 4.1.3
- Tom Leinster, Basic Category Theory, Theorem 2.3.6
- Emily Riehl, Category Theory in Context, 2nd ed., Lemma 4.7.1
- Tom Leinster, Basic Category Theory, Corollary 2.3.7
- Emily Riehl, Category Theory in Context, 2nd ed., Proposition 4.4.4
- Tom Leinster, Basic Category Theory, Section 4.2
- Emily Riehl, Category Theory in Context, 2nd ed., Lemma 4.7.1 and surrounding discussion
- Emily Riehl, Category Theory in Context, 2nd ed., Proposition 4.3.1
- Tom Leinster, Basic Category Theory, Section 2.3
- Emily Riehl, Category Theory in Context, 2nd ed., Proposition 4.3.4
- Saunders Mac Lane, Categories for the Working Mathematician, 2nd ed., Chapter IV.8
- Emily Riehl, Category Theory in Context, 2nd ed., Proposition 4.3.6
- Saunders Mac Lane, Categories for the Working Mathematician, 2nd ed., Chapter IV
- Emily Riehl, Category Theory in Context, 2nd ed., Proposition 4.3.7
- Saunders Mac Lane, Categories for the Working Mathematician, 2nd ed., Chapter IV.7
- Emily Riehl, Category Theory in Context, 2nd ed., Exercise 4.3.iv
- Emily Riehl, Category Theory in Context, 2nd ed., Lemma 4.2.11
- Emily Riehl, Category Theory in Context, 2nd ed., Lemma 4.6.11
- Saunders Mac Lane, Categories for the Working Mathematician, 2nd ed., Chapter IV.3
- Emily Riehl, Category Theory in Context, 2nd ed., Section 4.1
- Emily Riehl, Category Theory in Context, 2nd ed., Exercise 4.1.iv
- Emily Riehl, Category Theory in Context, 2nd ed., Definition 4.4.1
- Tom Leinster, Basic Category Theory, Example 2.2.7
- Emily Riehl, Category Theory in Context, 2nd ed., Corollary 4.2.10
- Emily Riehl, Category Theory in Context, 2nd ed., Theorem 4.6.2
- Tom Leinster, Basic Category Theory, Section 6.3
- Emily Riehl, Category Theory in Context, 2nd ed., Section 4.6
- Emily Riehl, Category Theory in Context, 2nd ed., Definition 4.6.7
- Emily Riehl, Category Theory in Context, 2nd ed., Definition 4.6.7 and Theorem 4.6.2
- Emily Riehl, Category Theory in Context, 2nd ed., Exercise 4.6.vi
- Emily Riehl, Category Theory in Context, 2nd ed., Proposition 4.6.1
- Tom Leinster, Basic Category Theory, Section 5.1
- Emily Riehl, Category Theory in Context, 2nd ed., Example 4.1.8
- Emily Riehl, Category Theory in Context, 2nd ed., Example 4.1.10
- Tom Leinster, Basic Category Theory, Example 2.1.3
- Emily Riehl, Category Theory in Context, 2nd ed., Example 4.1.6
- Tom Leinster, Basic Category Theory, Example 2.1.5
- Emily Riehl, Category Theory in Context, 2nd ed., Example 4.6.13
- Tom Leinster, Basic Category Theory, Example 6.3.14
- Emily Riehl, Category Theory in Context, 2nd ed., Example 4.1.12
- Tom Leinster, Basic Category Theory, Example 6.3.5
- Emily Riehl, Category Theory in Context, 2nd ed., Example 4.4.9
- Tom Leinster, Basic Category Theory, Example 2.1.6
- Emily Riehl, Category Theory in Context, 2nd ed., Definition 4.2.1
- Tom Leinster, Basic Category Theory, Definition 2.1.1