Alphabeta MathAdjunctions Units and Counits: Examples
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✓ 10 results · all verified · 5 also independently AI-judged
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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 Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
Category Theory
Linear Algebra
Real Analysis
2 · Logical flowchart
3 · Items
Example
The unit inserts generators as one-letter words and the counit evaluates words in the free-group adjunction
Example
The unit inserts basis vectors and the counit evaluates formal linear combinations in the free-vector-space adjunction
Example
Vanishing sets and vanishing ideals form a contravariant Galois connection
Example
Finite directed paths form the free category on a quiver
Example
The inclusion of groupoids into categories is left adjoint to the maximal-subgroupoid functor
Example
Frobenius reciprocity for group representations without tensor products
Counterexample
A componentwise family of morphisms need not be a natural transformation and hence need not be a unit
Counterexample
A wrong counit can be natural while both triangle identities fail
Example
A floor-division and multiplication adjunction between natural-number preorders
Example
Ceiling ⊣ inclusion ⊣ floor: an adjoint triple between (ℝ,≤) and (ℤ,≤)
Sources
- Emily Riehl, Category Theory in Context, 2nd ed., Example 4.2.4
- Tom Leinster, Basic Category Theory, Example 2.2.1
- Emily Riehl, Category Theory in Context, 2nd ed., Example 4.4.2
- Emily Riehl, Category Theory in Context, 2nd ed., Example 4.1.13
- Emily Riehl, Category Theory in Context, 2nd ed., Example 4.1.15
- Emily Riehl, Category Theory in Context, 2nd ed., Example 4.1.11
- Emily Riehl, Category Theory in Context, 2nd ed., Definition 4.2.1
- Tom Leinster, Basic Category Theory, Section 2.1
- Emily Riehl, Category Theory in Context, 2nd ed., Example 4.1.7