This page describes some examples of adjunctions. Adjunctions were introduced on the page here.
Example - Monoid
On this page is an example of an adjunction between set and a free monoid (List monad).
Example - Preordered Set
See this page for preordered set example.
Sum and Product
+Δ
×
The top triangle is the co-unit: The lower triangle is the unit: |
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The top triangle is the co-unit: The lower triangle is the unit: |
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This is explained by Bartosz Milewski on his blog here.
We can also relate this to logic:
existential![]() |
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weakening |
weakening | ![]() |
universal![]() |
Currying
This relates product to functions.
Some Other Very General Adjunctions
equality | ![]() |
contraction | ||
truth | ![]() |
comprehension or subset types | ||
equality | ![]() |
comprehension | ||
quotients | ![]() |
equality | ||
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Adjoint Pairs
The basis on a vector space
The free group on a set
G G / [G,G] commutator subgroup
universal enveloping algebra of a Lie algebra
completion of a metric space