Maths - Examples of Adjunctions

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

+left adjointΔleft adjoint×

The top triangle is the co-unit:
ξ : Δ • × -> I

The lower triangle is the unit:
η : I -> × • Δ

adjunction prod diagonal

The top triangle is the co-unit:
ξ : + • Δ -> I

The lower triangle is the unit:
η : I -> Δ • +

adjunction diagonal sum

This is explained by Bartosz Milewski on his blog here.

We can also relate this to logic:

existentialthere exists,sum Σ left adjoint

weakening
(adding an extra assumption)

weakening left adjoint universalfor all, product Π

Currying

This relates product to functions.

Some Other Very General Adjunctions

         
equality left adjoint contraction    
truth left adjoint comprehension or subset types    
equality left adjoint comprehension    
quotients left adjoint equality    
  left adjoint      

 

Adjoint Pairs

The basis on a vector space

The free group on a set

G mapElement G / [G,G] commutator subgroup

universal enveloping algebra of a Lie algebra

completion of a metric space

Category of Graphs

described on page here.

Between reflexive graph and set.

In reflexive graph: every node has loop to itself.

graph example

reflexive graph

irreflexive

Between irreflexive graph and set.

In irreflexive graph: every node does not have loop to itself.

Between dynamical system graph and set.

In dynamical system graph: every node has one outgoing arrow.

dynamical systems graph
dynamical systems and fixpoints

Different dynamical system graph and set.

This time the morphism to set defines the fixpoints.

Poset

An endofunctor on posets models closure. Posets don't have loops, therefore defined by fixpoints.

poset

Define T: P -> P

with

  • x <= T x
  • T² x <= T x

gives

T² <= T (that is it is idempotent)

poset

Which gives an adjunction: tleft adjointi

This is discussed as a monad on page here.

Implementing Posets in FriCas program is discussed on page here.

Category    
type theory quantifiers

substitution

(Cartesian maps)

 

 

For more information about these examples see Conceptual Mathematics book - Appendix II.

References

flag flag flag flag flag flag Conceptual Mathematics - This is a book about category theory that does not assume an extensive knowledge over a wide area of mathematics. The style of the book is a bit quirky though.
Implementing Graphs in FriCas program is discussed on page here.

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see also:

Other Sites

Youtube Videos

Adjunctions from Morphisms

Other Pages on this site

  • Category of graphs described on page here.
  • Implementing Graphs in FriCas program is discussed on page here.
  • Implementing Posets in FriCas program is discussed on page here.
Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Conceptual Mathematics - This is a book about category theory that does not assume an extensive knowledge over a wide area of mathematics. The style of the book is a bit quirky though.

 

Terminology and Notation

Specific to this page here:

 

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