# Maths - Examples of Adjunctions

## Sum and Product

+Δ×

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

This is explained by Bartosz Milewski on his blog here.

We can also relate this to logic:

 existential,sum Σ weakening (adding an extra assumption) weakening universal, product Π

## Currying

This relates product to functions.

## Some Other Very General Adjunctions

equality contraction
truth comprehension or subset types
equality comprehension
quotients equality

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

### Category of Graphs

described on page here.

 Between reflexive graph and set. In reflexive graph: every node has loop to itself.
 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.
 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.

 Define T: P -> P with x <= T x T² x <= T x gives T² <= T (that is it is idempotent)

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)

## References

 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.

Other Sites

### 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.

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.

 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.