Maths - Category Theory - Adjunctions

An adjunction is a looser relationship between structures than isomorphism or equivalence. On the previous page about equivalence we looked at 'equivalence as an isomorphism upto isomorphism':

We could look at adjunctions as a further weakening so that, instead of 2 isomorphisms of mappings we just have 2 mappings of mappings (natural transformations):

The topic of adjunctions is a very powerful concept but can be difficult to understand, it might therefore help to approach the subject from different directions. We give two definitions and show that they are the same.

Definition 1

adjunction An adjunction is given by a pair of functors, F and G.
adjunction triangle identity

Equipped with natural transformations

  • unit: μ : 1c =>GF
  • co-unit: ξ : FG => 1d

such that the diagrams on the left (known as the triangle identities) commutes.

adjuntion components

We can look at this in terms of the components. At first it may appear that we cannot use natural transformations because the arrows between C and D go in opposite directions and therefore the domains and codomains are not the same. However we can rearrange the diagram so that a natural transformation goes between 1 and GF and another one goes between FG and 1.

This (perhaps over-complicated) diagram is trying to represent the natuality squares of the two natural transformations and how they all link together..

For adjunctions there is a arrow in both directions between the two objects. It is not necessary that GF and FG are the identity elements but only that they have natural transformations to/from the identity elements.

FG is unit (does not change object - injective followed by surjective) but GF does change object (surjective followed by injective). Like equivalence but in one direction.

The above diagrams can also be represented as 2-cells:

  adjunction 2-cell

Definition 2

In this approach we are still using isomorphisms between morphisms but in an apparently different way that turns out to be the same.

We have an isomorphism between a morphism in C and a morphism in D. Until now, to compare morphisms, we have used natural transformations. But natural transformations can only compare morphisms with the same domain and codomain here the morphisms, being compared, are not even in the same category.

This comparison is called a 'natural isomorphism' as follows:

We have arrows in C which are isomorphic to arrows in D. Shown here as red arrows. adjunction
adjunctions

An adjunction is given by a natural isomorphism

morphisms
in D
  morphisms
in C
 
over
  over   over  
over
 
φ: D( F x , y ) → C( x , G y )
for every x∈C and y∈D
which is naturally isomorphic in both x and y

What are the functors that are naturally isomorphic?

Cop→ set

  D(F _,y)  
x
mapElement
D(F x,y)

Cop→ set

  C(_,G y)  
x
mapElement
C(x,G y)

D→ set

  D(F x,_)  
y
mapElement
D(F x,y)

D→ set

  C(x,G _)  
y
mapElement
C(x,G y)

So the naturality square in x is:

adjunction naturality in x

We precompose with F f to get:

fx -> fx' -> y

String Diagrams

String diagrams are the Poincaré dual of the usual category theoretic diagrams, they can be useful for 2-categories such as adjunctions. (more about string diagrams on this page).

  Category Diagram String Diagram
We start with a diagram like this (F is left adjoint to G) and add 2 natural transformations below (η and ξ) with the following axioms. adjunctions string diagram  
We take this diagram: adjunctions string diagram between functors and put it in a diagram with the objects. The string diagram has the identity functor omitted as explained above. Also, for simplicity, ξ is not shown in a circle. adjunctions string diagram adjunction string diagram
adjunction string diagram And again for η. adjunctions string diagram adjunction string diagram
adjunction string diagram Now for the triangle equalities. In the string diagram it just looks like we can pull the line straight. adjunction string diagram adjunction string diagram
adjunction string diagram The other triangle equality adjunction string diagram adjunction string diagram

Examples

Example 1 - Preordered Set

Here we work in the category of preordered sets (I have illustrated this as a directed graph, I find it helpful to use draw directed graphs as described in the box on the right)

preordered set 1

Here we look at a perordered set ≤(red) and a mapping to another perordered set (blue). The morphism between them (green) preserves the ordering. That is:

if a ≤ a' then F(a) ≤ F(a')

  So is there a mapping (G) back from B to A that preserves as much structure as possible in some unique way?
preordered set 2

Here is a first attempt, as we can see we can compose these morphisms and still preserve the order structure
:if a ≤ a' then GF(a) ≤ GF(a'). However, it does not do this in a unique way, there are many mappings for G that could have done this.

preordered set 3

So lets add another condition:

GF(a) ≤ a

This condition is not met for our first choice of G (on the diagram on the left this is interpreted as reversing one of the red arrows).

preordered set 4

However, given the morphism F then there is a way to choose a way to choose the morphism G so that the order is preserved relative to the original sequence.

For instance, if the number the elements of 'A' with ascending values. We then map these values onto 'B' using the morphism 'F'. We can then determine G by mapping the highest value for each element in 'B' to the corresponding number in 'A'.

G is then the adjoint of F.

 

So, for adjoint morphisms:

F(a) ≤ b if and only if a ≤ G(b)

Now we have a unique mapping G we can go on to show the unit and co-unit:

unit

μ : 1c =>GF

So the unit maps 1c to GF as in this diagram. In this diagram we can't derive mappings from movement of points, we just map the mappings directly.

I added in the purple arrows to make it into a graph, they are not really generated by GF.

counit

ξ : FG => 1d

In this case the co-unit is trivial.

Same Thing Using Definition 2

In this case we take an element 'A' of C and an element 'X' of D. We can then determine GX and FA by following the arrows.

The arrow A->GX is then isomorphic to the arrow FA->X.

This must be so for any 'A' in C and 'X' in D.

preorder adjunction definition 2

Example 2 - Monoid

This example looks at extremes such as the forgetful functor in one direction (which removes all structure and leaves the underlying set) and the free algebra in the other direction (see Presentation of a group. Free algebra has generators but not relations).

'term' or 'word' algebras determines an adjunction

list as free monoid

So,

therefore:

Question 1: What does the forgetful functor do to list? Does it return just the characters of the set? (in this case the characters A,B,C,D,E) or does it return an infinite set whose elements are every possible list?

Question 2: Is Monoid left or right adjoint to Set? What is the other 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

Examples

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: t left adjoint i

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)

Some very General Adjunctions

         
existentialthere exists,sum Σ left adjoint

weakening
(adding an extra assumption)

   
weakening left adjoint universalfor all, product Π    
equality left adjoint contraction    
truth left adjoint comprehension or subset types    
equality left adjoint comprehension    
quotients left adjoint equality    
  left adjoint      

 


metadata block
see also:

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.
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