# Maths - Category Theory - Adjunctions

An adjunction is a looser relationship between structures than isomorphism or equivalence. As with isomorphism and equivalence an adjunction applies to a pair of functors, F and G.

On the previous page about equivalence we looked at equivalence as an isomorphism upto isomorphism:

 Isomorphism axioms 1C GF FG 1D

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):

 Adjunction natural transformations: note: these are not the axioms. unit: μ : 1c => GF co-unit: ξ : FG => 1d Adjunction axioms: F FGF GFG G

This can intuitively make sense, for instance, if G looses information (is forgetful) then we may be able to choose a F such that: applying FG after G looses no more information than we have already lost doing G once. So going round the loop looses no more information than we have already lost.

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.

### Natural Transformations

My interpretation of these natural transformations would be:

 FG => 1d For every F there is one or more G that makes FG map to identity. 1c => GF For every F there is one or more G that makes GF a valid mapping (not necessarily identity). 1C= GF For every F there is a G that makes FG map to identity.

## Definition 1 An adjunction is given by a pair of functors, F and G (see diagram above) 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. 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: ## 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.  An adjunction is given by a natural isomorphism

 morphisms in D morphisms in C    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 D(F x,y)

Cop→ set

 C(_,G y) x C(x,G y)

D→ set

 D(F x,_) y D(F x,y)

D→ set

 C(x,G _) y C(x,G y)

So the naturality square in x is: We precompose with F f to get: fx -> fx' -> y

 Every adjunction gives rise to a monad. Every monad gives rise to many adjunctions   ### As 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. We take this 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.   And again for η.   Now for the triangle equalities. In the string diagram it just looks like we can pull the line straight.   The other triangle equality  ## Examples

existential ,sum Σ weakening

weakening universal , product Π
equality contraction
truth comprehension or subset types
equality comprehension
quotients equality ### Links to More Detailed Examples

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.

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

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.