Maths - Adjunctions - Examples Preordered Set

• F : A -> B
• G : B -> A

such that for all a in A and b in B, we have

F(a) ≤ b if and only if a ≤ G(b).

This works for composed morphisms like this:

(if we work through each rectangle the above condition holds)

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')

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.

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).

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:

μ : 1_c =>GF

So the unit maps 1_c 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.

ξ : FG => 1_d

In this case the co-unit is trivial.

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.

Define T: P -> P

with

• x <= T x
• T² x <= T x

gives

T² <= T (that is it is idempotent)

substitution

(Cartesian maps)