Definition and examples

Exercise 1-101(2)

Does \(\mathbb{R}\xrightarrow{\lceil x/3 \rceil}\mathbb{Z}\) have a left adjoint \(\mathbb{Z} \xrightarrow{L} \mathbb{R}\)? If not, why? If so, does its left adjoint have a left adjoint?

Solution(1)

Assume we have an arbitrary left adjoint, \(L\).
For \(x\) as it approaches \(0.0 \in \mathbb{R}\) from the right, we have ?\(R(x) \leq 1\), therefore ?\(L(1) \leq x\) because \(L\) is left adjoint.
Therefore \(L(1)\leq 0.0\), yet this implies ?\(R(0.0) \leq 1\).
This contradicts ?\(R(0.0)=0\), therefore no left adjoint exists.

Floor and ceil(1)

Consider the map \(\mathbb{Z} \xrightarrow{3z} \mathbb{R}\) which sends an integer to \(3z\) in the reals.
To find a left adjoint for this map, we write \(\lceil r \rceil\) for the smallest natural above \(r \in \mathbb{R}\) and \(\lfloor r \rfloor\) for the largest integer below \(r \in \mathbb{R}\)
The left adjoint is ?\(\lceil r/3 \rceil\)
Check: ?\(\lceil x/3 \rceil \leq y\) \(\iff x \leq 3y\)

A Galois connection between preorders \(P\) and \(Q\), and the left and right adjoints of a Galois connection

A pair of monotone maps \(P \xrightarrow{f} Q\) and \(Q \xrightarrow{g} P\) such that:

\(f(p) \leq q \iff p \leq g(q)\)
\(f\) is left adjoint and \(g\) is right adjoint of the Galois connection.

Linked by

Total order galois connections(1)

Consider the total orders \(P = Q = \underline{3}\) with the following monotone maps:
- These do form a Galois connection
These maps do not form a Galois connection:
- These do not because of ?\(p=2, q = 1\)
- ?\(f(p)=2 \not \leq q=1\) which is not the same as ?\(p = 1 \leq g(q)=2\)
In some sense that can be formalized, for total orders the notion of Galois connection corresponds to ?the maps not ‘crossing over’.