Functorial theory of Galois connections

We define the following category $C$ as an index category for monotone Galois connections: $C$ has four non-identity morphisms with the properties that $FG$ and $GF$ are idempotent, $FGF = F$ and $GFG = G$. A structure with these composition laws forms a category so $C$ is a well-defined category. We will use this category to examine the theory of Galois connections.

Galois connections as presheaves
Using the category $C$ we can define every monotone Galois connection as a presheaf of preorders in the functor category $[C,Ord]$. These are presheaves by the forgetful functor $F: Ord \to Sets$ from the category of preorders to the topos $Sets$.

Definition. let $(F,G)$ be a monotone Galois connection of preorders $A$ and $B$. Then their presheaf of preorders is defined by the diagram $C$ using the composed component arrows $FG$ and $GF$ as closure and interior operators.

This nicely encapsulates the entire datum of a monotone Galois connection into a single structure presheaf. The dual condition that all presheaves of preorders over $C$ is a monotone Galois connection is of course not true. Instead for that to happen the mappings of the presheaf need to have certain special conditions hold.

Properties of the component morphisms
The four morphisms in the Galois connection diagram $F$,$G$, $FG$, and $GF$ all belong to different types of categories and they all have their own theories associated to them:

• $F$: a residuated mapping (it reflects principal down sets)
• $G$: a coresiduated mapping (it reflects principal up sets)
• $GF$: a closure operator (idempotent, extensive, and monotone)
• $FG$: an interior operator (idempotent, decreasing, and monotone)

Each of these form different types of categories, because they are all closed under composition. The category of partial orders and residuated maps can be used to study the compositional properties of monotone Galois connections.

Presheaf perspective on order theory
It is increasingly my contention that the basic objects of order theory should be presheaves of preorders, and that this presheaf theoretic perspective should be applied to the subject. Let $S$ be a set, then it is associated to a lattice of preorders. A particular elegant construction is that we can associate instead to any presheaf $F: X \to Sets$ a lattice of presheaves of preorders.

I argue that the perspective of studying presheaves of preorders, which are functors $F: C \to Ord$ gives us the best theoretic footing on which to do order-theory from. In the same way that algebraic geometry studies certain presheaves of rings, order theory should be remade for presheaf foundations. In this context, a preorder is a presheaf over the trivial category, a monotone map is a presheaf over $T_2$, an order isomorphism is a presheaf over $K_2$, a Galois connection is a presheaf over $C$, and so on.

Presheaves and their topoi $Sets^C$ should be their basic object of study in any case in either logic or geometry. Topos theory is of the greastest foundational importance, however, when we get around to studying preorders, which are among the most fundamental objects of study then they should be considered by presheaves of preorders. I think the presheaf theoretic perspective will be getting greater acceptance and acknowledgement as time goes on.

Besides algebraic geometry, logic, and order theory it is desirable that presheaves should be used to reinterpret our understanding of computer science. Computation on the machine should be modeled by certain presheaves of memory locations, as this will produce the best results. So the presheaf perspective has the widest degree of applicability in different fields.

References:
Galois connection