
- Objects
- Hom classes
- Morphisms
Object preorder:
The object preorder is the most basic of the three. It fully determines thin categories.
Definition. let C be a category, Ob(C) its set of objects, and x,y \in Ob(C). Then x \subseteq y means that \exists f : x \to y.
Example. the total order T_4

The hom class preorder is simply the interval inclusion preorder of the object preorder.
Definition. let (A,B) and (C,D) be hom classes in C then (A,B) \subseteq (C,D) provided that A \subseteq C and B \subseteq D.
Example. the interval inlusion ordering on T_4 Morphic preordering:
The morphic preordering of a category is a generalisation of the Green's J preorder of a semigroup.
Definition. let C be a category, Arrows(C) its set or proper class of morphisms, x,y \in Arrows(C). Then x \subseteq y provided that \exists l,r : y = l\circ x \circ r.
Monotonicity:
We can now show that the object and morphism preorders of a category are related by monotone and antitone relatiosnhips.
Theorem.
- T: Arrows(C) \to Ob(C)^2 the map from any morphism to its hom class is monotone
- In : Arrows(C) \to Ob(C) the map from any morphism to its input object is antitone
- Out : Arrows(C) \to Ob(C) the map from any morphism to its output object is monotone

(2) C \subseteq A, so that the input object of y is less then that of x. It follows that the input object is antitone.
(3) B \subseteq D implies that the output object map is monotone. \square
These are the three properties of morphisms inherent to the definition of a category and they are all monotone. This defines the relationship between the object and morphism preorders of a category.
See also:
Categories for order theorists:
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