Monotonicity of morphism properties

Recall that every category is associated with a suborder of the lattice of partitions of morphisms. It will be shown that the members of this lattice can be represented by either monotone or antitone maps over the preorders of a category. In order to do this, we must establish the following trinity of preorders on the elements of a category:

• Objects
• Hom classes
• Morphisms

Then the monotonicity of the maps between these preorders will clarify the relationship between the object and morphism preorders and the order theoretic nature of categories.

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:
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.

1. $T: Arrows(C) \to Ob(C)^2$ the map from any morphism to its hom class is monotone
2. $In : Arrows(C) \to Ob(C)$ the map from any morphism to its input object is antitone
3. $Out : Arrows(C) \to Ob(C)$ the map from any morphism to its output object is monotone

Proof. (1) let $x : A \to B, y : C \to D \in Arrows(C)$ be morphisms, then $x \subseteq y$ implies that $y = l \circ x \circ r$. Thus, we have the following chain of morphisms. By simple inspection we have $C \subseteq A$ and $B \subseteq D$, so that $(A,B) \subseteq (C,D)$ in the hom class ordering.

(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: