The section preorder of the Hom bicopresheaf

Let $C$ be a category and $Hom : C^{op} \times C \to Sets$ its hom bicopresheaf. Then the sections of $Hom$ are precisely the morphisms of $C$, where each section can be represented by a tuple $((a,b), m : a \to b)$ with $m \in Hom(a,b)$. With this construction, we can produce a preordering on $Arrows(C)$ differently, which is interesting for studying the algebro-logical theory of categories.

Theorem. let $C$ be a category then the section preorder of $Hom$ is the $J$ preorder of $C$.

Proof. let $m: a \to b$ and $n: c \to d$ be morphisms in $C$. Then for $m \subseteq n$ in the section preorder of $Hom$ it must be the case that there exists an ordered pair: $$(i : c \to a, o : b \to d) \in C^{op} \times C$$ $$o \circ m \circ i = n$$ This is precisely the definition of $m \subseteq_J n$. It follows that the $J$ preorder of $C$ is the section preorder of $Hom$. $\square$

This constructs the overall morphic preordering of a category $C$ from its hom copresheaf $Hom : C^{op} \times C \to Sets$. There are two other morphic preorders associated with a category $C$: the $L$ and $R$ preorders, both of which are suborders of $J$. These can be constructed by restricting the hom bicopresheaf $Hom: C^{op} \times C \to Sets$ to certain wide subcategories of its index category $C^{op} \times C$. This is the subject of the following lemma.

Lemma. let $F : C \to Sets$ be a functor and let $S \subseteq C$ be a wide subcategory then the section preorder of $F_S$ is a subpreorder of the section preorder of $F$.

Proof. let $(t_1,x_1) \subseteq (t_2, x_2)$ in the section preorder of $F_S$ then $\exists m : t_1 \to t_2 \in S$ such that $F_S(m)(x_1) = x_2$. But since $S \subseteq C$ we have that $m : t_1 \to t_2$ in $C$ such that $F(m)(x_1) = x_2$. So $a \subseteq_{F_S} b$ implies that $a \subseteq_{F} b$, which means that the section preorder of $F_S$ is a subpreorder of that of $F$. $\square$

We can use this lemma to construct the specialized $L$ and $R$ preordering on the morphism set of a category as subpreorders of the $J$ preorder produced by the Hom bicopresheaf.

Theorem. let $C$ be a category and $Hom : C^{op} \times C \to Sets$ its hom bicopresheaf. Then the output action preorder $L$ is the the section preorder of the restriction of $Hom$ to the wide subcategory with only output actions and $R$ is the section preorder of the resrtiction of $Hom$ to the wide subcategory containing only input actions.

Proof. if $m: a \to b \subseteq_L n: a \to c$ this means that there exists $o : b \to c$ with $o \circ m = n$ which is the same as saying there is $(1_a, o)$ in the wide subcategory of $C^{op} \times C$ with only output actions. Similarily for the right preorder $\subseteq_R$, so the $L$ and $R$ preorders are formed by the section preorders of restrictions of the hom bicopresheaf. $\square$

This produces an interpretation of the morphic preorders of a category $C$ that is much more suitable for our purposes. We can now define the morphic preordering of $C$ to simply be the object preordering of the category of elements of the hom functor. We have now two types of preorders on a category, both of which are object preorders of their respective categories:

• The preorder on $Ob(C)$ is the object preorder of $C$.
• The preorder on $Arrows(C)$ is the object preorder of the category of elements of the hom functor.

This is better because it is much more natural to work with object preorderings then any other type of preorder. Preorders themselves are defined as object preorders, and the composition operation of a category can add extra context to the logic of a preorder. Categories can be used to describe the algebraic laws of motion of preorders.

References:
Hom functor