Questions about PO category

Question

I'm very new to category theory. And I have some basic questions:

Let $(X, \leq)$ be a preordered set ($\leq$ is reflexive and transitive).

Define a category $PO(X, \leq)$ by

• The objects of $PO(X, \leq)$ is the set $X$.
• The morphisms are defined to be $Hom(x,y) = \emptyset$ if $x \not\leq y$ and $Hom(x,y)= \{u_{x,y}\}$ if $x \leq y$.
• Composition is defined by $u_{y,z} \circ u_{x,y} = u_{x,z}$

Questions:

(1) Are the $u_{x,y}$ just formal symbols? What exactly are they?

(2) It looks like we only defined the composition map $\circ$ for the case $$\circ: Hom(x,y) \times Hom(y,z) \to Hom(x,z)$$ where $x \leq y$ and $y \leq z$.

Do we implicitely define the composition maps to be $\emptyset$ in all other cases?

Answer 1

(1) Are the $u_{x,y}$ just formal symbols? What exactly are they?

Yes, that's exactly what they are: just stand-ins. The situation is similar to when I tell you we are dealing with the group $C_4$; it doesn't matter whether this is the numbers $\{0, 1, 2, 3\}$, or equivalence classes of integers, or formal symbols $e, g, g^2, g^3$; you understand what to do with them.

(2) (...) Do we implicitely define the composition maps to be $\emptyset$ in all other cases?

Indeed, in all other cases the domain of the composition function is empty, so the only possible mapping is the empty map.

Answer 2

(1) Yes, $u_{x,y}$ is just the name given to the unique morphism $x \to y$ when $x \le y$.

(2) If $x \not\le y$ or $y \not\le z$, then the domain of $\circ : \mathrm{Hom}(x,y) \times \mathrm{Hom}(y,z) \to \mathrm{Hom}(x,z)$ is empty, so the only possible way that it can be defined is as the trivial map $\varnothing \to \mathrm{Hom}(x,z)$.

Put another way, the composition map $\circ$ on a category $\mathcal{C}$ can be specified by defining what value is taken by $g \circ f$ on all pairs $(f,g)$ of composable morphisms in $\mathcal{C}$. In this case, the only pairs composable morphisms are those of the form $(u_{x,y}, u_{y,z})$ for $x \le y \le z$, so it suffices to only define what $u_{y,z} \circ u_{x,y}$ is for $x \le y \le z$.