Slice Category of Order Category

Theorem
Let $\mathbf P$ be a order category, and denote its ordering by $\preceq$.

Let $p \in \mathbf P_0$ be an object of $\mathbf P$.

Then:


 * $\mathbf P \mathbin / p \cong \mathop{\bar \downarrow} \left({p}\right)$

where:


 * $\mathbf P \mathbin / p$ is the slice of $\mathbf P$ over $p$;
 * $\mathop{\bar \downarrow} \left({p}\right)$ is the order category defined by the weak lower closure of $p$.

Proof
The objects of $\mathbf P \mathbin / p$ are morphisms $q \to p$ of $\mathbf P$.

The morphisms are $q \to r$ fitting into a commutative diagram:


 * $\begin{xy}\xymatrix@C=1em{

q \ar[rr] \ar[rd] & & r \ar[ld]

\\ & p }\end{xy}$

Define a functor $U: \mathbf P \mathbin / p \to \mathbf P$ by:


 * $U \left({q \to p}\right) := q$
 * $U \left({q \to r}\right) := q \to r$

Because there is at most one morphism $q \to p$ for each $q$, $U$ is injective on objects.

That $U$ is faithful is trivial.

Hence by Functor is Embedding iff Faithful and Injective on Objects, $U$ is an embedding.

That is, $\mathbf P \mathbin / p$ is isomorphic to the image of $U$.

Now the objects of the image of $U$ in $\mathbf P$ are those $q$ such that $q \preceq p$.

That is, the image of $U$ has as objects $\mathop{\bar \downarrow} \left({p}\right)$.

By Commutative Diagram in Preorder Category, every morphism $q \to r$ in $\mathop{\bar \downarrow} \left({p}\right)$ fits into a commutative triangle as above.

Thus all morphisms of $\mathop{\bar \downarrow} \left({p}\right)$ are in the image of $U$, so that $\mathop{\bar \downarrow} \left({p}\right)$ is precisely the image of $U$.

Hence the result.