Generalisation of posets

Question

The notion of (set) monoid can be generalised to that of monoid object in a monoidal category. Can the notion of poset be generalised in a similar fashion? How?

Answer 1

The internalization of preorders are internal binary relations which are reflexive and transitive.

An internal relation $(d_0,d_1 : R \to X)$ is called reflexive if there is a morphism $\Delta : X \to R$ such that $d_0 \Delta = \mathrm{id}_X = d_1 \Delta$. Since $(d_0,d_1)$ are jointly monic, $\Delta$ is unique, if it exists.

An internal relation $(d_0,d_1 : R \to X)$ is called transitive if for all morphisms $u : T \to R$, $v : T \to R$ with $d_1 u = d_0 v$ there is some morphism $w : T \to R$ with $d_0 w = d_0 u$ and $d_1 w = d_1 v$. Again, $w$ is unique, if it exists.

This describes internal preorders. We get internal posets by adding antisymmetry:

An internal relation $(d_0,d_1 : R \to X)$ is called antisymmetric if for all morphisms $u : T \to R$, $v : T \to R$ with $d_1 u = d_0 v$ and $d_0 u = d_1 v$ we have $d_0 u = d_1 u$.