nLab partially ordered object

Idea

The notion of a partially ordered object is the generalization of that of partially ordered sets as one passes from the ambient category of sets into more general ambient categories with suitable properties.

Definitions

In a finitely complete category $C$, a partially ordered object is a preordered object $(X, R, s, t, \rho, \tau_p)$ such that the internal preorder $R\stackrel{(s,t)}\hookrightarrow X \times X$ is an internal antisymmetric relation.

See also

• preordered object

• partially ordered set

• internal antisymmetric relation

• meet-semilattice object

• join-semilattice object