Posetal category

In mathematics, a posetal category is a category whose homsets each contain at most one morphism. As such a posetal category amounts to a preordered set. The further requirement that the category be skeletal is often assumed, which in the case of a posetal category is equivalent to the requirement that the only isomorphisms are the identity morphisms, equivalently that the preordered set satisfies antisymmetry and hence is a partially ordered set.

All diagrams commute in a posetal category. When the commutative diagrams of a category are interpreted as a typed equational theory whose objects are the types, a posetal category corresponds to an inconsistent theory understood as one satisfying the axiom x = y.

All posetal categories are automatically 2-categories because they satisfy the requisite commutativity conditions vacuously when the 2-cells are taken to be the identity morphisms on the 1-cells.

Some lattice theoretic structures are definable as posetal categories of a certain kind, usually with the stronger assumption of being skeletal. For example a distributive lattice may be defined as a posetal distributive category, while a Heyting algebra may be defined as a posetal cartesian closed category.