A Category where morphisms are governed by a specific Logic Species (Ω). More...

Concept definition

template<typename T, typename Rel std::less_equal<T>, typename L Boole>
concept dedekind::category::IsPosetal = 
    IsThinCategory<T, Rel, L> &&  
                                  
    IsPartialOrder<T, Rel, typename L::Ω> && requires(Rel rel, T a, T b) {
      
      { rel(a, b) } -> std::same_as<typename L::Ω>;
    }

Detailed Description

A Category where morphisms are governed by a specific Logic Species (Ω).

This definition reifies the Poset as a skeletal category over a Topos L. By default, it assumes Classical (Boolean) logic, but it can be parameterized to support intuitionistic or fuzzy relations.

IsPosetal is the strict refinement of IsThinCategory (#698 row 1): a posetal category is a thin category that is also skeletal (antisymmetric). The faithful inclusion IsPosetal ⊊ IsThinCategory is encoded definitionally in the signature per the project's "faithful specialization in the type signature from day one" posture (#698).

Template Parameters

T	The Domain (Objects).
Rel	The Relation (Morphisms).
L	The Logic Species (The Subobject Classifier).