dedekind::category::IsThinCategory Concept Reference

A category in which every hom-set has at most one morphism. More...

Concept definition

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

Detailed Description

A category in which every hom-set has at most one morphism.

Equivalently, a preorder over T with relation Rel returning L::Ω. The axioms enforced:

• Closure: rel(a, b) yields a truth value in L::Ω.
• Reflexivity: ∀a. rel(a, a) = L::True (the identity morphism).
• Transitivity: ∀a,b,c. rel(a, b) ∧ rel(b, c) ⇒ rel(a, c) (composition).

Antisymmetry is not required; IsPosetal in :posetal is the strict refinement that adds it.

Template Parameters

T	The Domain (Objects).
Rel	The Relation (the unique-morphism witness).
L	The Logic Species (the Subobject Classifier Ω).