The Identity Morphism and Functorial Witness. More...

Public Types
using	Domain = T
	The source category species.
using	Codomain = T
	The target category species (invariant for identity).

Public Member Functions
constexpr T	operator() (const T &x) const noexcept
	The Morphic Action (Object -> Object).
constexpr bool	operator== (const Identity &) const =default
template<typename S> requires (!std::same_as<std::remove_cvref_t<S>, T>) && requires { T::template presents<std::remove_cvref_t<S>>; } && Presents<S, T>
constexpr S	operator() (const S &s) const noexcept
	The identity at a presentation of `T`, where `T` is an object that names its presentations (`morphism__Objects_And_Their_Presentations`): the presentation itself. A value-typed `T` keeps the overload above.
template<IsArrow Arrow> requires std::same_as<typename Arrow::Domain, T>
constexpr auto	operator() (Arrow f) const noexcept
	The Functorial Action (Morphism -> Morphism).

Detailed Description

template<typename T>
struct dedekind::category::Identity< T >

The Identity Morphism and Functorial Witness.

Factory Verification: Tagging and Species

Integrity.

morphism__The_Universal_Functor_Interface

In the Category of Species (Set), Identity acts as the unit morphism (id_T: T \to T). In the Category of Functors, it acts as the Identity Functor (Id: \mathcal{C} \to \mathcal{C}).

This reification is "Dual-Action": it preserves the structure of both individual elements (Species) and their transformations (Morphisms). It is the canonical "Zero-Length Highway" that connects a species to itself without distortion.

Template Parameters

T	The Species (Object) for which this identity is defined.

Member Function Documentation

◆ operator()() [1/2]

template<typename T>

template<IsArrow Arrow>
requires std::same_as<typename Arrow::Domain, T>

auto dedekind::category::Identity< T >::operator() ( Arrow f ) const

inlineconstexprnoexcept

The Functorial Action (Morphism -> Morphism).

Implements the Identity Functor lift (F(f) = f). This satisfies the Functorial Identity Law: mapping an arrow through the identity context must result in the original arrow.

Template Parameters

Arrow A type satisfying the IsArrow concept.

Parameters

f	The arrow to be lifted into the identity context.

Returns: The original arrow, preserved without transformation.

◆ operator()() [2/2]

template<typename T>

T dedekind::category::Identity< T >::operator() ( const T & x ) const

inlineconstexprnoexcept

The Morphic Action (Object -> Object).

morphism__Categorical_Actions

The following operators ensure that Identity acts as a structure-preserving map across different levels of the categorical hierarchy.

Implements the identity function (f(x) = x). This satisfies the "Identity on Set" requirement for Level 0 categorical grounding.