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Initial and terminal objects may also be characterized in terms of universal properties and adjoint functors. Let 1 be the discrete category with a single object (denoted by •), and let U : C → 1 be the unique (constant) functor to 1. Then An initial object I in C is a universal morphism from • to U.
A monoid object in [C, C] is a monad on C. For any category with a terminal object and finite products, every object becomes a comonoid object via the diagonal morphism Δ X : X → X × X. Dually in a category with an initial object and finite coproducts every object becomes a monoid object via id X ⊔ id X : X ⊔ X → X.
Formally, we start with a category C with finite products (i.e. C has a terminal object 1 and any two objects of C have a product). A group object in C is an object G of C together with morphisms. m : G × G → G (thought of as the "group multiplication") e : 1 → G (thought of as the "inclusion of the identity element")
If F and G are (covariant) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism η X : F(X) → G(X) in D such that for every morphism f : X → Y in C, we have η Y ∘ F(f) = G(f) ∘ η X; this means that the following diagram is commutative:
In object-oriented programming, the singleton pattern is a software design pattern that restricts the instantiation of a class to a singular instance. It is one of the well-known "Gang of Four" design patterns , which describe how to solve recurring problems in object-oriented software. [ 1 ]
Ind-objects in C form a category ind-C.. Two ind-objects : and : determine a functor I op x J Sets,. namely the functor ((), ()). The set of morphisms between F and G in Ind(C) is defined to be the colimit of this functor in the second variable, followed by the limit in the first variable:
The virtual method table is the same for all objects belonging to the same class, and is therefore typically shared between them. Objects belonging to type-compatible classes (for example siblings in an inheritance hierarchy) will have virtual method tables with the same layout: the address of a given method will appear at the same offset for ...
The unit type is the terminal object in the category of types and typed functions. It should not be confused with the zero or empty type, which allows no values and is the initial object in this category. Similarly, the Boolean is the type with two values. The unit type is implemented in most functional programming languages.