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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.
Let C be a category with finite products and a terminal object 1. A list object over an object A of C is: an object L A, a morphism o A : 1 → L A, and; a morphism s A : A × L A → L A; such that for any object B of C with maps b : 1 → B and t : A × B → B, there exists a unique f : L A → B such that the following diagram commutes:
Examples of limits and colimits in Ring include: The ring of integers Z is an initial object in Ring. The zero ring is a terminal object in Ring. The product in Ring is given by the direct product of rings. This is just the cartesian product of the underlying sets with addition and multiplication defined component-wise.
Note that because a nullary biproduct will be both terminal (a nullary product) and initial (a nullary coproduct), it will in fact be a zero object. Indeed, the term "zero object" originated in the study of preadditive categories like Ab , where the zero object is the zero group .
In the category of sets, morphisms are functions and the terminal objects are singletons.Therefore, a morphism : is a function from a singleton {} to the set : since a function must specify a unique element in the codomain for every element in the domain, we have that () is one specific element of .
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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.
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