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For example, the initial object in any concrete category with free objects will be the free object generated by the empty set (since the free functor, being left adjoint to the forgetful functor to Set, preserves colimits). Initial and terminal objects may also be characterized in terms of universal properties and adjoint functors.
In C++, a constructor of a class/struct can have an initializer list within the definition but prior to the constructor body. It is important to note that when you use an initialization list, the values are not assigned to the variable. They are initialized. In the below example, 0 is initialized into re and im. Example:
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 .
Universal constructions are functorial in nature: if one can carry out the construction for every object in a category C then one obtains a functor on C. Furthermore, this functor is a right or left adjoint to the functor U used in the definition of the universal property. [2] Universal properties occur everywhere in mathematics.
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.
Dually, a final coalgebra is a terminal object in the category of F-coalgebras. The finality provides a general framework for coinduction and corecursion. For example, using the same functor 1 + (−) as before, a coalgebra is defined as a set X together with a function f : X → (1 + X).
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.
However, LH does not have a terminal object, and thus is not Cartesian closed. If C has pullbacks and for every arrow p : X → Y, the functor p * : C/Y → C/X given by taking pullbacks has a right adjoint, then C is locally Cartesian closed. If C is locally Cartesian closed, then all of its slice categories C/X are also locally Cartesian closed.