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In category theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object (also called terminal element ): T is terminal if for every object X in C there exists exactly one morphism X → T .
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.
If A is an object of C, then the functor from C to Set that sends X to Hom C (X,A) (the set of morphisms in C from X to A) is an example of such a functor. If C is a small category (i.e. the collection of its objects forms a set), then the contravariant functors from C to Set, together with natural transformations as morphisms, form a new ...
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.
HTML element content categories. HTML documents imply a structure of nested HTML elements. These are indicated in the document by HTML tags, enclosed in angle brackets thus: < p >. [73] [better source needed] In the simple, general case, the extent of an element is indicated by a pair of tags: a "start tag" < p > and "end tag" </ p >. The text ...
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Any two objects X and Y of C have a product X ×Y in C. Any two objects Y and Z of C have an exponential Z Y in C. The first two conditions can be combined to the single requirement that any finite (possibly empty) family of objects of C admit a product in C, because of the natural associativity of the categorical product and because the empty ...