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  2. Initial and terminal objects - Wikipedia

    en.wikipedia.org/wiki/Initial_and_terminal_objects

    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 .

  3. Category of sets - Wikipedia

    en.wikipedia.org/wiki/Category_of_sets

    The empty set serves as the initial object in Set with empty functions as morphisms. Every singleton is a terminal object, with the functions mapping all elements of the source sets to the single target element as morphisms. There are thus no zero objects in Set. The category Set is complete and co-complete.

  4. Initial algebra - Wikipedia

    en.wikipedia.org/wiki/Initial_algebra

    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).

  5. Category of topological spaces - Wikipedia

    en.wikipedia.org/wiki/Category_of_topological_spaces

    Examples of limits and colimits in Top include: The empty set (considered as a topological space) is the initial object of Top; any singleton topological space is a terminal object. There are thus no zero objects in Top. The product in Top is given by the product topology on the Cartesian product.

  6. Category of rings - Wikipedia

    en.wikipedia.org/wiki/Category_of_rings

    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.

  7. Category of small categories - Wikipedia

    en.wikipedia.org/wiki/Category_of_small_categories

    The terminal object is the terminal category or trivial category 1 with a single object and morphism. [2] The category Cat is itself a large category, and therefore not an object of itself. In order to avoid problems analogous to Russell's paradox one cannot form the “category of all categories”. But it is possible to form a quasicategory ...

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