Search results
Results from the WOW.Com Content Network
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 ...
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.
As a rule of thumb, an equivalence of categories preserves all "categorical" concepts and properties. If F : C → D is an equivalence, then the following statements are all true: the object c of C is an initial object (or terminal object, or zero object), if and only if Fc is an initial object (or terminal object, or zero object) of D
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).
A (covariant) functor F from a category C to a category D, written F : C → D, consists of: for each object x in C, an object F(x) in D; and; for each morphism f : x → y in C, a morphism F(f) : F(x) → F(y) in D, such that the following two properties hold: For every object x in C, F(1 x) = 1 F(x);