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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 ...
For a category C, the following are all equivalent: C is finitely complete, C has equalizers and all finite products, C has equalizers, binary products, and a terminal object, C has pullbacks and a terminal object. The dual statements are also equivalent. A small category C is complete if and only if it is cocomplete. [1]
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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.
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