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The empty set is the unique initial object in Set, the category of sets. Every one-element set is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category.
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.
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. The coproduct is given by the disjoint union of topological spaces.
The empty set can be turned into a topological space, called the empty space, in just one way: by defining the empty set to be open. This empty topological space is the unique initial object in the category of topological spaces with continuous maps. In fact, it is a strict initial object: only the empty set has a function to the empty set.
A cone to an object X is just a morphism with codomain X. A morphism f : Y → X is a limit of the diagram X if and only if f is an isomorphism. More generally, if J is any category with an initial object i, then any diagram of shape J has a limit, namely any object isomorphic to F(i). Such an isomorphism uniquely determines a universal cone to F.
The point is the empty list, and the function is cons, taking a number and a finite list, and returning a new finite list with the number at the head. In categories with binary coproducts, the definitions just given are equivalent to the usual definitions of a natural number object and a list object, respectively.
The empty set (considered as a preordered set) is the initial object of Ord, and the terminal objects are precisely the singleton preordered sets.
1. An object A is terminal (also called final) if there is exactly one morphism from each object to A; e.g., singletons in Set. It is the dual of an initial object. 2. An object A in an ∞-category C is terminal if (,) is contractible for every object B in C. thick subcategory