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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 thi
In object-oriented computer programming, a null object is an object with no referenced value or with defined neutral (null) behavior.The null object design pattern, which describes the uses of such objects and their behavior (or lack thereof), was first published as "Void Value" [1] and later in the Pattern Languages of Program Design book series as "Null Object".
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.
Given a diagram F: J → C (thought of as an object in C J), a natural transformation ψ : Δ(N) → F (which is just a morphism in the category C J) is the same thing as a cone from N to F. To see this, first note that Δ(N)(X) = N for all X implies that the components of ψ are morphisms ψ X : N → F(X), which all share the domain N.
The object pool design pattern creates a set of objects that may be reused. When a new object is needed, it is requested from the pool. If a previously prepared object is available, it is returned immediately, avoiding the instantiation cost. If no objects are present in the pool, a new item is created and returned.
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 initial object of Cat is the empty category 0, which is the category of no objects and no morphisms. [1] 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.
Define the diagonal functor Δ : C → C J as follows: Δ(N) : J → C is the constant functor to N for all N in C. If F is a diagram of type J in C, the following statements are equivalent: ψ is a cone from N to F; ψ is a natural transformation from Δ(N) to F (N, ψ) is an object in the comma category (Δ ↓ F) The dual statements are also ...