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For example, the initial object in any concrete category with free objects will be the free object generated by the empty set (since the free functor, being left adjoint to the forgetful functor to Set, preserves colimits). Initial and terminal objects may also be characterized in terms of universal properties and adjoint functors.
Universal constructions are functorial in nature: if one can carry out the construction for every object in a category C then one obtains a functor on C. Furthermore, this functor is a right or left adjoint to the functor U used in the definition of the universal property. [2] Universal properties occur everywhere in mathematics.
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 ...
In mathematics, the category Ord has preordered sets as objects and order-preserving functions as morphisms.This is a category because the composition of two order-preserving functions is order preserving and the identity map is order preserving.
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.
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).
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 ...
A monoid object in [C, C] is a monad on C. For any category with a terminal object and finite products, every object becomes a comonoid object via the diagonal morphism Δ X : X → X × X. Dually in a category with an initial object and finite coproducts every object becomes a monoid object via id X ⊔ id X : X ⊔ X → X.