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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.
The partially ordered class of all ordinal numbers is cocomplete but not complete (since it has no terminal object). A group, considered as a category with a single object, is complete if and only if it is trivial. A nontrivial group has pullbacks and pushouts, but not products, coproducts, equalizers, coequalizers, terminal objects, or initial ...
Let T, η, μ be a monad over a category C.The Kleisli category of C is the category C T whose objects and morphisms are given by = (), (,) = (,).That is, every morphism f: X → T Y in C (with codomain TY) can also be regarded as a morphism in C T (but with codomain Y).
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 ...