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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.
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.
In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C that are mutually inverse to each other, i.e. FG = 1 D (the identity functor on D) and GF = 1 C. [1] This means that both the objects and the morphisms of C and D stand in a one-to-one correspondence to each other. Two isomorphic ...
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 ...
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.
As a rule of thumb, an equivalence of categories preserves all "categorical" concepts and properties. If F : C → D is an equivalence, then the following statements are all true: the object c of C is an initial object (or terminal object, or zero object), if and only if Fc is an initial object (or terminal object, or zero object) of D
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.
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 ...