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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.
That is, a category C is complete if every diagram F : J → C (where J is small) has a limit in C. Dually, a cocomplete category is one in which all small colimits exist. A bicomplete category is a category which is both complete and cocomplete. The existence of all limits (even when J is a proper class) is too strong to be practically relevant.
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.
Thus the zero ring is a terminal object in the category of rings. [8] If A is a nonzero ring, then there is no ring homomorphism from the zero ring to A. In particular, the zero ring is not a subring of any nonzero ring. [8] The zero ring is the unique ring of characteristic 1. The only module for the zero ring is the zero module.
However, LH does not have a terminal object, and thus is not Cartesian closed. If C has pullbacks and for every arrow p : X → Y, the functor p * : C/Y → C/X given by taking pullbacks has a right adjoint, then C is locally Cartesian closed. If C is locally Cartesian closed, then all of its slice categories C/X are also locally Cartesian closed.
Essentially, we create a category whose objects are cones, and where the limiting cone is a terminal object; then, each universal morphism for the limit is just the morphism to the terminal object. This works in the dual case, with a category of cocones having an initial object.
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. In order to avoid problems analogous to Russell ...