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Initial and terminal objects may also be characterized in terms of universal properties and adjoint functors. Let 1 be the discrete category with a single object (denoted by •), and let U : C → 1 be the unique (constant) functor to 1. Then An initial object I in C is a universal morphism from • to U.
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.
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.
If F and G are (covariant) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism η X : F(X) → G(X) in D such that for every morphism f : X → Y in C, we have η Y ∘ F(f) = G(f) ∘ η X; this means that the following diagram is commutative:
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's paradox one cannot form the “category of all categories”. But it is possible to form a quasicategory ...
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).
In category theory, the notion of final functor (resp. initial functor) is a generalization of the notion of final object (resp. initial object) in a category. A functor F : C → D {\displaystyle F:C\to D} is called final if, for any set-valued functor G : D → Set {\displaystyle G:D\to {\textbf {Set}}} , the colimit of G is the same as the ...
Suppose : and : are two functors such that for all objects a and a ′ of A and all objects b of B, the copowers (′,) exist in C. Then the functor X has a left Kan extension Lan F X {\displaystyle \operatorname {Lan} _{F}X} along F , which is such that, for every object b of B ,