Category Theory/(Co-)cones and (co-)limits

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Definition (cone):

Let be a category and let be a diagram in . A cone over is an object of , together with morphisms for each , so that for each (so that ) we have

.

Definition (limit):

Let be a category, let be a diagram in and let

Definition (diagonal):

Let be a category, and let be an object of . Further, suppose that the object and the morphisms and constitute a product of with itself in . Then the diagonal (also called "diagonal morphism") is the unique morphism

such that .

Definition (anti-diagonal):

Let be a category, and let be an object of . Further, suppose that the object and the morphisms and constitute a coproduct of with itself in . Then the anti-diagonal (also called "anti-diagonal morphism") is the unique morphism

such that .

Exercises[edit]

    1. Let be a category such that every two objects of have a product. Suppose further that is another category, and that is a functor. Let be an object of . Use the universal property of the product in order to show that there exists a functor that sends an object of to the object of .
    2. Prove that any morphism in gives rise to a natural transformation .
    3. Can we weaken the assumption that every two objects of have a product? (Hint: Consider the image of the class function on objects associated to the functor .)