Cones and cocones
There are a few auxiliary concepts we’ll need before the next major topic. Let’s start with two categories and , and the category of functors . If the following seems very complicated, consider to be any particular toy category you’d like, so this category of functors is a diagram category.
Now, for every object there is a constant functor that sends every object of to , and every morphism to the identity on . Actually, this assignment of the constant functor to an object of is a functor from to . Indeed, given a morphism we get a natural transformation from the one constant functor to the other, whose component at each object of is . We call this the “diagonal” functor . That is, is the constant functor with value .
Let be some particular functor in . A cone is an object in the comma category . Let’s unpack this definition. Since has as its source, and is a fixed object — the same thing as a functor from the category to , we know what objects of this category look like.
An object of the category consists of an object of and an arrow from to . But an arrow in is a natural transformation of functors. That is, for each object of we need an arrow in from to . But for all objects . So we just need an object and an arrow for each .
Of course, there’s also naturality conditions to be concerned with. If is an arrow in , and and are the required arrows from , then naturality requires that . So we need a collection of arrows from to the objects in the image of that are compatible with the arrows from . Such a collection defines an object in the comma category — a cone on .
Cocones are defined similarly. A cocone on is an object in the comma category . That is, it’s an object and a collection of arrows from the objects in the image of that are compatible with the arrows from .
The description may seem a little odd, but try writing it out for some very simple categories . For example, let be a set. Then try letting it be an ordinal, or another preorder. After you write down the definition of a cone and a cocone on some simple categories the general idea should seem to make more sense.