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cone, its category, and limit

1. What's a cone?

In category theory, cone is a kind of natural transformation, to describe a diagram formally. The setup of cone is starting from a diagram, which we will define it as a category ๐’Ÿ (usually is a finite one but no need to be) with two functors to the target category ๐’ž The two functors are

  1. \(\Delta_c\) sends every objects of ๐’Ÿ to a certain object \(c\) of ๐’ž and
  2. \(D\) sends the structure of ๐’Ÿ to ๐’ž, that is this functor existed iff we have same thing in ๐’ž we don't care where is it.

Now we can have picture of it, the ๐’ž(๐’Ÿ) is the ๐’Ÿ structure in ๐’ž

cone.png

Since \(c\) is just a point, we called it apex and \(D\) is a plane for us in this sense, we capture this and called the natural transformation from \(\Delta_c\) to \(D\) a cone. The brilliant part is that, with a fixed functor \(D\) to fixed the plane, different \(c\) brings different cones! What in my mind is like:

bird.gif

Image the head is apex and the body is the fixed ๐’ž(๐’Ÿ), you will slowly get the idea.

1.1. The category of cones

To describe previous idea, and in category theory, we make another category! By picking all apex of cone as objects, and picking morphism between apex in ๐’ž as morphisms, this is the category of cones (please check it does really a category). The funny part we care here, is the terminal object of the category of cones, that is the next section.

2. Limit (universal cone)

A limit or universal cone is a terminal object of category of cones. If you consider it carefully, and you will find this is the best cone of cones, because every other cones can be represented by composition of an addition morphism from itself to the limit and the limit! And since \(D\) is fixed, we also write limit as

\[ Lim D \]

The last thing about limit, is it might not exist, so you have to ensure there has one.

3. Conclusion

With concept of cone and limit, you might already find there has some concept can be replaced by cone and limit, you're right and it would be fun to rewrite them in this new sense. Have a nice day.

Date: 2023-03-20 Mon 00:00
Author: Lรฎm Tsรบ-thuร n