What is a Group Action? : A Group as a Category and The Skeleton Operation ☠

This week I try to take a more Categorical approach to answering and expanding upon the question of "what is a group action". Along the way I'll go over thinking about a group as a category and eventually hit on the skeleton operation on a category and use it to present an example of the categorification of the Orbit-Stabilizer theorem. Here are some videos that are "pre-reqs", that is, they introduce some topics in a bit more detail than I do here.

CORRECTIONS:
1. When introducing Groups, while glossing over the group axioms I forget to mention that a group, G, must be closed under the binary operation. That is for any a,b in G, ab must also be in G.

This video can be broken up into the following sections.

00:00 Intro
I once again fail to say "Hi I'm Nathan and Welcome to my YouTube Channel" but I still introduce the goal of the video lol.

00:32 What is a Group Action?
I briefly revisit the idea of what a group is before "complicating" it into the object that we'll spend most of the time interacting with, a group action. There are many different types of group actions but we will focus on 2 examples.

03:46 Example 1
The first of 2 group action examples. We look at probably the most accessible example of a group action, a group acting on itself with the action operating the same way as the group operation.

05:05 Example 2
In the second of 2 group action examples we look at another small but more complicated example of a group acting on itself, here we us the conjugate action and we'll continue to look at this example throughout the video.

After the second example, the video begins to bring in a lot more ideas from category theory and we'll spend a lot of time looking at categories and groups and how the two can correspond to one another.

09:30 A Group as a Category
Here I will begin translating the idea of what we have talked about with groups and group actions into a more category theoretic context. We won't dive too deep into the Category of G-Sets, but we do define what G-Sets are and how they can be described as functors.

11:21 The Translation Groupoid of A Group
Next we 'zoom in' on what our G-Set functor does to the categorical group and arrive at the directed graphs that we looked at previously by looking at the Translation Groupoid.

13:27 The Skeleton Operation
The last Category theoretic tool we will need for the categorification of the orbit-stabilizer theorem, the skeleton operator on a category fuses together identical (or isomorphic) objects, which allows us to see the information in the Translation Groupoid category more clearly.

15:50 The Orbit-Stabilizer Theorem
Here we introduce (not prove...) the theorem and then talk through how the skeleton operation on the translation groupoid generates a very nice picture of the theorem in a Category Theoretic context.

17:35 Another way of Thinking about the Skeleton Category
And lastly I pull away from the abstract nonsense of the skeleton construction to quickly walk through another way to interpret each of the orbits in the skeleton category that can help when trying to determine the size of other symmetry/dihedral groups.

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WHAT GEAR I USED FOR THIS VIDEO:

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#CHALK #CategoryTheory #Skeleton

[Aleph]

AH! When you said "categorification of the group stabilizer theorem", I felt such a sudden thrill of excitement. I love that you don't shy away from the hard abstract mathy bits. This is a great way for people like me to get a primer on more abstract math (group theory, category theory, etc.) Awesome video as usual!

[merbst]

Excellent topic selection!
Categorical Algebra is sweet.

[curiosway]

Introducing the idea was awsome as you explained it without getting too much in the math of approaching it
❤

[Emmanuel_]

Hey man I'm a 2nd year math major from South Africa. I really find your videos enjoyable and I hope you continue with this golden content.

[alexandersanchez]

I just read about this in Aluffi's book this morning and now I get recommended this video. On the one hand, I'm super pleased to be exposed to an awesome math channel that's new to me. On the other, though, the coincidence is a little spooky... I guess I must have tipped off Google with some of my searches earlier today?

[kylecompiles]

Awesome video, incredibly high quality and easy to follow. Definitely piques my interest in category theory. I recently found your channel and love the content, I’m hoping to pursue my own PhD in math once I round out some of my background so both the math-focused vids and school-focused vids are great :)

[aidansgarlato]

Bro I love math you are a god send please keep it up

[austintalksmath]

Did you forget to write the closure property of groups in the beginning of the video? I really love your videos and I hope you are able to continue making these awesome videos as you go on into your Phd program!

[jeffreyhowarth]

At 7:15 you where saying that you could get these 3 disjoint connected components, referring to the equilateral triangle. Could I say that I could get these 4 disjoint connected components referring to the square or is there a difference in these 2 geometric groups?

[joetursi]

Two thoughts: It would have ben alot easier to understand if you imply said " a group action is a homorphism from a grroup G to a set S. and you should write larger script, especially with ordred pairs, cross products etc.

[jeffreyhowarth]

At 7:15 you where saying that you could get these 3 disjoint connected components, referring to the tequila fetal triangle