Compact closed category
Okay kiddo, let's talk about something called a "compact closed category". Are you ready?
Imagine you have a bunch of boxes, and each box has a special toy inside. Each toy can do something different, like play music or light up. Now, you want to be able to play with these toys and make them do things, but you also want to keep them safe in their boxes when you're not playing with them. That's kind of like a compact closed category.
In a compact closed category, you have a bunch of things called "objects". These are like the boxes with toys inside. Each object has its own special properties, just like each toy has its own special abilities.
You also have things called "morphisms". These are like the instructions for playing with the toys. Each morphism tells you how to take one object and turn it into another one. It's kind of like taking the toy out of one box and putting it into another one so you can play with it in a different way.
But remember how we said that you want to keep the toys safe in their boxes when you're not playing with them? Well, in a compact closed category, you also have something called a "dual" for each object. The dual is like a mirror image of the object that you can use to put the toy back in the box and keep it safe. It's like making sure that you always have a way to put the toy away when you're done playing with it.
So, a compact closed category is kind of like a system for playing with special toys inside their boxes. You can take them out and put them into different boxes, but you always have a way to put them back and keep them safe. It's a way to have fun with the toys while also making sure they stay protected.
Imagine you have a bunch of boxes, and each box has a special toy inside. Each toy can do something different, like play music or light up. Now, you want to be able to play with these toys and make them do things, but you also want to keep them safe in their boxes when you're not playing with them. That's kind of like a compact closed category.
In a compact closed category, you have a bunch of things called "objects". These are like the boxes with toys inside. Each object has its own special properties, just like each toy has its own special abilities.
You also have things called "morphisms". These are like the instructions for playing with the toys. Each morphism tells you how to take one object and turn it into another one. It's kind of like taking the toy out of one box and putting it into another one so you can play with it in a different way.
But remember how we said that you want to keep the toys safe in their boxes when you're not playing with them? Well, in a compact closed category, you also have something called a "dual" for each object. The dual is like a mirror image of the object that you can use to put the toy back in the box and keep it safe. It's like making sure that you always have a way to put the toy away when you're done playing with it.
So, a compact closed category is kind of like a system for playing with special toys inside their boxes. You can take them out and put them into different boxes, but you always have a way to put them back and keep them safe. It's a way to have fun with the toys while also making sure they stay protected.