Monoidal category action
Okay, let's imagine that you have a bunch of toys, blocks, and Legos. You like to play with them separately, but sometimes you want to combine them and make new things. That's kind of like what happens in a monoidal category.
A monoidal category is a fancy way of talking about a collection of mathematical objects (like numbers, matrices, or vectors) that you can combine using a special operation (like addition or multiplication).
But what makes a monoidal category special is that it also has what's called an "action." This means that you can take some of the objects in the category and use them to do things to the other objects. It's like using your toys to build a tower or a castle.
For example, let's say you have some numbers and some vectors. You might be able to "act" on the vectors with the numbers by scaling them up or down. So if you have a vector (1,1) and a number 2, you can "act" on the vector by scaling it up to (2,2).
Or maybe you have some matrices that represent transformations in space, and you want to "act" on vectors to see what happens when you apply those transformations. This is kind of like building a Lego spaceship and pretending it's flying through space.
In general, the action in a monoidal category has to follow some rules, like being associative (meaning it doesn't matter which order you do things in) and having an identity (meaning there's something you can do that doesn't change anything). These rules make sure that everything "plays nicely" together, so you don't end up with weird results.
So, in summary, a monoidal category with an action is like a big toy box with lots of different things you can combine and play with. You can use the things in the box to do stuff to each other, like building towers or spaceships. And as long as you follow the rules, you can create all kinds of fun and interesting things!
A monoidal category is a fancy way of talking about a collection of mathematical objects (like numbers, matrices, or vectors) that you can combine using a special operation (like addition or multiplication).
But what makes a monoidal category special is that it also has what's called an "action." This means that you can take some of the objects in the category and use them to do things to the other objects. It's like using your toys to build a tower or a castle.
For example, let's say you have some numbers and some vectors. You might be able to "act" on the vectors with the numbers by scaling them up or down. So if you have a vector (1,1) and a number 2, you can "act" on the vector by scaling it up to (2,2).
Or maybe you have some matrices that represent transformations in space, and you want to "act" on vectors to see what happens when you apply those transformations. This is kind of like building a Lego spaceship and pretending it's flying through space.
In general, the action in a monoidal category has to follow some rules, like being associative (meaning it doesn't matter which order you do things in) and having an identity (meaning there's something you can do that doesn't change anything). These rules make sure that everything "plays nicely" together, so you don't end up with weird results.
So, in summary, a monoidal category with an action is like a big toy box with lots of different things you can combine and play with. You can use the things in the box to do stuff to each other, like building towers or spaceships. And as long as you follow the rules, you can create all kinds of fun and interesting things!