Fukaya category

Imagine you have a very special toy box filled with lots of different toys that are all related in some way. Each toy has its own unique features, but they are all connected to each other in some way that makes them special.

Now imagine this toy box is called the Fukaya category. Inside this box, there are different objects called Lagrangian submanifolds. These objects are like little toy houses inside the box.

Each Lagrangian submanifold is defined by some complicated mathematical equations, but for our ELI5 purposes, we can just think of them as different toy houses with different shapes, sizes, and colors.

But the coolest thing about these toy houses is that they are all connected to each other in some way. They are linked together by things called morphisms, which are like little nerf darts that connect one toy house to another.

These morphisms are important because they help us understand how the different Lagrangian submanifolds in the Fukaya category are related to each other. They let us see how each toy house fits into the larger picture and how they work together to create a bigger structure.

Now, this may all sound confusing and complicated, but it's actually a very important and useful tool for mathematicians and physicists who study things like symplectic geometry and string theory.

By looking at the different Lagrangian submanifolds and their morphisms in the Fukaya category, they can gain insights into the nature of space and time itself. They can explore the ways in which these objects interact and how they can be used to build complex mathematical models of our universe.

So, in short, the Fukaya category is a special toy box filled with Lagrangian submanifolds connected by morphisms. It's like a giant puzzle that helps us understand the structure of space and time in a deeper and more meaningful way.