A¹ homotopy theory
Okay kiddo, let's talk about something called homotopy theory. This is a special way of looking at shapes and spaces and how they can change into each other.
Imagine you have a toy car and you move it from one spot to another. Now imagine you have a bunch of different toys and you can smoosh them, stretch them, or twist them so they look different, but they are still the same toy.
In homotopy theory, we look at shapes and spaces like toys. We try to see if we can smoothly move one shape into another without breaking it. This process of smoothly moving one shape into another is called a homotopy.
Now think about a picture of a circle. You can draw it really small or really big, but it's still a circle. In homotopy theory, we call all different sized circles "homotopy equivalent" because we can use a homotopy to change one into the other without breaking the circle.
We also use this idea to look at more complicated shapes and spaces, like a donut or a twisted up pretzel. We can use homotopies to see if they are the same or different from each other.
So, homotopy theory is like playing with toys and seeing how we can change them without breaking them. It helps us understand shapes and spaces and how they relate to each other. Cool, right?
Imagine you have a toy car and you move it from one spot to another. Now imagine you have a bunch of different toys and you can smoosh them, stretch them, or twist them so they look different, but they are still the same toy.
In homotopy theory, we look at shapes and spaces like toys. We try to see if we can smoothly move one shape into another without breaking it. This process of smoothly moving one shape into another is called a homotopy.
Now think about a picture of a circle. You can draw it really small or really big, but it's still a circle. In homotopy theory, we call all different sized circles "homotopy equivalent" because we can use a homotopy to change one into the other without breaking the circle.
We also use this idea to look at more complicated shapes and spaces, like a donut or a twisted up pretzel. We can use homotopies to see if they are the same or different from each other.
So, homotopy theory is like playing with toys and seeing how we can change them without breaking them. It helps us understand shapes and spaces and how they relate to each other. Cool, right?