Path space (algebraic topology)
Okay kiddo, let's talk about something called "path space" in algebraic topology.
Imagine that you have a piece of paper and you draw a shape on it (like a circle or a square). Now, imagine that you want to find all the possible paths you could take inside that shape.
The path space is exactly what it sounds like - it's the space where all those paths live! It's like a big collection of all the different ways you could move around inside that shape.
But wait, there's more! In algebraic topology, we don't just care about the paths themselves - we also care about how we can connect them to each other.
For example, imagine that you drew a circle on your piece of paper. You could take one path that went around the circle once, or you could take a different path that went around the circle twice.
But these two paths are actually connected to each other in a certain way - you could take the first path and "smush" it down to look like the second path.
In algebraic topology, we describe this connection by saying that the two paths are "homotopic" to each other. That just means that we can smoothly transform one path into the other without lifting our pen off the paper.
So in the path space, we not only have all the different paths, but we also have information about how they can be connected to each other. This helps us understand the shape we started with in a deeper way.
Does that make sense, kiddo?
Imagine that you have a piece of paper and you draw a shape on it (like a circle or a square). Now, imagine that you want to find all the possible paths you could take inside that shape.
The path space is exactly what it sounds like - it's the space where all those paths live! It's like a big collection of all the different ways you could move around inside that shape.
But wait, there's more! In algebraic topology, we don't just care about the paths themselves - we also care about how we can connect them to each other.
For example, imagine that you drew a circle on your piece of paper. You could take one path that went around the circle once, or you could take a different path that went around the circle twice.
But these two paths are actually connected to each other in a certain way - you could take the first path and "smush" it down to look like the second path.
In algebraic topology, we describe this connection by saying that the two paths are "homotopic" to each other. That just means that we can smoothly transform one path into the other without lifting our pen off the paper.
So in the path space, we not only have all the different paths, but we also have information about how they can be connected to each other. This helps us understand the shape we started with in a deeper way.
Does that make sense, kiddo?