Chain homotopy
Chain homotopy is like a game of connect-the-dots but with shapes instead of dots. Let's say you have two shapes, which are made up of lines and corners. These shapes might be similar in some ways, but they might be different in others.
Chain homotopy helps you figure out if your shapes are "the same" - meaning they can be turned into each other without cutting or tearing any of the lines. Here's how it works:
Imagine you have two shapes, and you want to see if one can be turned into the other without any cutting or tearing. You start by putting some dots (or "nodes") on the lines of one of the shapes - let's call this Shape A. Then you draw lines connecting these nodes to each other, making a kind of scaffolding or framework.
Next, you take your other shape - Shape B - and put dots on its lines in the same way. But this time, instead of drawing lines between the dots, you "drag" the dots along the lines of Shape B. You can move the dots around as long as you don't cross any lines or leave the shape entirely.
Now you have two different scaffolds - one for Shape A and one for Shape B - but they might not look the same. The question is: can you "morph" the scaffold for Shape A into the one for Shape B?
Here's where chain homotopy comes in. If you can find a way to "slide" the dots and lines of the scaffold for Shape A around so that it looks exactly like the scaffold for Shape B, then the two shapes are said to be "chain homotopic". This means they are "the same" in a certain way - they can be smoothly transformed into each other without cutting or tearing any of the lines.
It's kind of like playing with play-doh. You can take a ball of play-doh and squish it and mold it into different shapes, but you haven't destroyed anything - it's still all the same play-doh. Chain homotopy is like seeing if two different shapes can be squished and molded into each other in the same way.
Chain homotopy helps you figure out if your shapes are "the same" - meaning they can be turned into each other without cutting or tearing any of the lines. Here's how it works:
Imagine you have two shapes, and you want to see if one can be turned into the other without any cutting or tearing. You start by putting some dots (or "nodes") on the lines of one of the shapes - let's call this Shape A. Then you draw lines connecting these nodes to each other, making a kind of scaffolding or framework.
Next, you take your other shape - Shape B - and put dots on its lines in the same way. But this time, instead of drawing lines between the dots, you "drag" the dots along the lines of Shape B. You can move the dots around as long as you don't cross any lines or leave the shape entirely.
Now you have two different scaffolds - one for Shape A and one for Shape B - but they might not look the same. The question is: can you "morph" the scaffold for Shape A into the one for Shape B?
Here's where chain homotopy comes in. If you can find a way to "slide" the dots and lines of the scaffold for Shape A around so that it looks exactly like the scaffold for Shape B, then the two shapes are said to be "chain homotopic". This means they are "the same" in a certain way - they can be smoothly transformed into each other without cutting or tearing any of the lines.
It's kind of like playing with play-doh. You can take a ball of play-doh and squish it and mold it into different shapes, but you haven't destroyed anything - it's still all the same play-doh. Chain homotopy is like seeing if two different shapes can be squished and molded into each other in the same way.