Fibration of simplicial sets
Okay, let's try to explain what a fibration of simplicial sets is!
Imagine you have a bunch of toys that you want to put away in a toy box. But you can't just throw them all in at once - you need to put them in one at a time and make sure they fit and are in the right place.
Similarly, a simplicial set is a collection of shapes that are made up of points, line segments, triangles, and so on - just like toys in a toy box are made up of different shapes and sizes.
Now, imagine you have two different collections of toys that you want to put in two different toy boxes. You might have to rearrange the toys so that they fit in the box properly - this is like how we might need to rearrange the shapes in the simplicial set so that they form a nice pattern or structure.
But, what if we want to move the toys from one box to another? We might need to adjust the way we arrange the toys so that they fit properly in the new box. This is like how we might need to change the structure of the simplicial set so that it can be transformed and moved around.
A fibration of simplicial sets is a way to describe how we can transform the structure of one simplicial set into another. Think of it like a set of instructions for how to move the toys from one toy box to another.
In more technical terms, a fibration of simplicial sets is a mapping between two simplicial sets that preserves certain properties - specifically, it preserves the "lifting" property. This means that if we have a certain structure in one simplicial set, we can transfer that structure to the other simplicial set while preserving its properties.
So, a fibration of simplicial sets is like a way to move structures around in a controlled and precise way. Just like how we need to arrange toys in a certain way to fit them in a toy box, we need to arrange simplicial elements in a certain way to create a specific structure. The fibration of simplicial sets helps us understand how to do this accurately and effectively.
Imagine you have a bunch of toys that you want to put away in a toy box. But you can't just throw them all in at once - you need to put them in one at a time and make sure they fit and are in the right place.
Similarly, a simplicial set is a collection of shapes that are made up of points, line segments, triangles, and so on - just like toys in a toy box are made up of different shapes and sizes.
Now, imagine you have two different collections of toys that you want to put in two different toy boxes. You might have to rearrange the toys so that they fit in the box properly - this is like how we might need to rearrange the shapes in the simplicial set so that they form a nice pattern or structure.
But, what if we want to move the toys from one box to another? We might need to adjust the way we arrange the toys so that they fit properly in the new box. This is like how we might need to change the structure of the simplicial set so that it can be transformed and moved around.
A fibration of simplicial sets is a way to describe how we can transform the structure of one simplicial set into another. Think of it like a set of instructions for how to move the toys from one toy box to another.
In more technical terms, a fibration of simplicial sets is a mapping between two simplicial sets that preserves certain properties - specifically, it preserves the "lifting" property. This means that if we have a certain structure in one simplicial set, we can transfer that structure to the other simplicial set while preserving its properties.
So, a fibration of simplicial sets is like a way to move structures around in a controlled and precise way. Just like how we need to arrange toys in a certain way to fit them in a toy box, we need to arrange simplicial elements in a certain way to create a specific structure. The fibration of simplicial sets helps us understand how to do this accurately and effectively.