If two functions are homotopic must their mapping cylinders be also?

[dumbQuestion]

Hey I feel like I understand the concept of two spaces being homotopic and I can "visualize" the concept because I think of one space kind of continuously morphing into the other. But when it comes to thinking of homotopy between functions, I have a harder time. I was trying to think of a way to visualize things and the way I know to visualize a function is by looking at its mapping cylinder. So I was wondering, is it the case that if we can show the mapping cylinders of two functions are homotopic, that the functions themselves are homotopic?

Does anyone have any good resources on things I can read that will help me understand homotopy between functions a little more intuitively? Or to understand the "link" between spaces being homotopic and functions being homotopic

(I know the definitions, but I guess I just have a hard time visualizing functions being homotopic :( )

 

[pat_connell]

Yes, showing that two functions' mapping cylinders are homotopic does imply that the functions themselves are homotopic. That is because the mapping cylinder captures the behavior of the function in a way that is easy to visualize and reason about. As for resources, you might find it helpful to read up on the homotopy extension property. That property states that if two spaces, X and Y, are homotopic, then for any space Z, any continuous map f : X → Z can be extended to a continuous map F : Y → Z. This gives us a way to interpret homotopy between two functions as an extension problem. Another good resource is the book "Algebraic Topology" by Allen Hatcher. Hatcher gives a very detailed overview of homotopy theory, including homotopy between functions. He also includes many examples and exercises that may help you gain more insight into the subject. Finally, you can also learn more about homotopy by studying related topics such as homology, cohomology, and homotopy groups. These topics are all closely related and they can give you a better understanding of how homotopy works.