This will be a more open question. I would like to have a better understanding about how the homotopy group functors behave with different constructions such as limits or colimitis. Some examples: ...
As many of you know, category theorists are currently doing, among other things, a great job in advertising their modern developments. And I must say, this works for me - in particular, I find myself ...
I am led to understand that spectra are some kind of $\infty$-analogue of (discrete) abelian groups, or perhaps more accurately, some kind of generalisation of chain complexes of abelian groups. How ...
The definition of a path as a continuous map $I \rightarrow X$ is a completely natural one. But this raises two questions in my mind. First, what properties of the interval give rise to useful ...
What are the most useful fibrations that one be familiar with in order to use spectral sequences effectively in algebraic topology? There's at least the four different Hopf fibrations and $S^1\to ...
Edit: After a discussion with t.b. we agreed that this question aims to a different answer from this one, for more information you can read the comment below. Many times I've heard people ...
I covered homotopy theory in a recent maths course. However I was never presented with any reasons as to why (or even if) it is useful. Is there any good examples of its use outside academia?