3
votes
2answers
108 views

Properties of $\pi_n$ from a category theoretical point of view

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: ...
6
votes
2answers
201 views

What is a (the?) good starting point for learning the modern “higher” mathematics?

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 ...
7
votes
0answers
123 views

How strong is the analogy between spectra and abelian groups?

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 ...
8
votes
0answers
261 views

The status of $\mathbb{R}$ in homotopy theory.

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 ...
13
votes
4answers
474 views

Useful fibrations

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 ...
8
votes
3answers
425 views

What is combinatorial homotopy theory?

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 ...
12
votes
7answers
1k views

Real world uses of homotopy theory

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?