9
votes
2answers
861 views

Status of Beilinson conjectures?

(I hesitate to post that question here, but I received on answer on FB:) Does anyone know how the current status of work on them is? And how the possible generalizations etc. which one thinks ...
20
votes
4answers
1k views

When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics?

Every once in a blue moon it actually matters that some mathematical entity which might a priori only be a class is in fact a set. For clarification, here are some examples of what I do not ...
6
votes
1answer
581 views

Analogue to Serre spectral sequence for cofiber sequences and homotopy

(This is a follow-up question to this one). As it is nicely outlined in an answer to this question, homotopy groups behave well with respect to (Serre)-fibrations and (co)homology groups behave well ...
8
votes
2answers
1k views

Why does homotopy behave well with respect to fibrations and homology with respect to cofibrations?

(I apologize that this is a vague question). I seems to me somehow that homotopy groups behave well with respect to (Serre)-fibrations. For example you get a long exact sequence of homotopy groups ...
24
votes
1answer
1k views

Are there any “homotopical spaces” ?

This is a somewhat vague question; I don't know how "soft" it is, and even if it makes sense. [Edit: in the light of the comments, we can state my question in a formally precise way, that is: "Is ...
5
votes
2answers
523 views

visualizing what's going on in based homotopy theory, et al.

I'm reading J.P. May's Concise Course in Algebraic Topology, and I'm having a lot of trouble visualizing how things work in Chapter 8, "Based cofiber and fiber sequences". Of course this is pretty ...