Another variation on the topic of homotopy

[Victor Porton]

I have the following (very preliminary) idea:

Replace “path” in the definition of homotopy with “monovalued funcoid with domain [0;1]”. It enables things like infinitely short paths. (Consider such things as a plane without a point and an infinitely short path around this point, yet to be formulated precisely.) This "generalized path" is a mapping from [0;1] to ultrafilters (with certain restriction these ultrafilters conform to).

This way we may get another HoTT possibly not equivalent to the “main” HoTT. Moreover, we may probably construct several non-equivalent theories (needs careful consideration).

I have not yet formulated this precisely, but call you as soon as the rough idea appeared, so that you become able for example learn my theory of funcoids and start to ponder about my idea. I am going to write again when this will be formulated exactly. But you are free to join my research and race with me who will first have enough time to formulate this in details.

This idea uses theory of funcoids (discovered by me). By the way, please consider to nominate me for Breakthrough Prize for discovery (and thorough research) of the concept of funcoid. I need money. (Not three millions, I would probably donate two of them to some charity.)

Disclaimer: I am in no way an expert in homotopy and HoTT. But this my idea is probably great.