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In directed homotopy theory, one replaces spheres by directed spheres and homotopy groups by homotopy monoids.

  • Is it known what are the first few homotopy monoids of directed spheres?
  • Do homotopy monoids also stabilise?
  • If so, what are the first few stable homotopy monoids of directed spheres?
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    $\begingroup$ Do you have a particular formalism for directed homotopy theory in mind? I don't know about much about it, but my impression is that there are several different frameworks out there known as "directed homotopy theory" which are not necessarily equivalent to one another. Within such a framework, what is your definition of a "directed sphere"? And what is the "homotopy monoid" of such a thing? $\endgroup$
    – Tim Campion
    Commented Jul 29, 2021 at 17:26
  • $\begingroup$ @TimCampion I'm also not very familiar with it, so I'm not really sure... FWIW, I think one possible formalism would be the one formulated by Grandis. Their book on the subject (libgen link) defines the directed $n$-sphere in page 54 and fundamental monoids in Section 3.2.5, while... $\endgroup$
    – Emily
    Commented Jul 30, 2021 at 3:51
  • $\begingroup$ ..."higher homotopy monoids" seem to be defined in this other paper. $\endgroup$
    – Emily
    Commented Jul 30, 2021 at 3:51
  • $\begingroup$ Incidentally, I just found that this paper by Grandis computes in Section 3.7, a) the "first homotopy monoid of the directed circle" to be (indeed) $\mathbb{N}$. $\endgroup$
    – Emily
    Commented Jul 30, 2021 at 3:51
  • $\begingroup$ There is this abstract of a seminar talk - math.upenn.edu/events/directed-homotopy-directed-spheres - which however, as far as my Google skills go, never led to any paper. Perhaps it would be nevertheless useful to contact the speaker. $\endgroup$
    – Vladimir Dotsenko
    Commented Jul 30, 2021 at 5:56

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