Grothendieck's strange fact

[Venkata Rayudu Posina <posinavrayudu@gmail.com>]

Dear All,

I hope and pray you and your families are all safe and well.

In a letter to Professor Ronnie Brown, Grothendieck notes "a strange
fact" (p. 26, https://webusers.imj-prg.fr/~georges.maltsiniotis/ps/agrb_web.pdf):

"One interesting question here which I did not clear up yet, is,
whether weak equivalence for a map of hemispherical complexes can be
explicitly tested in terms of the source and target maps, just the
same way as if we had actual ∞-groupoids or ∞-Gr-stacks (never mind
whether associativities are strict or not), when the homotopy groups
can be computed directly in terms of these extra structures. When you
write down the condition that you get isomorphisms for these, it turns
out though that the condition makes sense in terms of the
source-and-target structure alone, without having to use the
composition laws at all (nor even degeneracies). This is a strange
fact, which should be understood."
--Grothendieck, in a letter to Professor Ronnie Brown (p. 26,
https://webusers.imj-prg.fr/~georges.maltsiniotis/ps/agrb_web.pdf)

I was wondering if the strange fact:

"the source-and-target structure alone"

has been understood.  For instance, does "the source-and-target
structure alone" correspond to the theory of irreflexive directed
multigraphs (see Lawvere and Schanuel's Conceptual Mathematics, p.
150)?

Happy Tuesday :-)

thanking you,
posina


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]