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":
"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:
"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)?