Dear All, Loday's model is for pointed spaces, and Grothendieck was critical of this in a letter to me in 1983, of which I have quoted part in the Indag Paper on my preprint page. I did not worry about this in the 1980s since the immediate consequences were quite novel. For example, Ellis and Steiner solved the old problem of the critical group for (n+1)-ads, and the nonabelian tensor product of groups has been well developed by group theorists (see www.groupoids.org.uk/nonabtens.html). What has not been looked at is an input of crossed modules over groupoids, instead of over groups, and considering first the work of Ellis-Steiner in that light. (crossed n-cubes of groupoids?) We know from examples that strict 2-fold groupoids are more complicated than homotopy 2-types, see my preprint arXiv:0903.2627v2; and the van Kampen theorem with Loday has not so far been given a version with many base points, unlike the version in the book Nonabelian Algebraic Topology. The philosophy given in the Indag Paper has relatively recently been put in this form, and so no part of it was discussed with Grothendieck, except the idea that n-fold groupoids model homotopy n-types, which, as said above, is not quite correct, though he thought it "absolutely beautiful". At that time, 1985, he was starting to write "Recollte et Semaille", a task which seemed to lead him away from mathematics. The work with Loday shows in many explicit examples how low dimensional identifications in topology can give rise to high dimensional homotopy invariants, and there are explicit and precise calculations using the higher van Kampen theorems. Such calculation is not the only aim, but it does give a useful test. Best Ronnie ----Original message----
From : t.porter.maths@gmail.com Date : 15/07/2017 - 07:35 (GMTDT) To : camell.kachour@gmail.com Cc : categories@mta.ca Subject : categories: Re: Homotopy hypothesis for contractible operad definitions of weak n-categories
Dear All, Can I ask why Loday's cat^n groups are not mentioned? (They have been now.) I know they are not globular, but by spreading out the `weakness' of the higher groupoid structures the axioms end up being strict (and very simple as they are really just abstractions of classical commutator identities). Surely they deserve to be used as a reference point to compare some of the other candidates. Loday's models work for *all *n-types for finite n. (I do not know how to handle general homotopy types using any similar methodology.) Tim [For admin and other information see: http://www.mta.ca/~cat-dist/ ]