Ronnie, The point about pointed spaces is perhaps not really that difficult to get around at least partially. Using either the Bullejos-Cegarra-Duskin method or my approach the formulae for simplicial groups extends more or less to simplicially enriched groupoids and thus to all homotopy types. This works equally well for your point in the second paragraph, but the obscure part is the following: I will look back at the examples in arXiv:0903.2627v2, but I remember discussing this with you and one of the differences is that in a simplicially enriched groupoid, G, one has a simplicial groupoid with constant object of objects. What this means for translating the simplicial composition in G into structure in the corresponding cat^n objects is where interesting things start happening so writing L for Loday's construction one wants to use a composition G(x,y)xG(y,z)-> G(x,z) for give something at the level of LG(x,y)?LG(y,z)->LG(x,z), where ? is some construction not yet defined. This should be possible to analyse as the multiplication in a simplicial group leads to structures in the corresponding cat^n group and back again, so probably one could look at that as G(x,x)\times G(x,x)\to G(x,x) to see what is happening??? This is getting a bit off-topic for the original messages so I will stop, but would welcome any ideas either on a separate thread or via MathOverflow perhaps. Tim On 15 July 2017 at 21:59, RONALD BROWN <ronnie.profbrown@btinternet.com> wrote:

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

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