n-types and n-fold structures

n-types and n-fold structures

[Paoli, Simona (Dr.)]

Dear All,

   With regard to the recent postings about modelling general n-types and n-fold groupoids, I would like to point out the following. A proof of the modelling of general n-types (not only path-connected ones) using n-fold structures is given in the following research monograph, which I recently posted  on arVix:

S.Paoli, Segal-type models of higher categories, arXiv.1707.01868

The structure used there, called groupoidal weakly globular n-fold categories, is a subcategory of n-fold categories in which certain sub-structures are groupoidal, and satisfying several other conditions. This result is a proof of the homotopy hypothesis for a new model of weak n-categories, called  weakly globular n-fold categories, which is proved to be suitably equivalent to the Tamsamani-Simpson model.

In the above work there is also a proof that it is possible to use as fundamental functor from spaces to groupoidal weakly globular n-fold categories the construction found in the following paper

D. Blanc, S. Paoli, Segal-type algebraic models of n-types, Algebraic and Geometric Topology, 14  3419–3491.

This construction produces a functor from spaces to a subcategory of n-fold  groupoids.

Best wishes,

Simona.


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Re: n-types and n-fold structures

[RONALD BROWN]

Dear All. 

The philosophy I have followed in essence since 1974 is now articulated in my Indag Math. paper recently advertised on this list "Modelling and Computing Homotopy Types: I" (see my preprint page) and involves strict structures and homotopically defined functors on "Topological Data", satisfying a Seifert-van Kampen type theorem, which enables explicit computations relating  to classical methods in homotopy theory, e.g. relative and n-adic homotopy  groups. 

I spent the years 1965 -1974 trying to construct a homotopy double groupoid  of a space, and just getting confused, and even unsure what to do with it if I got it, so it was a great relief to get, as explained in that paper, to get with Philip Higgins a nice definition for a pointed pair of spaces, which did all we wanted! 

The extension to filtered spaces took another 3 or 4 years with Philip Higgins, and then with Loday for n-cubes of pointed spaces another 3-4 years. All this related well to classical homotopy theory (and algebraic K-theory!)  methods. 

Note that in Section 5 of Esquisses d'un Programme, Grothendieck argues that extra structure is needed on topological spaces (if you like, moving from  "bare" topological spaces) to make them suitable for geometric work. Perhaps this reflects the difficulty I had 1965-74 in expressing my homotopy intuitions in mathematical form. though the algebraic work in dim 2  went ahead with Chris Spencer, in the early 1970s. 

The work of Ellis and Steiner on crossed n-cubes of groups gives an amazingly precise, explicit,  strict model for n-types of pointed spaces, enabling  for example, with the van Kampen theorem for diagrams of spaces, the solution of an old problem (John Moore, Barratt-Whitehead) of the critical group  for n-ads. There seem some possibilities for crossed n-cubes of groupoids!  Further work is need on applications to excision. which should have been pursued by me (with students, for example)  years ago! 

One aspect of the work with Loday. a nonabelian tensor product, has been well taken up by group theorists. see my on-line bibliography. The application of that construction to the low dimensional Blakers-Massey theorem changed radically the views on this area of at least one well known algebraic topologist  (he had previously described the generalised van Kampen programme as "ridiculous"!).  

Best wishes

Ronnie
www.groupoids.org.uk





----Original message----
From : sp424@leicester.ac.uk
Date : 19/07/2017 - 15:55 (GMTDT)
To : categories@mta.ca
Subject : categories: n-types and n-fold structures

Dear All,

    With regard to the recent postings about modelling general n-types and n-fold groupoids, I would like to point out the following. A proof of the modelling of general n-types (not only path-connected ones) using n-fold structures is given in the following research monograph, which I recently posted  on arVix:

S.Paoli, Segal-type models of higher categories, arXiv.1707.01868

The structure used there, called groupoidal weakly globular n-fold categories, is a subcategory of n-fold categories in which certain sub-structures are groupoidal, and satisfying several other conditions. This result is a proof of the homotopy hypothesis for a new model of weak n-categories, called   weakly globular n-fold categories, which is proved to be suitably equivalent to the Tamsamani-Simpson model.

In the above work there is also a proof that it is possible to use as fundamental functor from spaces to groupoidal weakly globular n-fold categories the construction found in the following paper

D. Blanc, S. Paoli, Segal-type algebraic models of n-types, Algebraic and Geometric Topology, 14  3419–3491.

This construction produces a functor from spaces to a subcategory of n-fold   groupoids.

Best wishes,

Simona.



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