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    • CommentRowNumber1.
    • CommentAuthorronniegpd
    • CommentTimeSep 13th 2012
    I have just been glancing at the wikipedia entries on higher categories and they seem to suggest that only weak higher categories are of interest in homotopy theory. I have made a slight modification to one entry, but I would like to point out that strict homotopy _double groupoids with connection_ of a pair of spaces occurred in

    R. Brown and P.J. Higgins, ``On the connection between the second
    relative homotopy groups of some related spaces'', _Proc. London Math. Soc._ (3) 36 (1978) 193-212.

    and the construction was generalised to filtered spaces in two paper with Higgins in JPAA 1981. The point of these constructions is not _about_ homotopy theory, but providing new algebraic structures to be used as algebraic tools for understanding and computation of homotopical invariants. In particular, our book "Nonabelian algebraic topology" brings together in Part I lots of nonabelian constructions and calculations in 2-dimensional homotopy theory, while the work with Loday extends these nonabelian constructions and calculations to higher dimensions, using strict $n$-fold groupoids. One particular construction which came out of the latter work, the nonabelian tensor product of groups, has a current bibliography of 120 items, largely because of the interest in the construction by group theorists.

    One reason for these explicit results is the idea of computation in homotopy theory using strict colimits, enabled by Higher Homotopy Seifert-van Kampen Theorems. These work by dealing with structured spaces, i.e. filtered spaces or $n$-cubes of spaces. The philosophical implications need discussion, possibly in this forum.

    Note Grothendieck's Section 5 in his "Esquisses d'un Programme" on the limitations for geometry of the notion of topological space. He advocates sophisticated notions of stratification.

    For me, the fact that these strict structures can be defined for say filtered spaces is itself of significance, since the detailed proofs are not straightforward, and there is "just enough room" to make the proofs work.