Holmstrom Cubical set

http://nlab.mathforge.org/nlab/show/cubical+set

From ALG-TOP Sep 2009, very interesting

(there might also be further follow-up emails)

I am enquiring about the history of cubical and simplicial sets, and would be grateful for any corrections/additions/references for the following account. As I do not have sure contact details with Dan Kan I am posting this generally, and would be very happy to have his or other account.

1. Dan Kan’s first work on abstract homotopy was cubical and published as @article {Kan55, AUTHOR = {Kan, D. M.}, TITLE = {Abstract homotopy. {I}}, JOURNAL = {Proc. Nat. Acad. Sci. U.S.A.}, VOLUME = {41}, YEAR = {1955}, PAGES = {1092–1096}, MRCLASS = {55.0X}, % MRNUMBER = {18,142d}, MRREVIEWER = {J. C. Moore}, }

However his second paper was simplicial, and I remember being told in the 1950s as a research student that reasons for abandoning the cubical approach were (i) the geometric realisation of I x I in cubical terms had non trivial homotopy type, and (ii) John Milnor had successfully written on the geometric realisation of simplicial sets, and in particular on the product.

Was the example in (i) ever published or remarked on in some form near that time; it is in quite recent work by Rick Jardine in HHA.

Ronnie Brown

Thanks for all the helpful comments.

I talked to Dan Kan on the phone last night - we first met in Oxford over 50 years ago!

He confirmed that the reasons for giving up the cubical theory in the 1950s were 1. Normalisation of the chains was essential. 2. Cubical groups were not fibrant. 3. The geometric realisation of I x I had the wrong homotopy type.

No getting round 1. A relevant reference is @article {antolini-weist, AUTHOR = {Antolini, Rosa and Wiest, Bert}, TITLE = {The singular cubical set of a topological space}, JOURNAL = {Math. Proc. Cambridge Philos. Soc.}, FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society}, VOLUME = {126}, YEAR = {1999}, NUMBER = {1}, PAGES = {149–154}, ISSN = {0305-0041}, CODEN = {MPCPCO}, MRCLASS = {55U10 (55U15)}, MRNUMBER = {MR1681660 (2000e:55014)}, MRREVIEWER = {Stanley Kochman}, }

A relevant reference for 2. is @article {tonks-cubgps, AUTHOR = {Tonks, A. P.}, TITLE = {Cubical groups which are {K}an}, JOURNAL = {J. Pure Appl. Algebra}, FJOURNAL = {Journal of Pure and Applied Algebra}, VOLUME = {81}, YEAR = {1992}, NUMBER = {1}, PAGES = {83–87}, ISSN = {0022-4049}, CODEN = {JPAAA2}, MRCLASS = {55U10 (18D35 18G30)}, MRNUMBER = {MR1173825 (93h:55013)}, MRREVIEWER = {Marek Golasi{'n}ski}, } which shows cubical groups with `connections' (see below) are Kan. Various extra structures on cubical sets are discussed in @article {Grandis-cubsite, AUTHOR = {Grandis, M. and Mauri, L.}, TITLE = {Cubical sets and their site}, JOURNAL = {Theory Applic. Categories}, FJOURNAL = {Theory and Applications of Categories}, VOLUME = {11}, YEAR = {2003}, PAGES = {185-201}, MRCLASS = {}, % MRNUMBER = {39 \#952}, MRREVIEWER = {}, } including the $connections' introduced by Higgins and me, in order to formulate the notion of $commutative shell' (or $cube'). (You get extra types of $degenerate' cubes using the monoid structure max on the unit interval.) A major reason for using cubes was the easy notion of multiple compositions, tricky in simplicial or globular theories. Another reason was easy homotopies. See our papers in JPAA.

1. now is OK according to recent work of Maltsiniotis on cubical sets with connection, which develops work of section 8.4 of @article {cisinski-ast, AUTHOR = {Cisinski, Denis-Charles}, TITLE = {Les pr'efaisceaux comme mod`eles des types d’homotopie}, JOURNAL = {Ast'erisque}, FJOURNAL = {Ast'erisque}, NUMBER = {308}, YEAR = {2006}, PAGES = {xxiv+390}, ISSN = {0303-1179}, ISBN = {978-2-85629-225-9}, MRCLASS = {55-02 (18F20 18G50 55P60 55U35)}, MRNUMBER = {MR2294028 (2007k:55002)}, MRREVIEWER = {Philippe Gaucher}, } H. Federer had some useful `Lecture Notes on Algebraic Topology' (Brown University, 1962) which it would be good to have available as a scan. Of course cubical homology was used by Serre, and in books by Hilton and Wylie, and by Massey.

See also I. Patchkoria on derived functors: arXiv: 0907.1905, which uses `pseudo-connections'.

Ronnie Brown

nLab page on Cubical set