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Someone anonymous has raised the question of subdivision at cellular approximation theorem. I do not have a source here in which I can check this. Can anyone else check up?
Anonymous is right; I removed the question mark, and expanded a little on the precise statement of cellular approximation. I may get around to writing simplicial approximation a little later (to take care of the gray link).
Todd: Thanks. I thought anonymous was right but could not double check.
Hello Tim, and everyone,
Just a quick message: one has to be a little careful with the wording of the simplicial approximation theorem. The proof is for simplicial complexes. It is a rather subtle matter to deduce the theorem for simplicial sets from the theorem for simplicial complexes.
Barycentrically subdividing any semi-simplicial set twice gives a simplicial complex, and thus deducing the result for semi-simplicial sets from that for simplicial complexes is straightforward. However, no number of barycentric subdivisions of a simplicial set need give a simplicial complex. The standard example is the model for the 2-sphere in which there is a single 0-simplex, a single (degenerate) 1-simplex, and a single non-degenerate 2-simplex.
It is possible to subdivide a simplicial set to obtain a simplicial complex, but, for instance, one has to use a different notion of subdivision first, before barycentrically subdividing.
I have been working with a few others on a ’cubical Milnor theorem’ via a ’cubical approximation theorem’. Using a notion of subdivision which is not the ’obvious’ one, it is possible to obtain a ’cubical complex’ from an arbitrary cubical set in a nice way.
I am not aware of a proof of the simplicial approximation theorem for simplicial sets (or cubical sets) which does not reduce to the case of simplicial complexes. I have thought quite a bit about it, and I believe it to be possible to give a direct proof, but using simplicial or cubical complexes gives a straightforward way to construct the homotopy.
This topic for arbitrary simplicial sets has a fraught history, especially if one asks for ’naturality’ of the homeomorphism between the simplicial set and its subdivision after taking geometric realisations. There is an instance of a paper with an error, which led to a paper by a different author which was itself erroneous, noticed by Rudolf Fritsch, who wrote a third paper about it! If you are interested in this curiosity, I would recommend to begin with Some remarks on S. Weingram: On the triangulation of a semisimplicial complex by Fritsch, or The simplicial extension theorem by Brian Sanderson.
As an aside: in my opinion, there is no way that the simplicial approximation theorem is a ’lemma’! Not only was it highly significant in the history of algebraic topology, it is for me the heart of the Quillen equivalence between the model structure on simplicial or cubical sets, and the Serre model structure on topological spaces. It is fashionable not to emphasise this nowadays, and instead to use, say, minimal Kan complexes. The theory of the latter relies indispensably on the axiom of choice. It is the analogue for simplicial sets of replacing categories by their skeletons!
I have written something on simplicial approximation theorem. Richard Williamson and others: please have a look.
In cellular approximation theorem, I changed each instance of “simplicial set” to “simplicial complex”.
I have not worked through the details of what you have written, Todd, but the essential idea is clearly present: one way or another, one reduces to writing down a ’linear homotopy’.
For me, it is more convenient to view a simplicial complex as a simplicial (or semi-simplicial, etc) set by definitiion. For instance: a ’simplicial set such that every simplex is uniquely determined by its vertices’. I would then just use the ordinary geometric realisation of simplicial sets.
I have added a section “Applications” with a subsection “Finite-dimensional universal bundles” (here) with a comment on how the cellular approximation theorem serves to show that the restriction of $E G$ to $sk_{d+1} B G$ is universal for $G$-principal bundles over $\leq d$-dimensional spaces.
I have added pointer to what seems to be the original proof of cellular approximation in equivariant homotopy theory:
and to the textbook account in
and to the review
For the equivariant case I have added further pointer to:
Sören Illman, Prop. 2.4 of: Equivariant singular homology and cohomology for actions of compact lie groups (doi:10.1007/BFb0070055) In: H. T. Ku, L. N. Mann, J. L. Sicks, J. C. Su (eds.), Proceedings of the Second Conference on Compact Transformation Groups Lecture Notes in Mathematics, vol 298. Springer 1972 (doi:10.1007/BFb0070029)
Sören Illman, p. 90 (5 of 6) in: Equivariant algebraic topology, Annales de l’Institut Fourier, Tome 23 (1973) no. 2, pp. 87-91 (doi:10.5802/aif.458)
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