Showing changes from revision #17 to #18:
Removed | Chan
related by the Dold-Kan correspondence
fibrations are Hurewicz fibrations,
cofibrations are closed Hurewicz cofibrations
and the role of weak equivalences is played by (strong) homotopy equivalences
The theorem might have been a
folklore at the time, but the actual paper has a number of subtleties.
Strøm’s proofs are not that well-known today and use techniques better known to the topologists of that time, and there is consequently a slight controversy among topologists now. One of these is that there are modern reproofs, but these modern techniques essentially use compactly generated spaces, while Strøm’s proofs succeeded in avoiding that assumption.
However, for many applications nowadays, one is mainly interested in the analogous model structure on the category of k-spaces, or of compactly generated spaces (weak Hausdorff k-spaces). Note that any cofibration in the latter category is closed.
This is a most rare property for a non-trivial model structure.
The Strøm model structure on the category of compactly generated spaces is a monoidal model category. This is proven in section 6.4 of A Concise Course in Algebraic Topology (without that language) using the fact that a subspace inclusion is a Hurewicz cofibration if and only if it is an NDR-pair.
The identity functor is left Quillen from the Quillen model structure (or the mixed model structure) to the Strøm model structure, and of course right Quillen in the other direction. This is just the observation that any Hurewicz fibration is a Serre fibration, and any homotopy equivalence is a weak homotopy equivalence—or dually, that any relative cell complex is a Hurewicz cofibration.
It follows, by composition, that the (geometric realization
\dashvsingular simplicial complex)-adjunction is Quillen between the standard model structure on simplicial sets and the Strøm model structure.
If denotes the category of compactly generated spaces, then geometric realization preserves finite products, and hence is a strong monoidal functor. Therefore, in this case the adjunction is a strong monoidal Quillen adjunction, and hence makes the Strøm model structure into a simplicial model category.
For any simplicial topological space, there is a homeomorphism between the geometric realizaton of the simplicial topological space and the ordinary geometric realization of the simplicial set that is the diagonal of the bisimplicial set
Moreover, the degreewise -counit yields a morphism
See geometric realization of simplicial topological spaces for more details.
The main article is
but it depends on earlier results of several authors and mostly his own earlier papers
One modern re-proof can be found in
A review is in section 2 and a generalizatin in section 5 of