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Introductions
Definitions
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Basic facts
Theorems
Arne Strøm has proven that the category Top of all topological spaces has a structure of a Quillen model category where
fibrations are Hurewicz fibrations,
cofibrations are closed Hurewicz cofibrations
and the role of weak equivalences is played by (strong) homotopy equivalences
(as opposed to the weak homotopy equivalences of the “standard” Quillen model structure on topological spaces).
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.
In the Strøm model structure, every object is both a fibrant object and a cofibrant object.
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 $id \colon Top \to Top$ is left Quillen from the Quillen model structure on topological spaces (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 retract of a relative cell complex inclusion is a Hurewicz cofibration.
It follows, by composition, that the (geometric realization $\dashv$ singular simplicial complex)-adjunction ${\vert-\vert} \colon sSet \leftrightarrows Top : Sing$ is a Quillen adjunction between the classical model structure on simplicial sets and the Strøm model structure.
If $Top$ denotes the category of compactly generated spaces, then geometric realization ${|-|} \colon sSet \to Top$ preserves finite products, and hence is a strong monoidal functor. Therefore, in this case the adjunction ${|-|} \dashv Sing$ is a strong monoidal Quillen adjunction, and hence makes the Strøm model structure into a simplicial model category.
Write $({\vert- \vert} \dashv Sing) : Top\stackrel{\overset{{|-|}}{\leftarrow}}{\underset{Sing}{\to}}$ sSet for the ordinary geometric realization/singular simplicial complex adjunction (see homotopy hypothesis).
For $S_{\bullet,\bullet} : \Delta^{op} \times \Delta^{op} \to Set$ a bisimplicial set, write $d S$ for its diagonal $d X : \Delta^{op} \to \Delta^{op} \times \Delta^{op} \stackrel{S}{\to} Set$.
For $X_\bullet$ any simplicial topological space, there is a homeomorphism between the geometric realization of the simplicial topological space $[n] \mapsto |Sing(X_n)|$ and the ordinary geometric realization of the simplicial set that is the diagonal of the bisimplicial set $Sing(X_\bullet)_\bullet$
Moreover, the degreewise $(|-| \dashv Sing)$-counit yields a morphism
and this is a cofibrant resolution in the Reedy model structure $[\Delta^{op}, Top_{Strom}]_{Reedy}$ relative to the Strøm model structure.
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
Arne Strøm,
One modern re-proof can be found in
A review is in section 2 and a generalization in section 5 of