nLab Strøm model structure

Idea

Arne Strøm proved 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 paper (Strøm 1972) 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.

Properties

General

Observation

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.

Monoidal 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.

Quillen adjunctions

The identity functor $id \colon Top \to Top$ is left Quillen from the classical model structure on topological spaces (or the mixed model structure) to the Strøm model structure, and of course right Quillen in the other direction.

$$Top_{Strom} \stackrel{\overset{id}{\longleftarrow}}{\underset{id}{\longrightarrow}} Top_{Quillen} \,.$$

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 ${\vert-\vert} \colon sSet \leftrightarrows Top \colon Sing$ is a Quillen adjunction between the classical model structure on simplicial sets and the Strøm model structure.

$$Top_{Strom} \stackrel{\overset{id}{\longleftarrow}}{\underset{id}{\longrightarrow}} Top_{Quillen} \stackrel{\overset{{\vert-\vert}}{\longleftarrow}}{\underset{Sing}{\longrightarrow}} sSet_{Quillen} \,.$$

Simplicial 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.

Geometric realization is a Reedy cofibrant replacement

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 $d X : \Delta^{op} \to \Delta^{op} \times \Delta^{op} \stackrel{S}{\to} Set$.

Proposition

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 $Sing(X_\bullet)_\bullet$

$$\left|[n] \mapsto |Sing(X_n)|\right| \simeq_{iso} | d Sing(X_\bullet)_\bullet | \,.$$

Moreover, the degreewise $(|-| \dashv Sing)$-counit yields a morphism

$$([n] \mapsto |Sing(X_n)|) \to X_\bullet$$

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.

Related concepts

• homotopy hypothesis

• model structure on topological spaces

  • classical model structure on topological spaces

    • classical model structure on pointed topological spaces

  • Strøm model structure

  • model structure on Delta-generated topological spaces

• model structure on simplicial sets, model structure on topological spaces

• Thomason model structure

• Hurewicz model structure on chain complexes

References

The model structure was originally established in

• Arne Strøm, The homotopy category is a homotopy category, Archiv der Mathematik 23 (1972) (pdf, pdf)

using results on Hurewicz cofibrations from:

• Arne Strøm, Note on cofibrations, Math. Scand. 19 (1966) 11-14 (jstor:24490229, dml:165952, MR0211403)

• Arne Strøm, Note on cofibrations II, Math. Scand. 22 (1968) 130–142 (1969) (jstor:24489730, dml:166037, MR0243525)

A new proof using algebraic weak factorization systems, and its generalization to any bicomplete category which is powered, copowered and enriched in TopSp is due to:

• Tobias Barthel, Emily Riehl, On the construction of functorial factorizations for model categories, Algebr. Geom. Topol. 13 (2013) 1089-1124 (arXiv:1204.5427, doi:10.2140/agt.2013.13.1089, euclid:agt/1513715550)

Beware that a proof of the Strøm model structure was also claimed in

• May, Sigurdsson, Parametrized Homotopy Theory,

but relying on

• Michael Cole, Prop. 5.3 in Many homotopy categories are homotopy categories, Topology and its Applications 153 (2006) 1084–1099 (doi:10.1016/j.topol.2005.02.006)

• Peter May, Kate Ponto, Lemma 17.1.7 in: More concise algebraic topology – Localization, Completion, and Model Categories, University of Chicago Press (2012) (ISBN:9780226511795, pdf)

which later was noticed to be false, by Richard Williamson, see Barthel & Riehl, p. 2 and Rem 5.12 and Sec. 6.1 for details.