nLab
Strøm model structure

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Homotopy theory

Contents

Idea

Arne Strøm has proven that the category Top of all topological spaces has a structure of a Quillen model category where

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.

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 TopTopTop \to Top 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 \dashv singular simplicial complex)-adjunction :sSetTop:Sing {|-|} \colon sSet \leftrightarrows Top : Sing is Quillen between the standard model structure on simplicial sets and the Strøm model structure.

Simplicial structure

If TopTop denotes the category of compactly generated spaces, then geometric realization :sSetTop {|-|} \colon sSet \to Top preserves finite products, and hence is a strong monoidal functor. Therefore, in this case the adjunction Sing{|-|} \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 (Sing):TopSing({\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 ,:Δ op×Δ opSetS_{\bullet,\bullet} : \Delta^{op} \times \Delta^{op} \to Set a bisimplicial set, write dSd S for its diagonal dX:Δ opΔ op×Δ opSSetd X : \Delta^{op} \to \Delta^{op} \times \Delta^{op} \stackrel{S}{\to} Set.

Proposition

For X X_\bullet any simplicial topological space, there is a homeomorphism between the geometric realizaton of the simplicial topological space [n]Sing(X n)[n] \mapsto |Sing(X_n)| and the ordinary geometric realization of the simplicial set that is the diagonal of the bisimplicial set Sing(X ) Sing(X_\bullet)_\bullet

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

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

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

and this is a cofibrant resolution in the Reedy model structure [Δ op,Top Strom] Reedy[\Delta^{op}, Top_{Strom}]_{Reedy} relative to the Strøm model structure.

See geometric realization of simplicial topological spaces for more details.

References

The main article is

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

but it depends on earlier results of several authors and mostly his own earlier papers

  • Arne Strøm,

    • Note on cofibrations, Math. Scand. 19 1966 11–14 (file MR0211403 (35 #2284));

    • Note on cofibrations II, Math. Scand. 22 1968 130–142 (1969) (file MR0243525 (39 #4846))

One modern re-proof can be found in

  • May and Sigurdsson, Parametrized homotopy theory (web)

A review is in section 2 and a generalizatin in section 5 of

Revised on February 11, 2013 00:10:23 by Urs Schreiber (89.204.135.60)