model category, model $\infty$-category
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homotopy theory, (∞,1)-category theory, homotopy type theory
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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.
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 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.
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.
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 model structure was originally established in
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:
Beware that a proof of the Strøm model structure was also claimed in
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.
Last revised on September 20, 2021 at 09:30:44. See the history of this page for a list of all contributions to it.