Strøm model structure in nLab
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Model category theory
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Homotopy theory
homotopy theory, (∞,1)-category theory, homotopy type theory
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models: topological, simplicial, localic, …
see also algebraic topology
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Idea
Arne Strøm proved that the category Top of all topological spaces has a structure of a Quillen model category where
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fibrations are Hurewicz fibrations,
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cofibrations are closed Hurewicz cofibrations
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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
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 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 singular simplicial complex)-adjunction is a Quillen adjunction between the classical model structure on simplicial sets and the Strøm model structure.
Simplicial 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.
Geometric realization is a Reedy cofibrant replacement
Write sSet for the ordinary geometric realization/singular simplicial complex adjunction (see homotopy hypothesis).
For a bisimplicial set, write for its diagonal .
See geometric realization of simplicial topological spaces for more details.
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:
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Arne Strøm, Note on cofibrations, Math. Scand. 19 (1966) 11-14 (jstor:24490229, dml:165952, MR0211403)
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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
but relying on
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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)
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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.