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semi-left-exact left Bousfield localization

Semi-left-exact left Bousfield localizations

Semi-left-exact left Bousfield localizations

Idea

When constructing a general left Bousfield localization of a cofibrantly generated model category CC at a set of maps SS, it is fairly straightforward to construct a localization functor, i.e. a fibrant replacement functor for the putative localized model category L SCL_S C, by a small object argument. We represent the maps in SS by cofibrations between cofibrant objects, take their pushout products with all boundary inclusions Δ nΔ n\partial \Delta^n \hookrightarrow \Delta^n (assuming for simplicity that CC is a simplicial model category), and add all the generating acyclic cofibrations of CC. Then an object ZZ has the right lifting property with respect to this set if and only if it is fibrant in CC (because we added the generating acyclic cofibrations of CC) and for all f:ABf:A\to B in SS the induced map Map(B,Z)Map(A,Z)Map(B,Z) \to Map(A,Z) of simplicial mapping spaces (which is a fibration since ff is a cofibration and ZZ is fibrant) is an acyclic fibration, i.e. ZZ is SS-local. However, it is much harder to construct a factorization of an arbitrary map as an SS-acyclic cofibration followed by an SS-fibration — one has to use a cardinality argument to obtain a set of generating SS-acyclic cofibrations — and accordingly the SS-fibrations have no explicit description.

This can be seen as a homotopy-theoretic analogue of the construction of a reflective factorization system. The localization functor exhibits the SS-local objects as a reflective subcategory, while the SS-acyclic cofibrations and SS-fibrations are a model-categorical representation of the corresponding reflective factorization system, whose left class consists of the morphisms inverted by the reflector (here, the SS-local equivalences) and whose right class is defined by orthogonality to these. Even in 1-category theory?, constructing reflective factorizations requires finicky cardinality or size-based arguments as well. However, there are some reflections, such as the semi-left-exact reflections, for which the reflective factorization system can be constructed by a direct argument in one step. The homotopy-theoretic analogue of these is a semi-left-exact left Bousfield localization.

Construction

The following is Theorem 1.1 of Stanculescu, which is an improved version of Theorems 9.3 and 9.7 from Bousfield 2001, which are in turn an improved version of Appendix A of Bousfield-Friedlander 78.

Theorem

Let CC be a model category and Q:CCQ:C\to C a functor equipped with a natural transformation α:IdQ\alpha:Id\to Q such that

  1. QQ is a homotopical functor, i.e. preserves weak equivalences.
  2. For each XCX\in C, the map Qα X:QXQQXQ \alpha_X:Q X \to Q Q X is a weak equivalence, and the map α QX:QXQQX\alpha_{Q X}:Q X \to Q Q X becomes a monomorphism in the homotopy category.
  3. Define an object XX to be QQ-local if it is fibrant and XQXX\to Q X is a weak equivalence, and define a morphism ff to be a QQ-equivalence if QfQ f is a weak equivalence. Then pullback along fibrations between QQ-local objects preserves QQ-equivalences.

Then there is a new model structure C QC^Q on CC whose cofibrations are those of CC, whose weak equivalences are the QQ-equivalences, and whose fibrations are the maps whose α\alpha-naturality-square is a homotopy pullback. Moreover, C QC^Q is right proper, and simplicial if CC is.

The first two conditions say essentially that QQ is a homotopical reflection into some subcategory (namely the QQ-local objects). The third condition says that it is semi-left-exact (a 1-categorical reflection of CC into BCB\subseteq C is semi-left-exact if and only if pullback along morphisms in BB preserves morphisms that are inverted by the reflector).

For details of the proof, see the references. The central point is the construction of the factorization into an SS-acyclic cofibration and an SS-fibration, which proceeds by first applying QQ along with fibrant replacement, then taking a homotopy pullback: the same way that a reflective factorization system is constructed from a semi-left-exact reflection in 1-category theory.

Remarks

References

  • Aldridge Bousfield, Eric Friedlander, def. 1.1.6 in Homotopy theory of Γ\Gamma-spaces, spectra, and bisimplicial sets, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (pdf)

  • Aldridge Bousfield, On the telescopic homotopy theory of spaces, Trans. Amer. Math. Soc. 353 (2001), 2391-2426, web with fulltext

  • Alexandru E. Stanculescu?, Note on a theorem of Bousfield and Friedlander, Topology and its Applications 155(13), arxiv:0806.4547.

  • Philip Hirschhorn, chapter 13 of Model Categories and Their Localizations, 2003 (AMS, pdf toc, pdf)

  • Cassidy and Hébert and Kelly, “Reflective subcategories, localizations, and factorization systems”. J. Austral. Math Soc. (Series A) 38 (1985), 287–329 (pdf)

  • David Gepner and Joachim Kock, Univalence in locally cartesian closed categories, arxiv:1208.1749

Last revised on January 21, 2019 at 04:52:08. See the history of this page for a list of all contributions to it.