nLab semi-left-exact left Bousfield localization

Idea

When constructing a general left Bousfield localization of a cofibrantly generated model category $C$ at a set of maps $S$, it is fairly straightforward to construct a localization functor, i.e. a fibrant replacement functor for the putative localized model category $L_S C$, by a small object argument. We represent the maps in $S$ by cofibrations between cofibrant objects, take their pushout products with all boundary inclusions $\partial \Delta^n \hookrightarrow \Delta^n$ (assuming for simplicity that $C$ is a simplicial model category), and add all the generating acyclic cofibrations of $C$. Then an object $Z$ has the right lifting property with respect to this set if and only if it is fibrant in $C$ (because we added the generating acyclic cofibrations of $C$) and for all $f:A\to B$ in $S$ the induced map $Map(B,Z) \to Map(A,Z)$ of simplicial mapping spaces (which is a fibration since $f$ is a cofibration and $Z$ is fibrant) is an acyclic fibration, i.e. $Z$ is $S$-local. However, it is much harder to construct a factorization of an arbitrary map as an $S$-acyclic cofibration followed by an $S$-fibration — one has to use a cardinality argument to obtain a set of generating $S$-acyclic cofibrations — and accordingly the $S$-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 $S$-local objects as a reflective subcategory, while the $S$-acyclic cofibrations and $S$-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 $S$-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 $C$ be a model category and $Q:C\to C$ a functor equipped with a natural transformation $\alpha:Id\to Q$ such that

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

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

The first two conditions say essentially that $Q$ is a homotopical reflection into some subcategory (namely the $Q$-local objects). The third condition says that it is semi-left-exact (a 1-categorical reflection of $C$ into $B\subseteq C$ is semi-left-exact if and only if pullback along morphisms in $B$ 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 $S$-acyclic cofibration and an $S$-fibration, which proceeds by first applying $Q$ 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

• Right properness of semi-left-exact left Bousfield localizations was also shown in Gepner-Kock, Prop. 7.8, with special attention paid to type-theoretic model categories.

• Nullification, i.e. localization at a family of maps $A\to \ast$, is always semi-left-exact. (Indeed, nullification is what in homotopy type theory is called a higher modality, and has reflection with stable units?, a stronger condition.)

• Left exact? reflections are always semi-left exact. In particular, the left-exact localizations that present Grothendieck (∞,1)-toposes can be constructed in this way.

References

• Aldridge Bousfield, Eric Friedlander, def. 1.1.6 in Homotopy theory of $\Gamma$-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 (2008) 1434-1438 [arxiv:0806.4547, doi:10.1016/j.topol.2008.05.003]

• 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