# nLab 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 $C$ 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), 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.