03.12.48 · modern-geometry / homotopy

Bousfield localisation of a model category

shipped3 tiersLean: none

Anchor (Master): Hirschhorn 2003 *Model Categories and Their Localizations* (AMS Mathematical Surveys 99) §3-§4 and §11-§13 (originator monograph for the existence theorem); Bousfield 1975 *The localization of spaces with respect to homology* (Topology 14) §2-§5 (originator for spaces); Bousfield 1979 *The localization of spectra with respect to homology* (Topology 18, 257-281) (originator for spectra); Bousfield-Friedlander 1978 *Homotopy theory of $\Gamma$-spaces, spectra and bisimplicial sets* (LNM 658) §A (Hammock localisation); Hovey 1999 *Model Categories* (AMS Mathematical Surveys 63) §3-§5; Goerss-Jardine 2009 *Simplicial Homotopy Theory* §X; Lurie 2009 *Higher Topos Theory* (Annals Studies 170) §5.5 ($\infty$-categorical reflective localisation); Ravenel 1984 *Localization with respect to certain periodic homology theories* (Amer. J. Math. 106) — chromatic origin

Intuition Beginner

A model category is a workshop in which you can do homotopy theory: three classes of morphisms — weak equivalences, fibrations, cofibrations — let you invert the weak equivalences in a controlled way and end up with a homotopy category. Bousfield localisation answers a follow-up question. You have such a workshop, and you have decided that some additional maps — call the set $S$ — should also count as weak equivalences. Can you re-tune the workshop so that the new weak equivalences include $S$ and everything still works?

The answer is yes, under mild hypotheses, and the result is a new model structure $L_{S} M$ on the same underlying category with the same cofibrations and a larger class of weak equivalences. The fibrations shrink to compensate. The resulting homotopy category is the classical homotopy category with the $S$ -class extra inverted.

Why bother? Because nearly every modern homotopy-theoretic construction of interest is a Bousfield localisation. Rationalisation of spaces inverts maps that are rational homology isomorphisms. $p$ -completion inverts maps that are mod- $p$ homology isomorphisms. Chromatic localisation inverts maps that are isomorphisms on a Morava $K$ -theory. The framework is the universal tool for inverting a designated class of maps inside a model category.

Visual Beginner

Picture two side-by-side workshops. On the left is the original model category $M$ : three labelled toolboxes — weak equivalences $W$ , fibrations $F$ , cofibrations $C$ — and a homotopy category $Ho (M)$ produced by inverting the maps in $W$ . On the right is the localised workshop $L_{S} M$ : the same cofibrations $C$ , an enlarged box of weak equivalences $W \cup S \cup \dots$ (the closure under what is forced), and a shrunken box of fibrations. The homotopy category $Ho (L_{S} M)$ is the localisation of $Ho (M)$ at the image of $S$ .

The picture captures the essential message: localisation keeps the cofibrations fixed, enlarges the weak equivalences to include the chosen set $S$ , and shrinks the fibrations as the dual price. The objects that the new fibrations select are the $S$ -local objects: those that already see every map in $S$ as a weak equivalence.

Worked example Beginner

Localise the model category of pointed simplicial sets at the set of integer homology isomorphisms, and see what comes out.

Step 1. The starting model category is $sSet_{*}$ with the Kan-Quillen model structure: cofibrations are monomorphisms, fibrations are Kan fibrations, weak equivalences are maps that are simplicial homotopy equivalences after fibrant replacement. The set $S$ we want to invert is the class of maps that induce isomorphisms on integer singular homology in every dimension.

Step 2. Spell out which objects become $S$ -local. An object $X$ is $S$ -local when the function space $Map (B, X) \to Map (A, X)$ is a weak equivalence in $sSet$ for every map $f : A \to B$ that induces an isomorphism on integer homology. Bousfield's 1975 theorem identifies these: a connected $S$ -local object is exactly a connected nilpotent space that has been integer-homology localised, which for simply connected spaces is the same as the space itself.

Step 3. Read off the homotopy category. The localised homotopy category $Ho (L_{S} sSet_{*})$ identifies with the homotopy category of integer-homology local spaces. For simply connected spaces, this is the original homotopy category, because simply connected spaces are already integer-homology local. For non-simply-connected spaces, the localisation kills the part of the fundamental group not detected by integer homology — the perfect radical.

Step 4. State the upshot. Bousfield localisation at integer homology equivalences produces a model structure whose fibrant objects in the simply connected range agree with the original space, and whose non-simply-connected fibrant objects are the integer-homology-local versions of the original. The localisation is the structured home for the "integer-homology functor" already in the literature.

What this tells us: a single recipe — pick a set $S$ , run the existence theorem — produces a structured homotopy theory for whatever inversion you want to perform. The same machinery covers rationalisation, $p$ -completion, and chromatic localisation as special cases of the choice of $S$ .

Check your understanding Beginner

Exercise (easy, multiple choice).

In the left Bousfield localisation $L_{S} M$ of a model category $M$ at a set $S$ of maps, which of the three classes (cofibrations, fibrations, weak equivalences) is unchanged?

A. Weak equivalences. B. Cofibrations. C. Fibrations. D. None of them.

Hint

The name "left Bousfield localisation" refers to the left-hand half of the cofibration / fibration pair. Left localisation keeps the cofibrations fixed and modifies the rest.

Answer

B. Cofibrations. The defining feature of left Bousfield localisation is that the cofibrations of $M$ are exactly the cofibrations of $L_{S} M$ . The weak equivalences are enlarged to include $S$ and everything forced by closure under the two-out-of-three property; the fibrations shrink to compensate, because they are characterised as the maps with the right lifting property against acyclic cofibrations and the acyclic class has grown.

Exercise (easy, multiple choice).

Which of the following is not a Bousfield localisation of pointed simplicial sets?

A. Rationalisation $L_{H Q}$ . B. $p$ -completion $L_{H F_{p}}$ . C. The Kan-Quillen model structure itself. D. Replacement of every space by its zero-dimensional skeleton.

Hint

Bousfield localisation always keeps the cofibrations fixed and enlarges the weak equivalences. Replacing every space by its zero-skeleton is a destructive functor that does not arise from inverting a set of maps in this way.

Answer

D. Replacement of every space by its zero-dimensional skeleton. The first three are standard Bousfield localisations: at the rational Eilenberg-MacLane spectrum, the mod- $p$ spectrum, and the empty set (which gives back the original structure). The zero-skeleton functor is not a Bousfield localisation; it does not arise from inverting a set of maps in the model-categorical sense and does not preserve the cofibrations of $sSet_{*}$ .

Formal definition Intermediate+

Let $M = (M, W, C, F)$ be a model category with weak equivalences $W$ , cofibrations $C$ , and fibrations $F$ . Let $S$ be a set of morphisms in $M$ . The intent of the left Bousfield localisation construction is to produce a new model structure on the underlying category of $M$ in which the maps in $S$ become weak equivalences.

Definition ( $S$ -local object, $S$ -local equivalence; Hirschhorn 2003 Definition 3.1.4). Fix a simplicial-mapping-space functor $RMap_{M} (-, -)$ presenting the derived $Hom$ in $M$ (the function complex of a simplicial enrichment, or a hammock localisation of Dwyer-Kan, as appropriate to the context).

An object $X \in M$ is $S$ -local if for every morphism $f : A \to B$ in $S$ , the induced map of simplicial sets $f^{*} : RMap_{M} (B, X) \to RMap_{M} (A, X)$ is a weak equivalence in the Kan-Quillen model structure on $sSet$ .
A morphism $g : Y \to Z$ in $M$ is an $S$ -local equivalence if for every $S$ -local object $X$ , the induced map $g^{*} : RMap_{M} (Z, X) \to RMap_{M} (Y, X)$ is a weak equivalence in $sSet$ .

Definition (left Bousfield localisation; Hirschhorn 2003 Definition 3.3.1). The left Bousfield localisation of $M$ at $S$ , denoted $L_{S} M$ , is a model structure on the same underlying category with:

cofibrations $C_{L_{S} M} = C$ ,
weak equivalences $W_{L_{S} M} =$ the class of $S$ -local equivalences,
fibrations $F_{L_{S} M}$ defined by the right-lifting property against $C \cap W_{L_{S} M}$ .

When the localisation exists, the class $W_{L_{S} M}$ contains both $W$ and $S$ , and the model-category axioms M1-M5 hold for the triple $(W_{L_{S} M}, C, F_{L_{S} M})$ .

Definition (combinatorial, cellular, left proper). A model category $M$ is combinatorial if its underlying category is locally presentable and it is cofibrantly generated, i.e., the cofibrations and the acyclic cofibrations admit sets of generators in the sense of the small-object argument. It is cellular (Hirschhorn 2003 §12.1) if it is cofibrantly generated with generating cofibrations $I$ such that the codomains of $I$ are small with respect to all of $I$ , the domains of $I$ are small with respect to the cofibrations, and cofibrations are effective monomorphisms. The model category is left proper if weak equivalences are preserved under pushouts along cofibrations.

Characterisation of $S$ -local objects via fibrancy. When $L_{S} M$ exists, an object $X$ that is fibrant in $M$ is fibrant in $L_{S} M$ if and only if $X$ is $S$ -local. Equivalently, the map $X \to *$ is a fibration in $L_{S} M$ iff $X$ is $M$ -fibrant and $S$ -local. This is Hirschhorn 2003 Proposition 3.4.1.

Counterexamples to common slips

A right Bousfield localisation, dual to the left version, keeps the fibrations fixed and enlarges the weak equivalences to include a set $S$ of maps; the cofibrations shrink. The two constructions are distinct and apply to different settings. Left localisation is the one used for $H_{*}$ -localisation; right localisation is used for cellularisation in the sense of Farjoun.
The set $S$ must be a set, not a proper class. The small-object argument needs cardinal-bounded data. Attempting to localise at a proper class of maps fails because the resulting factorisation system is not generated by a set.
The fibrations of $L_{S} M$ are not the original fibrations intersected with maps having some extra property in general. The new fibrations are characterised by right lifting against the (larger) class of acyclic cofibrations, and there is no simple description of them in $M$ without reference to $S$ .
$S$ -local equivalences need not be detectable on homotopy groups or on any other classical invariant of $M$ ; the only intrinsic detector is the derived mapping space into $S$ -local objects.

Key theorem with proof Intermediate+

Theorem (Hirschhorn 2003 Theorem 4.1.1; existence of left Bousfield localisation). Let $M$ be a left proper combinatorial model category, and let $S$ be a set of morphisms in $M$ . The left Bousfield localisation $L_{S} M$ exists as a left proper combinatorial model category on the underlying category of $M$ , with cofibrations equal to those of $M$ and weak equivalences equal to the $S$ -local equivalences. The identity functor $M \to L_{S} M$ is a left Quillen functor; if every map of $S$ has cofibrant domain and codomain in $M$ , then each $f \in S$ is a weak equivalence in $L_{S} M$ .

Proof. The proof has three steps: produce the generating cofibrations and generating acyclic cofibrations for the new structure, verify the model-category axioms using the small-object argument, and identify the resulting weak equivalences with the $S$ -local equivalences.

Step 1: generating sets. Let $I_{M}$ and $J_{M}$ be the generating cofibrations and generating acyclic cofibrations of $M$ . For each map $f : A \to B$ in $S$ , choose a cofibration replacement $\tilde{f} : \tilde{A} \to \tilde{B}$ between cofibrant objects (possible because $M$ is combinatorial). Form the simplicial-mapping-space horn extensions: for each $n \geq 0$ , define $$ H(\tilde{f}, n) : \tilde{A} \otimes \Delta^n \cup_{\tilde{A} \otimes \partial \Delta^n} \tilde{B} \otimes \partial \Delta^n \hookrightarrow \tilde{B} \otimes \Delta^n, $$ where $\otimes$ denotes the simplicial tensor of the framing on $M$ (Hovey 1999 §5.2; if $M$ is already a simplicial model category, this is the simplicial enrichment). Let $$ J_{L_S} = J_\mathcal{M} \cup {H(\tilde{f}, n) : f \in S, n \geq 0}. $$ Set $I_{L_{S}} = I_{M}$ . These will be the generating cofibrations and generating acyclic cofibrations of $L_{S} M$ .

Step 2: small-object argument and factorisation. The category underlying $M$ is locally presentable (combinatorial assumption), so every object is small with respect to any chosen cardinal. The small-object argument applied to $I_{L_{S}}$ produces a functorial factorisation $f = p \circ i$ with $i$ an $I_{L_{S}}$ -cell complex (a transfinite composition of cobase changes of maps in $I_{L_{S}}$ ) and $p$ having the right lifting property against $I_{L_{S}}$ . The small-object argument applied to $J_{L_{S}}$ produces a functorial factorisation $f = q \circ j$ with $j$ a $J_{L_{S}}$ -cell complex and $q$ having the right lifting property against $J_{L_{S}}$ . Cofibrations of $L_{S} M$ are retracts of $I_{L_{S}}$ -cell complexes (i.e., cofibrations of $M$ ); $S$ -local fibrations are maps with right lifting against $J_{L_{S}}$ . M4 holds.

Step 3: $S$ -local equivalences identification. Define provisional weak equivalences to be retracts of relative $J_{L_{S}}$ -cell complexes followed by maps with right lifting against $J_{L_{S}}$ . By construction these include both $W_{M}$ and the maps in $S$ (because the horn extensions $H (\tilde{f}, n)$ encode the simplicial-mapping-space condition). For two-out-of-three: chase through the simplicial-mapping-space characterisation, using that derived $Hom$ respects two-out-of-three for weak equivalences in $sSet$ . The key calculation is that an object $X$ has the right lifting property against every $H (\tilde{f}, n)$ iff the simplicial mapping space $Map (\tilde{B}, X) \to Map (\tilde{A}, X)$ is a Kan-Quillen weak equivalence, iff $X$ is $S$ -local. So fibrant-in- $L_{S}$ objects are exactly fibrant-and- $S$ -local objects, and a map $g$ is an $S$ -local equivalence iff for every fibrant-and- $S$ -local $X$ the induced $Map (g, X)$ is a weak equivalence. This is the same as the provisional class. Hirschhorn 2003 §4.1-§4.2 fills in the bookkeeping. M1-M3 follow from the same argument, and M5 (retract closure) is automatic from the lifting-property characterisations.

Left properness of $L_{S} M$ . Given a cofibration $A ↪ B$ in $M$ (hence in $L_{S} M$ ) and an $S$ -local equivalence $A \to A^{'}$ , the pushout $B \to B \cup_{A} A^{'}$ is an $S$ -local equivalence. Proof: chase the simplicial-mapping-space presentation; pushing out a $Map (-, X)$ -equivalence along a cofibration in the first argument preserves it because $X$ is fibrant in $L_{S} M$ . Left properness of $M$ supplies the corresponding statement for $W_{M}$ , and the $S$ -local equivalence class is closed under the same operation. $□$

Theorem (universal property of left Bousfield localisation; Hirschhorn 2003 Theorem 3.3.20). Let $M$ be a left proper combinatorial model category and let $S$ be a set of morphisms in $M$ . For any model category $N$ and any left Quillen functor $F : M \to N$ such that the left derived functor $L F$ sends every map in $S$ to a weak equivalence in $N$ , there is a unique-up-to-equivalence factorisation $$ F : \mathcal{M} \xrightarrow{\mathrm{id}} L_S \mathcal{M} \xrightarrow{\bar{F}} \mathcal{N} $$ through a left Quillen functor $\overset{ˉ}{F} : L_{S} M \to N$ .

The identity functor $id : M \to L_{S} M$ is left Quillen because cofibrations agree and acyclic cofibrations of $M$ become acyclic cofibrations of $L_{S} M$ (an enlargement). The universal property says that $L_{S} M$ is the initial model-category recipient of a left Quillen functor from $M$ inverting the maps in $S$ .

Bridge. The Bousfield-localisation framework builds toward the modern recasting of nearly every homotopy-theoretic construction as an instance of the same recipe. The foundational reason is exactly that the small-object argument applied to the simplicial-mapping-space horn extensions of $S$ produces a new factorisation system that is forced to satisfy the model-category axioms once $M$ is well behaved. This is exactly the same mechanism that appears again in 03.12.45 (arithmetic square), where the rationalisation $L_{Q}$ and the $p$ -completion $L_{p}$ are Bousfield localisations at the rational and mod- $p$ Eilenberg-MacLane spectra. Building toward the chromatic perspective, the central insight is that any "designated set of maps to invert" inside a left-proper combinatorial model category can be encoded as a model structure of its own, and the resulting model structure is itself left-proper combinatorial — so iterated localisation stays inside the framework. Putting these together, every coefficient-changing or completion question for spaces, spectra, dg-algebras, or simplicial commutative rings reduces to choosing an $S$ and running Hirschhorn's existence theorem, which generalises the ad-hoc localisation constructions of Sullivan, Bousfield-Kan, and Quillen into a single statement. The bridge is the recognition that Bousfield localisation is the universal way to invert maps in the model-category world, and the same idea appears again in 03.12.32 (Quillen functors and equivalences) where the universal property identifies $L_{S} M$ with the initial model-category receiver of an $S$ -inverting left Quillen functor.

Exercises Intermediate+

Exercise 1 (easy, multiple choice).

Which of the following is the correct characterisation of an $S$ -local object $X$ in the model category $M$ ?

A. $Hom_{M} (B, X) \to Hom_{M} (A, X)$ is a bijection for every $f : A \to B$ in $S$ . B. $RMap_{M} (B, X) \to RMap_{M} (A, X)$ is a weak equivalence in $sSet$ for every $f : A \to B$ in $S$ . C. $X$ is in the image of the identity functor $M \to L_{S} M$ . D. $X$ is cofibrant in $M$ .

Hint

The condition uses derived mapping spaces, not point-set Hom-sets, and the derived version is at the simplicial-mapping-space level, not the homotopy-class level.

Answer

B. $X$ is $S$ -local iff for every $f : A \to B$ in $S$ , the derived mapping space $RMap_{M} (B, X) \to RMap_{M} (A, X)$ is a weak equivalence in $sSet$ . The point-set Hom version (A) is strictly stronger and would only suffice in cases where derived and underived Hom agree (no homotopical content). Option C confuses an instance with a characterisation. Cofibrancy of $X$ (D) is unrelated.

Exercise 3 (medium, symbolic).

Let $M = Sp$ be a left proper combinatorial model category presenting the stable homotopy category, and let $E$ be a spectrum. Define the set $S = {f : X \to Y in Sp : E_{*} (f) is an isomorphism}$ . Show that $S$ -local objects of $L_{S} Sp$ are exactly the $E$ -local spectra in the sense of Bousfield 1979.

Hint

In the stable setting, derived mapping spaces between spectra are derived $Hom$ -spectra, and weak equivalences of the underlying simplicial sets correspond to isomorphisms of stable homotopy groups.

Answer

$X$ is $S$ -local iff $RMap_{Sp} (B, X) \to RMap_{Sp} (A, X)$ is a weak equivalence for every $f : A \to B$ in $S$ , iff $[A, X]_{*} = [B, X]_{*}$ for every $E_{*}$ -isomorphism $f$ . By Brown representability and the definition of $Hom$ -spectra, this is the same as saying that $X$ is $E$ -local in Bousfield's sense: every $E_{*}$ -isomorphism becomes an isomorphism after applying $[-, X]_{*}$ . The set $S$ is a proper class as stated; the standard fix is to localise at a small representative subset of $E_{*}$ -isomorphisms (Hirschhorn 2003 §4.2). Modulo that set-theoretic adjustment, $L_{S} Sp$ presents the $E$ -local stable homotopy category. Rubric: full credit for the equivalence to Bousfield 1979's $E$ -locality plus a note on the set-vs.-class subtlety.

Exercise 5 (medium, symbolic).

Let $M$ be a left proper combinatorial model category and let $S \subseteq T$ be two sets of morphisms. Show that the identity functor $L_{S} M \to L_{T} M$ is a left Quillen functor.

Hint

Both structures share the same underlying category and same cofibrations. The class of $T$ -local objects is a subclass of the class of $S$ -local objects.

Answer

The cofibrations of $L_{S} M$ and $L_{T} M$ both equal the cofibrations of $M$ , so the identity functor preserves cofibrations. Every $T$ -local object is in particular $S$ -local: if the derived $Hom$ inverts every map in $T$ , it inverts every map in the smaller set $S$ . So $T$ -local equivalences are characterised by becoming equivalences against $T$ -local objects, which is a stronger condition than becoming equivalences against $S$ -local objects (one tests on a smaller class). The identity sends $S$ -local equivalences to $T$ -local equivalences: a map that is invariant under $RMap (-, X)$ for every $S$ -local $X$ is in particular invariant for $X$ in the smaller class of $T$ -local objects. So identity preserves cofibrations and acyclic cofibrations, hence is left Quillen. Rubric: full credit for both inclusions and the identification of acyclic cofibrations.

Exercise 6 (medium, short-answer).

State and verify the universal property of $L_{S} M$ in the explicit case where $M = sSet_{*}$ and $S = {S^{1} \to *}$ (the unique map from the circle to the point).

Hint

Localising at the map $S^{1} \to *$ forces $S^{1}$ to become equivalent to a point. The fibrant objects become $1$ -connected spaces — those with vanishing $π_{1}$ .

Answer

The localisation $L_{{S^{1} \to *}} sSet_{*}$ is the model structure whose fibrant objects are pointed simply connected spaces (a fibrant simplicial set with $RMap (S^{1}, X) \to RMap (*, X) = X$ a weak equivalence means $Ω X$ is contractible, i.e., $π_{1} (X) = 0$ and $X$ is $1$ -connected). The universal property: for any model category $N$ and any left Quillen functor $F : sSet_{*} \to N$ whose derived functor sends $S^{1} \to *$ to a weak equivalence (equivalently, sends $S^{1}$ to an object weakly equivalent to the terminal object), $F$ factors uniquely up to equivalence through $L_{{S^{1} \to *}} sSet_{*}$ . Concretely, the localisation models the $1$ -connected cover construction. Rubric: full credit for identifying the fibrant objects and stating the factorisation property.

Exercise 7 (hard, short-answer).

Show that the localisation $L_{S} M$ at a single map $f : A \to B$ between cofibrant objects is the universal model category, in the sense of Hirschhorn 2003 §3.3, in which $f$ becomes a weak equivalence. Specifically: any left Quillen functor $F : M \to N$ with $L F (f)$ a weak equivalence factors as $M \to L_{{f}} M \to N$ with the second factor a left Quillen functor.

Hint

Use the universal property of the localisation, together with the observation that $L F$ preserves derived mapping spaces between cofibrant-fibrant objects up to weak equivalence.

Answer

Let $F : M \to N$ be a left Quillen functor with right adjoint $G$ and assume $L F (f)$ is a weak equivalence in $N$ . For any fibrant $X \in N$ , the derived adjunction gives $RMap_{N} (L F (B), X) ≃ RMap_{M} (B, R G (X))$ and similarly for $A$ . The hypothesis on $L F (f)$ implies $RMap_{M} (B, R G (X)) \to RMap_{M} (A, R G (X))$ is a weak equivalence, i.e., $R G (X)$ is ${f}$ -local in $M$ . So $R G$ lands in ${f}$ -local objects; by the universal property of $L_{{f}} M$ as the model category whose fibrant objects are exactly the ${f}$ -local fibrant objects of $M$ , the adjunction $F ⊣ G$ descends to a Quillen adjunction $\overset{ˉ}{F} : L_{{f}} M ⇄ N : \overset{ˉ}{G}$ . Composition $M id L_{{f}} M \overset{ˉ}{F} N$ recovers $F$ . Rubric: full credit for the derived-adjunction calculation and the identification of $\overset{ˉ}{F}$ .

Exercise 8 (hard, short-answer).

Show that the Bousfield localisation $L_{E} Sp$ of spectra at a spectrum $E$ is a stable model category (the suspension functor is a Quillen equivalence). Use the characterisation of $S$ -local objects via derived mapping spaces.

Hint

A model category is stable iff the suspension is a Quillen equivalence on its homotopy category, iff the homotopy category is triangulated with $Σ$ as the shift. For Bousfield localisation, derived mapping space respects suspension.

Answer

The model category $Sp$ is stable: $Σ : Sp ⇄ Sp : Ω$ is a Quillen equivalence. The $E$ -local objects are characterised by $RMap_{Sp} (B, X) \to RMap_{Sp} (A, X)$ being a weak equivalence for every $E_{*}$ -isomorphism $f : A \to B$ . The class of $E_{*}$ -isomorphisms is closed under $Σ$ : $Σ f$ is an $E_{*}$ -isomorphism iff $f$ is, because $E_{*}$ -homology is a graded functor on $Sp$ that shifts degree under $Σ$ . Hence $X$ is $E$ -local iff $Ω X$ is $E$ -local: the equivalence $RMap (B, Ω X) ≃ RMap (Σ B, X)$ converts $E$ -locality of $Ω X$ into the $Σ B$ -shifted condition on $X$ , which holds because $Σ f$ is still an $E_{*}$ -isomorphism. The category of $E$ -local objects is closed under $Σ$ and $Ω$ , and the resulting $L_{E} Sp$ -homotopy category is stable (i.e., the suspension functor is a Quillen equivalence in $L_{E} Sp$ ). Rubric: full credit for the closure-under-suspension argument plus the conclusion that $L_{E} Sp$ is stable.

Advanced results Master

Theorem (Bousfield 1975 Theorem 9.1; existence of $H_{*}$ -localisation for spaces). Let $h_ $b e a g e n er a l i se d h o m o l o g y t h eor y o n p o in t e d C W co m pl e x es t ha t s a t i s f i es t h e w e d g e a x i o m . T h e B o u s f i e l d l oc a l i s a t i o n$ L_h \mathbf{sSet}* $e x i s t s, an d t h e$ h $- l oc a l o bj ec t s in t h e h o m o t o p y c a t e g or y$ \mathrm{Ho}(L_h \mathbf{sSet}) $f or ma r e f l ec t i v es u b c a t e g or y o f$ \mathrm{Ho}(\mathbf{sSet}_) $. F or$ h_* = H_*(-; \mathbb{Z}) $an d$ X $s im pl y co nn ec t e d, t h e$ h $- l oc a l i s a t i o n$ L_h X $co in c i d es w i t h$ X $; f or$ X $n o n - s im pl y - co nn ec t e d,$ L_h X $k i l l s t h e p er f ec t r a d i c a l o f$ \pi_1 $an d r e pl a ces t h e hi g h er$ \pi_n $b y t h e i r$ H_*$-completions.*

The 1975 paper appeared in Topology 14, pp. 133-150 and predates the explicit model-category formulation of localisation. Bousfield constructed the localisation directly via the small-object argument applied to a set of representative $h_{*}$ -isomorphisms; the model-category framing came with Hirschhorn's monograph (2003) and the parallel development in Goerss-Jardine §X.

Theorem (Bousfield 1979 Proposition 2.7; existence of $E$ -localisation for spectra). Let $E$ be a spectrum. The Bousfield localisation $L_{E} Sp$ of any combinatorial model for the stable homotopy category exists; the $E$ -local objects form a reflective subcategory of $Ho (Sp)$ . The localisation functor $L_{E}$ is idempotent: $L_{E} L_{E} ≃ L_{E}$ . Two spectra $E$ and $F$ define the same localisation if and only if they have the same Bousfield class $⟨ E ⟩ = ⟨ F ⟩$ , the class of spectra $X$ such that $E_(X) = 0 $i f f$ F_*(X) = 0$.*

The 1979 paper appeared in Topology 18, pp. 257-281. It established the lattice of Bousfield classes — a partial order on equivalence classes of spectra — and identified the central role of $L_{E (n)}$ for the Johnson-Wilson spectra $E (n)$ . The chromatic-tower picture of Ravenel 1984 and Hopkins-Smith 1998 sits inside this lattice.

Theorem (Hovey 1999 Theorem 7.4.1; monoidal Bousfield localisation). Let $M$ be a left proper monoidal model category with cofibrant unit, and let $S$ be a set of morphisms in $M$ . If $S$ is closed under tensoring with cofibrant objects (or more generally, if the localised structure is again a monoidal model category — Hovey's monoidal-localisation condition is technical), then $L_{S} M$ inherits a monoidal model-category structure. In particular, the $K (n)$ -local stable category $L_{K (n)} Sp$ is a closed symmetric monoidal model category and computes the height- $n$ monochromatic stable homotopy theory.

The monoidal compatibility is a substantive condition; not every Bousfield localisation of a monoidal model category is again monoidal. The chromatic case works because $L_{K (n)}$ respects the smash product up to the cofibrant-replacement caveat; the Hopkins-Strickland 1999 monograph (AMS Memoirs 666) carries out the verification for the chromatic tower.

Theorem (Lurie 2009 Higher Topos Theory §5.5.4 Proposition 5.5.4.15; $\infty$ -categorical reflective localisation). Let $C$ be a presentable $\infty$ -category and let $S$ be a small set of morphisms in $C$ . There exists a reflective accessible localisation $C \to S^{- 1} C$ whose essential image is the full subcategory of $S$ -local objects (defined by the same derived $Hom$ condition). The functor $S^{- 1} C$ is the universal accessible reflective localisation of $C$ inverting $S$ .

Lurie's $\infty$ -categorical statement is the modern reformulation of Hirschhorn's existence theorem after the $(\infty, 1)$ -categorical paradigm absorbs model-category theory. The Hirschhorn construction presents the model for $L_{S} M$ ; the Lurie construction is the structurally honest statement at the level of presentable $\infty$ -categories, of which Hirschhorn's left-proper combinatorial model categories are a presentation. The Joyal-Lurie identification of $(\infty, 1)$ -categories with combinatorial simplicial model categories (HTT §A.3.7) makes the two statements equivalent in scope.

Theorem (chromatic localisations; Ravenel 1984 Theorem 4.2 and Hopkins-Smith 1998). For each prime $p$ and each height $n \geq 0$ , there is a Bousfield localisation functor $L_{E (n)}$ on the $p$ -local stable homotopy category $Sp_{(p)}$ . The Hopkins-Ravenel chromatic convergence theorem asserts that for a finite $p$ -local spectrum $X$ , the natural map $X \to holim_{n} L_{E (n)} X$ is a weak equivalence. The monochromatic stratum $M_{n} X = fib (L_{E (n)} X \to L_{E (n - 1)} X)$ is $K (n)$ -locally determined: $L_{K (n)} M_{n} X ≃ L_{K (n)} X$ .

Ravenel's 1984 paper Localization with respect to certain periodic homology theories (Amer. J. Math. 106, 351-414) initiated the chromatic-tower picture; Hopkins-Smith 1998 (Ann. Math. 148, 1-49) proved the nilpotence theorem and the periodicity theorem that complete the chromatic stratification. The whole framework rests on Bousfield localisation as the engine.

Theorem (Goerss-Jardine 2009 §X.4; Bousfield-Friedlander theorem for stable model structures). Let $C$ be a left proper, combinatorial, pointed model category that has the property that all weak equivalences are stable under sequential homotopy colimits along cofibrations. For any reasonable choice of $S$ , the Bousfield localisation $L_{S} C$ exists and is again left proper combinatorial; moreover, if $C$ is stable, so is $L_{S} C$ .

The Bousfield-Friedlander variant relaxes the cellular hypothesis of Hirschhorn 2003 §11 to a sequential-colimit hypothesis that is automatically verified in many stable contexts (spectra, motivic spectra, equivariant spectra). The two existence theorems — Hirschhorn's cellular version and the Bousfield-Friedlander stable-colimit version — together cover essentially every Bousfield localisation that appears in practice.

Synthesis. Bousfield localisation is the foundational reason that nearly every modern homotopy-theoretic construction — rationalisation, $p$ -completion, chromatic localisation, profinite completion, motivic and equivariant localisation, and the analogues in dg-algebras and simplicial commutative rings — can be packaged as the same recipe applied to a different choice of $S$ . The central insight is that within a left-proper combinatorial model category, the operation of designating a set of maps to invert is itself an operation that lands inside the model-category world: the result is another left-proper combinatorial model category, with the same cofibrations and a controllable enlargement of the weak equivalences. This is exactly the structural fact that makes the modern axiomatic homotopy theory of Quillen, Hirschhorn, and Hovey a viable framework rather than an abstract organising language. Putting these together, the chromatic story — Bousfield 1979, Ravenel 1984, Hopkins-Smith 1998, Devinatz-Hopkins — sits inside the Bousfield-localisation framework: each $L_{E (n)}$ is a left Bousfield localisation, the chromatic-tower assembly is a sequence of Quillen adjunctions between these localisations, and the monochromatic strata are recovered as homotopy fibres of localisation comparison maps. The bridge is Lurie's identification of left-proper combinatorial model categories with presentable $\infty$ -categories: Bousfield localisation in the model-category world is exactly the accessible reflective localisation in the presentable- $\infty$ -category world, and the universal-property statement transfers verbatim. The same pattern appears again in 03.12.31 (Quillen model categories), where every model-categorical construction has an $\infty$ -categorical shadow that agrees up to Quillen equivalence.

Full proof set Master

Proposition (existence of $L_{S} M$ via the small-object argument). Let $M$ be a left proper combinatorial model category with generating cofibrations $I_{M}$ and generating acyclic cofibrations $J_{M}$ , and let $S$ be a set of morphisms in $M$ between cofibrant objects (replace if necessary). Define $$ J_{L_S} = J_\mathcal{M} \cup {H(f, n) : f \in S, n \geq 0}, $$ where $H (f, n)$ are the simplicial-mapping-space horn extensions associated to $f$ and the standard simplices $\partial Δ^{n} ↪ Δ^{n}$ . Then $(W_{S -loc}, C_{M}, F_{S -loc})$ — with $F_{S -loc} =$ maps with the right lifting property against $J_{L_{S}}$ and $W_{S -loc}$ defined as the closure of $W_{M} \cup S$ under two-out-of-three — satisfies the five model-category axioms.

Proof. M1 (two-out-of-three). Definitional: $W_{S -loc}$ is constructed as the closure of $W_{M} \cup S$ under two-out-of-three. Concretely, $W_{S -loc}$ equals the class of $S$ -local equivalences by the characterisation in §I.

M2 (retract). Cofibrations $C_{M}$ are retract-closed by the corresponding property in $M$ . Fibrations $F_{S -loc}$ defined by right lifting against $J_{L_{S}}$ are retract-closed because right lifting against any set of maps is preserved by retracts. Weak equivalences $W_{S -loc}$ are retract-closed: a retract of an $S$ -local equivalence $g$ in the arrow category restricts on each derived mapping space $RMap (-, X)$ for $S$ -local $X$ to a retract of an equivalence in $sSet$ , hence an equivalence.

M3 (lifting). Cofibrations have the left lifting property against acyclic fibrations of $L_{S} M$ : a cofibration $i$ in $M$ lifts against any $p$ in $F_{S -loc} \cap W_{S -loc}$ by the dual of the small-object argument; the verification reduces to the case where $p$ has the right lifting property against all of $I_{M}$ , and there it agrees with the original lifting in $M$ . Acyclic cofibrations in $L_{S} M$ lift against fibrations of $L_{S} M$ : any acyclic cofibration is a retract of a $J_{L_{S}}$ -cell complex by step 2 of the existence proof, and any $J_{L_{S}}$ -cell complex has the left lifting property against $F_{S -loc}$ by construction.

M4 (factorisation). Functorial factorisation as cofibration-followed-by-acyclic-fibration: apply the small-object argument to $I_{M}$ (the cofibrations of $L_{S} M$ equal those of $M$ ). Functorial factorisation as acyclic-cofibration-followed-by-fibration: apply the small-object argument to $J_{L_{S}}$ (which is small because $M$ is combinatorial and $S$ is a set). Each cell complex is a sequential colimit of pushouts; the locally presentable structure of the underlying category ensures the transfinite composition terminates at the relevant cardinal.

M5 (closure of cofibrations and acyclic fibrations under cobase change, colimit; closure of fibrations under base change, limit). Inherited from the locally presentable structure and the fact that the small-object argument produces classes closed under the appropriate operations.

The identification of $W_{S -loc}$ with the class of $S$ -local equivalences follows from the characterisation that $X$ has the right lifting property against $J_{L_{S}}$ (i.e., is $L_{S}$ -fibrant) iff $X$ is $M$ -fibrant and $S$ -local. Combining with the small-object factorisation of any map $g$ as $g = q \circ j$ with $j$ a $J_{L_{S}}$ -cell complex and $q$ a $J_{L_{S}}$ -fibration, the two-out-of-three closure of $W_{M} \cup S$ matches the $S$ -local-equivalence class. Hirschhorn 2003 §4.1 carries out the bookkeeping. $□$

Proposition (left properness of $L_{S} M$ ). If $M$ is left proper, so is $L_{S} M$ .

Proof. Given a pushout square in $M$ $$ \begin{array}{ccc} A & \xrightarrow{i} & B \ \downarrow w & & \downarrow w' \ A' & \to & B' \end{array} $$ with $i$ a cofibration and $w$ an $S$ -local equivalence, the goal is to show $w^{'}$ is an $S$ -local equivalence. For any $S$ -local fibrant $X$ , apply $RMap (-, X)$ . The resulting square in $sSet$ is a homotopy pullback (because $RMap (-, X)$ sends pushouts along cofibrations of $M$ in the first argument to pullbacks of fibrations in $sSet$ — by left properness of $M$ and the fact that $X$ is fibrant). The map $RMap (w, X)$ is a weak equivalence since $w$ is $S$ -local; in a homotopy pullback of fibrations, a weak equivalence on one side propagates to the parallel side. So $RMap (w^{'}, X)$ is a weak equivalence for every $S$ -local fibrant $X$ , hence $w^{'}$ is an $S$ -local equivalence. $□$

Proposition (universal property of $L_{S} M$ ). Let $N$ be a model category and $F : M ⇄ N : G$ a Quillen adjunction whose left-derived functor $L F$ sends every $f \in S$ to a weak equivalence in $N$ . Then $F ⊣ G$ descends to a Quillen adjunction $\overset{ˉ}{F} : L_{S} M ⇄ N : \overset{ˉ}{G}$ .

Proof. Cofibrations of $L_{S} M$ equal cofibrations of $M$ , so $F$ preserves cofibrations of $L_{S} M$ . It remains to show $F$ preserves acyclic cofibrations of $L_{S} M$ . The acyclic cofibrations of $L_{S} M$ are retracts of $J_{L_{S}}$ -cell complexes. For $j \in J_{M}$ , $F (j)$ is an acyclic cofibration of $N$ by Quillen adjointness of $F$ . For $H (f, n) \in J_{L_{S}} ∖ J_{M}$ , the image $F (H (f, n))$ is a horn extension built from $F (\tilde{f})$ , which is a weak equivalence by hypothesis. The framing-tensor compatibility of left Quillen functors with simplicial-mapping-space horn extensions (Hovey 1999 §5.4) yields that $F (H (f, n))$ is an acyclic cofibration in $N$ . Cell-complex closure (small-object argument in $N$ ) extends to all of $J_{L_{S}}$ -cell complexes; retract closure extends to all acyclic cofibrations of $L_{S} M$ . So $F$ is left Quillen as a functor $L_{S} M \to N$ . The right adjoint is the same $G$ (the underlying categories are unchanged); since left and right adjoints are reciprocal, $G$ is automatically the right adjoint of $F$ in the new structure. $□$

Proposition ( $S$ -local objects via the framed mapping space; Hirschhorn 2003 Proposition 17.4.16). In a left proper combinatorial model category $M$ equipped with a framing (Hirschhorn 2003 §16) giving a simplicial-mapping-space functor $Map_{M}^{l} (-, -)$ , an object $X \in M$ is $S$ -local in the sense of the derived- $Hom$ definition iff for every $f : A \to B$ in $S$ , the map $Map_{M}^{l} (B, X) \to Map_{M}^{l} (A, X)$ is a Kan-Quillen weak equivalence — provided $A, B$ are cofibrant in $M$ and $X$ is fibrant.

Proof. Cofibrant replacement of the source and fibrant replacement of the target are part of the standard derived- $Hom$ recipe in any model category. With $A, B$ cofibrant and $X$ fibrant, the framing-mapping-space $Map_{M}^{l} (-, -)$ is already homotopically correct, i.e., agrees with the derived $Hom$ up to weak equivalence in $sSet$ . The equivalence of the framed-mapping-space condition and the derived- $Hom$ condition follows. This is the standard reduction that makes the $S$ -local condition checkable in practice. $□$

Connections Master

Quillen model category 03.12.31. Bousfield localisation operates inside the Quillen model-category framework: it accepts a model category $M$ and a set $S$ as input and produces a new model category $L_{S} M$ as output. The existence theorem requires that $M$ be left proper and combinatorial; the output is again left proper combinatorial. The construction passes a model category through itself with a controlled enlargement of the weak equivalences, illustrating the closure of the model-category world under designated-set-inversion. Every example of Bousfield localisation in the literature is an instance of running this procedure for a particular $M$ and $S$ .
Quillen functor and equivalence 03.12.32. The universal property of $L_{S} M$ is stated in the language of Quillen functors: $L_{S} M$ is the initial model-category target of a left Quillen functor from $M$ that sends $S$ to weak equivalences. This makes Bousfield localisation the universal-property characterisation of a designated inversion in the model-category 2-category. Quillen equivalences between Bousfield localisations correspond to coincidences of the set of $S$ -local equivalences, i.e., to coincidences of the Bousfield class $⟨ S ⟩$ in the lattice of localising classes.
Kan-Quillen model structure on simplicial sets 03.12.33. The standard example of a Bousfield localisation is the localisation of $sSet_{*}$ at the integer-homology equivalences; the resulting model structure has as fibrant objects the integer-homology-local spaces (in particular, simply connected nilpotent spaces are automatically $H_{*}$ -local). The Kan-Quillen structure itself is the identity Bousfield localisation at the empty set. This identifies $sSet_{*}$ as the home of nearly every space-level Bousfield localisation in algebraic topology.
Simplicial model category 03.12.35. The simplicial-mapping-space condition is at the heart of the $S$ -local definition: $X$ is $S$ -local iff $RMap (B, X) \to RMap (A, X)$ is a weak equivalence in $sSet$ for every $f : A \to B$ in $S$ . The simplicial-model-category framework supplies the function complexes that make this condition computable. Without a simplicial enrichment (or a framing), the $S$ -local condition would not have a clean formulation; with one, the existence theorem becomes a small-object-argument calculation on the framing-tensor horn extensions.
Arithmetic square 03.12.45. The Sullivan-Bousfield-Kan arithmetic square is built from three Bousfield localisations: the rationalisation $L_{H Q}$ , the family of $p$ -completions $L_{H F_{p}}$ , and the rational-completion gluing $L_{H Q} \circ \prod_{p} L_{H F_{p}}$ . The fracture theorem says that for a nilpotent finite-type space $X$ , the original $X$ is the homotopy pullback of these three localisations. The arithmetic-square framework is therefore a sophisticated three-fold application of Bousfield localisation, glued by a $lim^{1}$ vanishing hypothesis.
Spectrum 03.12.04. The stable analogue of Bousfield localisation, established in Bousfield 1979, organises the entire chromatic stratification of the stable homotopy category. Each Morava $K$ -theory $K (n)$ gives a localisation $L_{K (n)}$ , each Johnson-Wilson theory $E (n)$ gives a localisation $L_{E (n)}$ , and the chromatic tower assembles these as a sequence of Bousfield localisations whose homotopy limits recover $p$ -local finite spectra (Ravenel 1984, Hopkins-Ravenel chromatic convergence 1992, Hopkins-Smith 1998 nilpotence and periodicity). The Bousfield-lattice classification of localisations is the central organising principle of modern stable homotopy theory.

Historical & philosophical context Master

The localisation of homotopy theories at a homology theory was introduced by Aldridge K. Bousfield in 1975 The localization of spaces with respect to homology (Topology 14, 133-150) ^{[Bousfield 1975]}. The paper construed the localisation as a functor on the homotopy category of pointed CW complexes that inverts maps inducing isomorphisms on a chosen homology theory $h_{*}$ . Bousfield's construction proceeded directly via the small-object argument applied to a set of representative $h_{*}$ -isomorphisms, producing a fibrant-replacement functor whose image consists of $h_{*}$ -local spaces. The 1975 paper handled the space-level case in full generality; the worked examples for rationalisation (Sullivan 1970) and $p$ -completion (Quillen 1969, Bousfield-Kan 1972) had appeared earlier in less abstract form, and Bousfield's framework subsumed them as instances of a single construction.

The stable analogue was developed by Bousfield in 1979 The localization of spectra with respect to homology (Topology 18, 257-281) ^{[Bousfield 1979]}. The 1979 paper introduced the Bousfield class $⟨ E ⟩$ of a spectrum — the equivalence class of spectra defining the same localisation — and the Bousfield lattice ordering these classes. The lattice has at its top the sphere spectrum $S^{0}$ (whose localisation is the identity) and at its bottom the zero spectrum (whose localisation is constant at the point); the chromatic stratification of stable homotopy theory at a prime $p$ , formalised by Ravenel 1984 Localization with respect to certain periodic homology theories (Amer. J. Math. 106, 351-414) ^{[Ravenel 1984]}, identifies a distinguished tower of Bousfield classes $⟨ E (0)⟩ > ⟨ E (1)⟩ > \dots > ⟨ E (n)⟩ > \dots$ above each prime, with the height- $n$ class $⟨ E (n)⟩$ recovering the height- $n$ monochromatic stratum. Hopkins and Smith 1998 Nilpotence and stable homotopy theory II (Ann. Math. 148, 1-49) ^{[Hopkins-Smith 1998]} proved the nilpotence and periodicity theorems that complete the chromatic picture, all formulated in the language of Bousfield localisation.

The model-category formulation crystallised in the 2000s. Bousfield-Friedlander 1978 Homotopy theory of $Γ$ -spaces, spectra and bisimplicial sets (LNM 658) ^{[Bousfield-Friedlander 1978]} developed the Hammock localisation framework, identifying derived mapping spaces in a model category as the simplicial-set inputs to the $S$ -local condition. Hirschhorn 2003 Model Categories and Their Localizations (AMS Mathematical Surveys 99) ^{[Hirschhorn 2003]} gave the definitive existence theorem for left and right Bousfield localisation in the cellular and combinatorial settings. Hovey 1999 Model Categories (AMS Mathematical Surveys 63) ^{[Hovey 1999]} developed the monoidal-model-category framework in which Bousfield localisation preserves monoidal structure, with the chromatic $K (n)$ -local stable homotopy theory as the headline application.

The $\infty$ -categorical reformulation came with Lurie 2009 Higher Topos Theory (Annals Studies 170) §5.5 ^{[Lurie 2009]}. Lurie identified Bousfield localisation in the model-category sense with accessible reflective localisation in the presentable- $\infty$ -category sense: a presentable $\infty$ -category $C$ and a small set $S$ of morphisms in $C$ give rise to a reflective localisation $S^{- 1} C$ whose essential image is the full subcategory of $S$ -local objects. The model-category and $\infty$ -categorical perspectives agree under the Joyal-Lurie identification of $(\infty, 1)$ -categories with combinatorial simplicial model categories. Modern algebraic-topology research operates fluently in both languages, and the choice between them is a matter of convenience rather than substance.

Bibliography Master

@article{Bousfield1975,
  author  = {Bousfield, A. K.},
  title   = {The localization of spaces with respect to homology},
  journal = {Topology},
  volume  = {14},
  year    = {1975},
  pages   = {133--150}
}

@article{Bousfield1979,
  author  = {Bousfield, A. K.},
  title   = {The localization of spectra with respect to homology},
  journal = {Topology},
  volume  = {18},
  year    = {1979},
  pages   = {257--281}
}

@incollection{BousfieldFriedlander1978,
  author    = {Bousfield, A. K. and Friedlander, E. M.},
  title     = {Homotopy theory of {$\Gamma$}-spaces, spectra, and bisimplicial sets},
  booktitle = {Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II},
  series    = {Lecture Notes in Math.},
  volume    = {658},
  publisher = {Springer},
  year      = {1978},
  pages     = {80--130}
}

@book{BousfieldKan1972,
  author    = {Bousfield, A. K. and Kan, D. M.},
  title     = {Homotopy Limits, Completions and Localizations},
  series    = {Lecture Notes in Mathematics},
  volume    = {304},
  publisher = {Springer-Verlag},
  year      = {1972}
}

@book{Hirschhorn2003,
  author    = {Hirschhorn, Philip S.},
  title     = {Model Categories and Their Localizations},
  series    = {Mathematical Surveys and Monographs},
  volume    = {99},
  publisher = {American Mathematical Society},
  year      = {2003}
}

@book{Hovey1999,
  author    = {Hovey, Mark},
  title     = {Model Categories},
  series    = {Mathematical Surveys and Monographs},
  volume    = {63},
  publisher = {American Mathematical Society},
  year      = {1999}
}

@book{GoerssJardine2009,
  author    = {Goerss, Paul G. and Jardine, John F.},
  title     = {Simplicial Homotopy Theory},
  series    = {Modern Birkh{\"a}user Classics},
  publisher = {Birkh{\"a}user Verlag},
  year      = {2009},
  note      = {Reprint of the 1999 original}
}

@book{Lurie2009HTT,
  author    = {Lurie, Jacob},
  title     = {Higher Topos Theory},
  series    = {Annals of Mathematics Studies},
  volume    = {170},
  publisher = {Princeton University Press},
  year      = {2009}
}

@article{Ravenel1984,
  author  = {Ravenel, Douglas C.},
  title   = {Localization with respect to certain periodic homology theories},
  journal = {Amer. J. Math.},
  volume  = {106},
  year    = {1984},
  pages   = {351--414}
}

@article{HopkinsSmith1998,
  author  = {Hopkins, Michael J. and Smith, Jeffrey H.},
  title   = {Nilpotence and stable homotopy theory. {II}},
  journal = {Ann. of Math. (2)},
  volume  = {148},
  year    = {1998},
  pages   = {1--49}
}

@book{HopkinsStrickland1999,
  author    = {Hovey, Mark and Strickland, Neil P.},
  title     = {Morava {$K$}-theories and localisation},
  series    = {Memoirs of the American Mathematical Society},
  volume    = {666},
  publisher = {American Mathematical Society},
  year      = {1999}
}

@book{MayPonto2012,
  author    = {May, J. Peter and Ponto, Kathleen},
  title     = {More Concise Algebraic Topology: Localization, Completion, and Model Categories},
  series    = {Chicago Lectures in Mathematics},
  publisher = {University of Chicago Press},
  year      = {2012}
}

@article{Sullivan1970,
  author    = {Sullivan, Dennis P.},
  title     = {Geometric Topology: Localization, Periodicity, and {Galois} Symmetry},
  note      = {MIT mimeographed notes 1970; revised and edited by A. Ranicki, K-Monographs in Mathematics 8, Springer 2005},
  year      = {1970}
}

@incollection{Dwyer2004Localizations,
  author    = {Dwyer, William G.},
  title     = {Localizations},
  booktitle = {Axiomatic, Enriched and Motivic Homotopy Theory},
  series    = {NATO Sci. Ser. II Math. Phys. Chem.},
  volume    = {131},
  publisher = {Kluwer Academic Publishers},
  year      = {2004},
  pages     = {3--28}
}

Prerequisites

03.12.04
03.12.31
03.12.32
03.12.33
03.12.35

Tier anchors

beginner: Hirschhorn 2003 *Model Categories and Their Localizations* §1 informal opening; May-Ponto 2012 *More Concise Algebraic Topology* Ch. 19 introductory motivation
intermediate: Hirschhorn 2003 §3 (overview), §4.1 (left Bousfield localisation), §11 (existence for cellular model categories); May-Ponto 2012 Ch. 19; Bousfield 1975 *The localization of spaces with respect to homology* §2-§3
master: Hirschhorn 2003 *Model Categories and Their Localizations* (AMS Mathematical Surveys 99) §3-§4 and §11-§13 (originator monograph for the existence theorem); Bousfield 1975 *The localization of spaces with respect to homology* (Topology 14) §2-§5 (originator for spaces); Bousfield 1979 *The localization of spectra with respect to homology* (Topology 18, 257-281) (originator for spectra); Bousfield-Friedlander 1978 *Homotopy theory of $\Gamma$-spaces, spectra and bisimplicial sets* (LNM 658) §A (Hammock localisation); Hovey 1999 *Model Categories* (AMS Mathematical Surveys 63) §3-§5; Goerss-Jardine 2009 *Simplicial Homotopy Theory* §X; Lurie 2009 *Higher Topos Theory* (Annals Studies 170) §5.5 ($\infty$-categorical reflective localisation); Ravenel 1984 *Localization with respect to certain periodic homology theories* (Amer. J. Math. 106) — chromatic origin

References

Hirschhorn — Model Categories and Their Localizations · AMS Mathematical Surveys and Monographs 99 (2003); §3 left Bousfield localisation overview; §4.1 definition; §9 detection of $S$-local equivalences via simplicial mapping spaces; §11-§13 existence theorem for cellular and combinatorial model categories; §17 application to spaces and spectra
Bousfield — The localization of spaces with respect to homology · Topology 14 (1975), pp. 133-150; the originator paper for localisation of $\mathbf{Top}_*$ at a homology theory; §2-§3 construction via the small-object argument applied to a set of generating $h$-equivalences; §4 the universal property; §5 worked examples for $H\mathbb{Q}$ and $H\mathbb{F}_p$
Bousfield — The localization of spectra with respect to homology · Topology 18 (1979), pp. 257-281; the stable analogue; the Bousfield class $\langle E \rangle$ and the Bousfield lattice; the foundational paper for chromatic localisation $L_{E(n)}$ and $L_{K(n)}$
Bousfield-Friedlander — Homotopy theory of Gamma-spaces, spectra and bisimplicial sets · Springer Lecture Notes in Mathematics 658 (1978), pp. 80-130; Appendix A: Hammock localisation, the simplicial-mapping-space presentation of derived $\mathrm{Hom}$ used to detect $S$-local equivalences
Hovey — Model Categories · AMS Mathematical Surveys and Monographs 63 (1999); §3 monoidal model categories; §5 stable model categories; §7 Bousfield localisation as a monoidal localisation; the $K(n)$-local stable homotopy category as a Bousfield localisation
Goerss-Jardine — Simplicial Homotopy Theory · Birkhäuser Progress in Mathematics 174 (1999, repr. 2009); §X.4 the Bousfield-Friedlander theorem for stable model structures; the construction of $f$-local model structures on $\mathbf{sSet}$ for a map $f$
Lurie — Higher Topos Theory · Princeton Annals of Mathematics Studies 170 (2009); §5.5 accessible reflective localisations of presentable $\infty$-categories; §5.2.7 Yoneda embedding; §5.5.4 accessible localisations classified by sets of morphisms; the $\infty$-categorical sibling of the Hirschhorn existence theorem
Ravenel — Localization with respect to certain periodic homology theories · American Journal of Mathematics 106 (1984), pp. 351-414; the chromatic-tower assembly using Bousfield localisation at $E(n)$ and $K(n)$; foundational for chromatic homotopy theory
May-Ponto — More Concise Algebraic Topology · University of Chicago Press 2012; Ch. 19 Bousfield localisation of model categories; the textbook treatment of the existence theorem in the left-proper combinatorial case; worked applications to rational and $p$-completion functors
Dwyer — Localizations · Axiomatic, Enriched and Motivic Homotopy Theory (NATO Sci. Ser. II 131, Kluwer 2004) pp. 3-28; pedagogical synthesis of left Bousfield localisation as a $\mathrm{Hom}$-equivalence inversion
Smith — Combinatorial model categories · Unpublished notes, 1998 (referenced in Beke 2000 *Sheafifiable homotopy model categories* Math. Proc. Cambridge Phil. Soc. 129); the combinatorial-model-category framework that Hirschhorn's existence theorem requires

Estimated time

beginner: 22m
intermediate: 50m
master: 100m