Bousfield localizations provide the categorical language for reflective subcategories. This section introduces them, gives basic examples, and proves the recognition criterion of Proposition 21.8.9.
Definition 21.8.1. A functor \(L\colon C \to D\) is called a Bousfield localization if it admits a fully faithful right adjoint \(R\colon D \hookrightarrow C\). In this case, we also call \(D\) a reflective subcategory of \(C\).
A Bousfield colocalization is instead a functor \(C \to D\) that admits a fully faithful left adjoint.
Example 21.8.2 (Rationalization). The inclusion \(\Vect _{\Q } \hookrightarrow \Ab \) of rational vector spaces into abelian groups admits a left adjoint \[ \Q \otimes _{\Z } -\colon \Ab \to \Vect _{\Q }, \] called the rationalization functor. It is a Bousfield localization.
Example 21.8.3 (Abelianization). The inclusion \(\Ab \hookrightarrow \Grp \) of abelian groups into all groups admits a left adjoint \[ (-)^{\mathrm {ab}}\colon \Grp \to \Ab , \] given by the abelianization functor.
Example 21.8.4 (Group completion). The inclusion \(\Grp \hookrightarrow \Mon \) of groups into monoids admits a left adjoint \[ (-)^{\grp }\colon \Mon \to \Grp \] called the (discrete) group completion functor.
Example 21.8.5 (Path components). The inclusion \(\Set \hookrightarrow \An \) admits a left adjoint \[ \pi _0\colon \An \to \Set \] given by the set of path components.
Example 21.8.6 (Geometric realization). The inclusion \(\An \hookrightarrow \Cat _{\infty }\) has a left adjoint \[ \geom {-}\colon \Cat _{\infty } \to \An \] given by the geometric realization functor. It also admits a right adjoint \[ (-)^{\simeq }\colon \Cat _{\infty } \to \An \] given by the groupoid core functor.
Example 21.8.7 (Homotopy category). The inclusion \(\Cat \hookrightarrow \Cat _{\infty }\) has a left adjoint \[ \Ho (-)\colon \Cat _{\infty } \to \Cat _1 \] given by the homotopy category functor.
The terminology is justified by the fact that every Bousfield (co)localization is a localization in the sense of Definition 1.5.19:
Lemma 21.8.8. Let \(L\colon E \to C\) be a Bousfield localization, and let \(W := L^{-1}(C^{\simeq })\) be the class of morphisms in \(E\) inverted by \(L\). Then \(L\) exhibits \(C\) as a localization of \(E\) at \(W\). The same conclusion holds if \(L\) is a Bousfield colocalization.
Proof. See Reference ? of [Cisinski et al. (2026)]. □
Here is a useful criterion for checking that a full subcategory is a reflective subcategory:
Proposition 21.8.9 (cf. [Lurie (2009), Proposition 5.2.7.4]). Let \(L\colon C \to C\) be a functor equipped with a natural transformation \(\eta \colon \id _C \Rightarrow L\) such that both maps \[ \eta _{Lx} \colon Lx \to LLx \qquadtext { and } L\eta _x\colon Lx \to LLx \] are isomorphisms for all \(x \in C\). Then the functor \(L\colon C \to \im (L)\) is left adjoint to the inclusion \(\im (L) \hookrightarrow C\), with unit \(\eta \). In particular, \(L\colon C \to \im (L)\) is a Bousfield localization.
Proof. See Reference ? of [Cisinski et al. (2026)]. □
Exercises
Exercise 21.1 (Composition of adjunctions). Suppose that \(F\colon C\rightleftarrows D\noloc G\) and \(F'\colon D\rightleftarrows E\noloc G'\) are adjunctions. Construct the unit and counit of the composite adjunction \[ F'F\colon C\rightleftarrows E\noloc GG' \] from the units and counits of the given adjunctions, and verify the triangle identities.
Exercise 21.2 (Kan extension along an endpoint). Let \(i\colon \{0\}\hookrightarrow [1]\) and let \(x\in C\), where \(C\) has an initial and a terminal object. Compute both \(i_!(x)\) and \(i_*(x)\) as arrows in \(C\). Identify the unit and counit maps, and repeat the calculation for the inclusion \(\{1\}\hookrightarrow [1]\).
Exercise 21.3 (Cofinal subsequences). Let \(u\colon \N \to \N \) be an increasing and unbounded function, regarded as a functor between posets. Show that \(u\) is final. Deduce that for every sequence \[ X_0\longrightarrow X_1\longrightarrow X_2\longrightarrow \dots \] in an \(\infty \)-category, passing to a cofinal subsequence does not change its colimit. In particular, compare the colimit of this sequence with the colimit of the subsequence \((X_{2n})_{n\geq 0}\).
Exercise 21.4 (A Bousfield localization of abelian groups). Fix a prime \(p\) and consider \[ L\colon \Ab \longrightarrow \Ab , \qquad A\longmapsto A\otimes _{\Z }\Z [1/p]. \] Show that the unit \(A\to L(A)\) exhibits \(L\) as a Bousfield localization. Identify the local abelian groups and the \(L\)-equivalences.
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