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6.3 Localization

Let $\operatorname{\mathcal{C}}$ be a category and let $W$ be a collection of morphisms in $\operatorname{\mathcal{C}}$. One can then construct a new category by formally adjoining an inverse to each morphism of $W$.

Definition 6.3.0.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We say that $F$ exhibits $\operatorname{\mathcal{D}}$ as a strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$ if, for every category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a bijection

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$} \} \ar [d] \\ \{ \textnormal{Functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ carrying each $w \in W$ to an isomorphism in $\operatorname{\mathcal{E}}$} \} .} \]

Remark 6.3.0.2 (Existence and Uniqueness). Let $\operatorname{\mathcal{C}}$ be a category and let $W$ be a collection of morphisms in $\operatorname{\mathcal{C}}$. Then there exists a category $W^{-1} \operatorname{\mathcal{C}}$ and a functor $F: \operatorname{\mathcal{C}}\rightarrow W^{-1} \operatorname{\mathcal{C}}$ which exhibits $W^{-1} \operatorname{\mathcal{C}}$ as a strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Moreover, the category $W^{-1} \operatorname{\mathcal{C}}$ is determined uniquely up to isomorphism. In what follows, we will sometimes abuse terminology by referring to $W^{-1} \operatorname{\mathcal{C}}$ as the strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Explicitly, the category $W^{-1} \operatorname{\mathcal{C}}$ can be constructed from $\operatorname{\mathcal{C}}$ by adjoining a new morphism $w^{-1}: Y \rightarrow X$ for each morphism $w: X \rightarrow Y$ of $W$, and imposing the relations $w^{-1} \circ w = \operatorname{id}_{X}$ and $w \circ w^{-1} = \operatorname{id}_{Y}$. From this description, we see that the functor $F$ induces a bijection $\operatorname{Ob}(\operatorname{\mathcal{C}}) \simeq \operatorname{Ob}(W^{-1} \operatorname{\mathcal{C}})$.

Example 6.3.0.3. Let $\operatorname{Kan}$ denote the category of Kan complexes and let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes (Construction 3.1.5.10). Then the quotient functor $\operatorname{Kan}\rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ exhibits $\mathrm{h} \mathit{\operatorname{Kan}}$ as a strict localization of $\operatorname{Kan}$ with respect to the collection of all homotopy equivalences (see Corollary 3.1.7.7).

Warning 6.3.0.4. Let $\operatorname{\mathcal{C}}$ be a category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. If $\operatorname{\mathcal{C}}$ is small (Definition ), then the strict localization $W^{-1} \operatorname{\mathcal{C}}$ is also small. Beware that if $\operatorname{\mathcal{C}}$ is only assumed to be locally small (Definition ), then $W^{-1} \operatorname{\mathcal{C}}$ need not be locally small. However, one can often ensure that $W^{-1} \operatorname{\mathcal{C}}$ is small by imposing additional assumptions on the collection of morphisms $W$.

Remark 6.3.0.5. Let $\operatorname{\mathcal{C}}$ be a category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $F: \operatorname{\mathcal{C}}\rightarrow W^{-1} \operatorname{\mathcal{C}}$ be a functor which exhibits $W^{-1} \operatorname{\mathcal{C}}$ as a strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Then, for every category $\operatorname{\mathcal{E}}$, the precomposition functor $\operatorname{Fun}( W^{-1} \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ induces an isomorphism from $\operatorname{Fun}( W^{-1} \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ to the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ spanned by those functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ which carry each element $w \in W$ to an isomorphism in $\operatorname{\mathcal{E}}$. Bijectivity at the level of objects follows immediately from the definition. At the level of morphisms, it follows from the bijectivity of the map

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors $W^{-1} \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}([1],\operatorname{\mathcal{E}})$} \} \ar [d] \\ \{ \textnormal{Functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}([1],\operatorname{\mathcal{E}})$ carrying $W$ to isomorphisms} \} .} \]

Beware that Definition 6.3.0.1 is not invariant under equivalence. If $\operatorname{\mathcal{C}}$ is a category, $W$ is a collection of morphisms in $\operatorname{\mathcal{C}}$, and $\operatorname{\mathcal{D}}$ is a category which is equivalent but not isomorphic to the strict localization $W^{-1} \operatorname{\mathcal{C}}$, then $\operatorname{\mathcal{D}}$ is not a strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$. We can remedy the situation by introducing a more liberal notion of localization.

Definition 6.3.0.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We will say that $F$ exhibits $\operatorname{\mathcal{D}}$ as a $1$-categorical localization of $\operatorname{\mathcal{C}}$ with respect to $W$ if, for every category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a fully faithful functor $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$, whose essential image consists of those functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ which carry each $w \in W$ to an isomorphism in $\operatorname{\mathcal{E}}$.

Example 6.3.0.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. If $F$ exhibits $\operatorname{\mathcal{D}}$ as a strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$, then $F$ exhibits $\operatorname{\mathcal{D}}$ as a $1$-categorical localization of $\operatorname{\mathcal{C}}$ with respect to $W$ (see Remark 6.3.0.5). The converse is false (except in the trivial case where $\operatorname{\mathcal{C}}$ is empty).

Example 6.3.0.8. Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets, and let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes (Construction 3.1.5.10). Then the fibrant replacement functor $\operatorname{Ex}^{\infty }: \operatorname{Set_{\Delta }}\rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ exhibits $\mathrm{h} \mathit{\operatorname{Kan}}$ as a $1$-categorical localization of $\operatorname{Set_{\Delta }}$ with respect to the collection $W$ of weak homotopy equivalences (see Variant 3.1.7.8). However, it does not exhibit $\mathrm{h} \mathit{\operatorname{Kan}}$ as a strict localization of $\operatorname{Set_{\Delta }}$ with respect to $W$ (since it is not bijective on objects).

Remark 6.3.0.9. Let $\operatorname{\mathcal{C}}$ be a category, let $W$ be a collection of morphisms in $\operatorname{\mathcal{C}}$, and let $F: \operatorname{\mathcal{C}}\rightarrow W^{-1} \operatorname{\mathcal{C}}$ be a functor which exhibits $W^{-1} \operatorname{\mathcal{C}}$ as a strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be another functor. Then $G$ exhibits $\operatorname{\mathcal{D}}$ as a $1$-categorical localization of $\operatorname{\mathcal{C}}$ with respect to $W$ if and only if the following conditions are satisfied:

  • The functor $G$ carries each $w \in W$ to an isomorphism in $\operatorname{\mathcal{D}}$, and therefore factors uniquely as a composition $\operatorname{\mathcal{C}}\xrightarrow {F} W^{-1} \operatorname{\mathcal{C}}\xrightarrow {G'} \operatorname{\mathcal{D}}$.

  • The functor $G': W^{-1} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of categories.

Our goal in this section is to adapt the notion of localization to the setting of $\infty $-categories. We begin in §6.3.1 by introducing an $\infty $-categorical counterpart of Definition 6.3.0.6. Given an $\infty $-category $\operatorname{\mathcal{C}}$ and a collection $W$ of morphisms of $\operatorname{\mathcal{C}}$, we say that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$ if, for every $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a fully faithful functor of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \xrightarrow {\circ F} \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$, whose essential image consists of those functors which carry each element of $W$ to an isomorphism in $\operatorname{\mathcal{E}}$ (Definition 6.3.1.9). In §6.3.2, we show that such a localization always exists (Proposition 6.3.2.1) and is uniquely determined up to equivalence (Remark 6.3.2.2); we will often emphasize this uniqueness by denoting the $\infty $-category $\operatorname{\mathcal{D}}$ by $\operatorname{\mathcal{C}}[W^{-1}]$.

Let $\operatorname{\mathcal{C}}$ be an ordinary category, and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Then $W$ can also be regarded as a collection of morphisms of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. By virtue of Proposition 6.3.2.1, there exists a functor of $\infty $-categories $F: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ with respect to $W$. In this case, it is not hard to see that the induced map $\operatorname{\mathcal{C}}\simeq \mathrm{h} \mathit{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \xrightarrow { \mathrm{h} \mathit{F} } \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ exhibits the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ as a $1$-categorical localization of $\operatorname{\mathcal{C}}$ with respect to $W$, in the sense of Definition 6.3.0.6 (Example 6.3.1.16). Beware that, in this situation, the unit map $\operatorname{\mathcal{D}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{D}}} )$ is generally not an equivalence. In other words, the formation of localizations (in the $\infty $-categorical setting) generally does not carry ordinary categories to ordinary categories, even up to equivalence. In fact, we prove in §6.3.7 that every $\infty $-category $\operatorname{\mathcal{D}}$ can be obtained by localizing (the nerve of) a partially ordered set (Theorem 6.3.7.1). The proof will make use of some basic stability properties for the class of localizations, which we establish in §6.3.4.

In general, it is very difficult to give an explicit description of the localization of an $\infty $-category $\operatorname{\mathcal{C}}$ with respect to a class of morphisms $W$. In §6.3.3, we study a special case in which such a description is available. We will say that a localization functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ is reflective if it admits a right adjoint. In this case, the right adjoint $G: \operatorname{\mathcal{C}}[W^{-1}] \rightarrow \operatorname{\mathcal{C}}$ is automatically fully faithful, and its essential image is a reflective subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ (Proposition 6.3.3.13). In this case, we can identify $\operatorname{\mathcal{C}}[W^{-1}]$ with $\operatorname{\mathcal{C}}'$, which can be characterized as the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects (Definition 6.3.3.1). Reflective localizations are extremely common in practice, and will play a central role in the theory of locally presentable $\infty $-categories which we develop in §.

Warning 6.3.0.10. It also is possible to contemplate a version of Definition 6.3.0.1 in the $\infty $-categorical setting. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Let us say that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ exhibits $\operatorname{\mathcal{D}}$ as a strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$ if, for every $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a bijection

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$} \} \ar [d] \\ \{ \textnormal{Functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ carrying each $w \in W$ to an isomorphism in $\operatorname{\mathcal{E}}$} \} }. \]

However, this definition is useless. One can show that an $\infty $-category $\operatorname{\mathcal{C}}$ admits a strict localization with respect to $W$ only in the trivial case where every element of $W$ is already an isomorphism in $\operatorname{\mathcal{C}}$ (in which case we can take $F$ to be the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$). Roughly speaking, the problem is that if $w: X \rightarrow Y$ is an isomorphism in an $\infty $-category $\operatorname{\mathcal{C}}$, then the homotopy inverse isomorphism $w^{-1}: Y \rightarrow X$ is only well-defined up to homotopy (or up to a contractible space of choices), in contrast with classical category theory where the inverse isomorphism $w^{-1}$ is unique.

Structure

  • Subsection 6.3.1: Localizations of $\infty $-Categories
  • Subsection 6.3.2: Existence of Localizations
  • Subsection 6.3.3: Reflective Localizations
  • Subsection 6.3.4: Stability Properties of Localizations
  • Subsection 6.3.5: Fiberwise Localization
  • Subsection 6.3.6: Universal Localizations
  • Subsection 6.3.7: Subdivision and Localization