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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, 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 (Variant 4.7.8.6), then $W^{-1} \operatorname{\mathcal{C}}$ need not be locally small. However, one can often ensure that $W^{-1} \operatorname{\mathcal{C}}$ is locally 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 6.3.1.19). 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 §6.3.6 that every $\infty $-category $\operatorname{\mathcal{D}}$ can be obtained by localizing (the nerve of) a partially ordered set (Theorem 6.3.6.1).

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 6.3.3.6). 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: Fiberwise Localization
  • Subsection 6.3.5: Universal Localizations
  • Subsection 6.3.6: Subdivision and Localization