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5.3 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 5.3.0.1. 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 5.3.0.2 (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 $\operatorname{\mathcal{C}}[W^{-1}]$ and a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ which exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as a strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Moreover, the category $\operatorname{\mathcal{C}}[W^{-1}]$ (and the functor $F$) are determined uniquely up to isomorphism. In what follows, we will sometimes abuse terminology by referring to $\operatorname{\mathcal{C}}[W^{-1}]$ as the strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Explicitly, the category $\operatorname{\mathcal{C}}[W^{-1}]$ 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 $\operatorname{Ob}(\operatorname{\mathcal{C}}) \simeq \operatorname{Ob}(\operatorname{\mathcal{C}}[W^{-1}])$.

Warning 5.3.0.3. 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 $\operatorname{\mathcal{C}}[W^{-1}]$ is also small. Beware that if $\operatorname{\mathcal{C}}$ is only assumed to be locally small (Definition ), then $\operatorname{\mathcal{C}}[W^{-1}]$ need not be locally small. However, one can often ensure that $\operatorname{\mathcal{C}}[W^{-1}]$ is small by imposing additional assumptions on the collection of morphisms $W$: we will return to this point (in the setting of $\infty $-categories) in Chapter .

Remark 5.3.0.4. Let $\operatorname{\mathcal{C}}$ be a category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ be a functor which exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as a strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Then, for every category $\operatorname{\mathcal{E}}$, the precomposition functor $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ induces an isomorphism from $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$ 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

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

Beware that Definition 5.3.0.1 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 $\operatorname{\mathcal{C}}[W^{-1}]$, 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 5.3.0.5. 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 classical 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 5.3.0.6. 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 classical localization of $\operatorname{\mathcal{C}}$ with respect to $W$ (see Remark 5.3.0.4). The converse is false (except in the trivial case where $\operatorname{\mathcal{C}}$ is empty).

Remark 5.3.0.7. Let $\operatorname{\mathcal{C}}$ be a category, let $W$ be a collection of morphisms in $\operatorname{\mathcal{C}}$, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ be a functor which exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ 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 classical 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 $\operatorname{\mathcal{C}}$, and therefore factors uniquely as a composition $\operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{C}}[W^{-1}] \xrightarrow {G'} \operatorname{\mathcal{D}}$.

  • The functor $G': \operatorname{\mathcal{C}}[W^{-1}] \rightarrow \operatorname{\mathcal{D}}$ 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 §5.3.1 by introducing an $\infty $-categorical counterpart of Definition 5.3.0.5. 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 5.3.1.9). In §5.3.2, we show that such a localization always exists (Proposition 5.3.2.1) and is uniquely determined up to equivalence (Remark 5.3.2.2).

Warning 5.3.0.8. One could also consider a version of Definition 5.3.0.1 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

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

Beware that this definition has very limited utility: one can show that $\operatorname{\mathcal{C}}$ 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 issue is that if $u: X \rightarrow Y$ is an isomorphism in an $\infty $-category, then the inverse isomorphism $u^{-1}: Y \rightarrow X$ is only well-defined up to homotopy (or at best up to a contractible space of choices), in contrast with classical category theory where the inverse isomorphism $u^{-1}$ is uniquely determined.

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 5.3.2.1, 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 classical localization of $\operatorname{\mathcal{C}}$ with respect to $W$, in the sense of Definition 5.3.0.5 (Example 5.3.1.16). 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 show in §5.3.4 that every $\infty $-category $\operatorname{\mathcal{D}}$ can be obtained by localizing (the nerve of) an ordinary category with respect to a suitably chosen collection of morphisms (Proposition 5.3.4.1). The proof will make use of some basic stability properties for the class of localizations, which we establish in §5.3.3.

Warning 5.3.0.9. Throughout most of this book, we will often employ the following conventions:

  • If $\operatorname{\mathcal{C}}$ is an $\infty $-category and $W$ is a collection of morphisms of $\operatorname{\mathcal{C}}$, then we let $\operatorname{\mathcal{C}}[W^{-1}]$ denote a localization of $\operatorname{\mathcal{C}}$ with respect to $W$ (Remark 5.3.2.2).

  • If $\operatorname{\mathcal{C}}$ is an ordinary category, we abuse terminology by identifying $\operatorname{\mathcal{C}}$ with the associated $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.

Beware that these conventions are in conflict with one another. If $W$ is a collection of morphisms in an ordinary category $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}[W^{-1}]$ denotes the strict localization of Remark 5.3.0.2, then the $\infty $-categorical localization $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})[W^{-1}]$ is generally not equivalent to the nerve $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}[W^{-1}] )$. To avoid confusion, we will henceforth use the notation $\operatorname{\mathcal{C}}[W^{-1}]$ only for the $\infty $-categorical notion of localization, unless otherwise specified.

Structure

  • Subsection 5.3.1: Localizations of $\infty $-Categories
  • Subsection 5.3.2: Existence of Localizations
  • Subsection 5.3.3: Stability Properties of Localizations
  • Subsection 5.3.4: The Category of Simplices