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7.1.1 Limits and Colimits in $\infty $-Categories

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. For each object $X \in \operatorname{\mathcal{C}}$, we let $\underline{X} \in \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ denote the constant diagram $K \rightarrow \{ X\} \hookrightarrow \operatorname{\mathcal{C}}$. Note that the construction $X \mapsto \underline{X}$ determines a functor of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}})$, carrying each morphism $f: X \rightarrow Y$ to a natural transformation $\underline{f}: \underline{X} \rightarrow \underline{Y}$.

Definition 7.1.1.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $Y$, let $K$ be a simplicial set, and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. We say that a natural transformation $\alpha : \underline{Y} \rightarrow u$ exhibits $Y$ as a limit of $u$ if the following condition is satisfied:

$(\ast )$

For each object $X \in \operatorname{\mathcal{C}}$, the composition

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})}( \underline{X}, \underline{Y} ) \xrightarrow { [ \alpha ] \circ } \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}})}( \underline{X}, u ) \]

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, where the second map is described in Notation 4.6.7.15.

We will say that a natural transformation $\beta : u \rightarrow \underline{y}$ exhibits $Y$ as a colimit of $u$ if the following dual condition is satisfied:

$(\ast ')$

For each object $Z \in \operatorname{\mathcal{C}}$, the composition

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, Z) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})}( \underline{Y}, \underline{Z} ) \xrightarrow { \circ [ \alpha ]} \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}})}( u, \underline{Z} ) \]

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

Remark 7.1.1.2. Stated more informally, a natural transformation $\alpha : \underline{Y} \rightarrow u$ exhibits $Y$ as a limit of $u$ if and only if postcomposition with $\alpha $ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})}( \underline{X}, u )$ for each object $X \in \operatorname{\mathcal{C}}$. Similarly, a natural transformation $\beta : u \rightarrow \underline{Y}$ exhibits $Y$ as a colimit of $u$ if and only if precomposition with $\beta $ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})}( u, \underline{Z} )$ for each object $Z \in \operatorname{\mathcal{C}}$.

Remark 7.1.1.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $Y$ and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then a natural transformation $\alpha : \underline{Y} \rightarrow u$ exhibits $Y$ as a limit of $u$ if and only if exhibits $Y$ as a colimit of the induced diagram $u^{\operatorname{op}}: K^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$, when regarded as a morphism in the $\infty $-category $\operatorname{Fun}( K^{\operatorname{op}}, \operatorname{\mathcal{C}}^{\operatorname{op}} ) \simeq \operatorname{Fun}(K, \operatorname{\mathcal{C}})^{\operatorname{op}}$.

Example 7.1.1.4. Let $\operatorname{\mathcal{C}}$ be an ordinary category, let $K$ be a simplicial set, and suppose we are given a diagram $u: K \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, which we can identify with a functor of ordinary categories $U: \mathrm{h} \mathit{K} \rightarrow \operatorname{\mathcal{C}}$ (see Proposition 1.3.5.7). If $Y$ is an object of $\operatorname{\mathcal{C}}$, then we can use Corollary 1.4.3.5 to identify natural transformations $\underline{Y} \rightarrow u$ (of diagrams in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$) with natural transformations $\underline{Y} \rightarrow U$ (of diagrams in the ordinary category $\operatorname{\mathcal{C}}$). Under this identification, a natural transformation $\underline{Y} \rightarrow u$ exhibits $Y$ as a limit of $u$ (in the $\infty $-categorical sense of Definition 7.1.1.1) if and only if it exhibits $Y$ as a limit of $U$ (in the classical sense of Definition 7.1.0.1).

Example 7.1.1.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

  • The morphism $f$ is an isomorphism from $X$ to $Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (Definition 1.3.6.1).

  • The morphism $f$ exhibits $X$ as a limit of the diagram $\{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$.

  • The morphism $f$ exhibits $Y$ as a colimit of the diagram $\{ X\} \hookrightarrow \operatorname{\mathcal{C}}$.

Remark 7.1.1.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let $Y \in \operatorname{\mathcal{C}}$ be an object. If $\alpha : \underline{Y} \rightarrow u$ is a natural transformation, then the condition that $\alpha $ exhibits $Y$ as a limit of $u$ depends only on its homotopy class $[\alpha ]$ (as a morphism in the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$). Similarly, if $\beta : u \rightarrow \underline{Y}$ is a natural transformation, then the condition that $\beta $ exhibits $Y$ as a colimit of $u$ depends only on its homotopy class $[\beta ]$.

Remark 7.1.1.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $Y$, let $K$ be a simplicial set, and let $\beta : u \rightarrow u'$ be an isomorphism in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Suppose we are given a natural transformation $\alpha : \underline{Y} \rightarrow u$, and let $\alpha ': \underline{Y} \rightarrow u'$ be any composition of $\alpha $ with $\beta $. Then $\alpha $ exhibits $Y$ as a limit of $u$ if and only if $\alpha '$ exhibits $Y$ as a limit of $u$. Similarly, if $\gamma ': u' \rightarrow \underline{Y}$ is a natural transformation and $\gamma : u \rightarrow \underline{Y}$ is a composition of $\beta $ with $\gamma '$, then $\gamma $ exhibits $Y$ as a colimit of $u$ if and only if $\gamma '$ exhibits $Y$ as a colimit of $u'$.

Remark 7.1.1.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. Suppose we are given a natural transformation of diagrams $\beta : \underline{Y} \rightarrow u$, and let $\alpha : \underline{X} \rightarrow u$ be a composition of $\beta $ with the constant natural transformation $\underline{f}: \underline{X} \rightarrow \underline{Y}$. Then any two of the following three properties imply the third:

  • The natural transformation $\alpha $ exhibits $X$ as a limit of the diagram $u$.

  • The natural transformation $\beta $ exhibits $Y$ as a limit of the diagram $u$.

  • The morphism $f: X \rightarrow Y$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.

Remark 7.1.1.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories, let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let $Y \in \operatorname{\mathcal{C}}$ be an object equipped with a natural transformation $\alpha : \underline{Y} \rightarrow u$. If $F(\alpha ): \underline{F(Y)} \rightarrow (F \circ u)$ exhibits $F(Y)$ as a limit of the diagram $(F \circ u): K \rightarrow \operatorname{\mathcal{D}}$, then $\alpha $ exhibits $Y$ as a limit of $u$. The converse holds if $F$ is an equivalence of $\infty $-categories.

Definition 7.1.1.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. We say that an object $Y \in \operatorname{\mathcal{C}}$ is a limit of $u$ if there exists a natural transformation $\alpha : \underline{Y} \rightarrow u$ which exhibits $Y$ as a limit of $u$, in the sense of Definition 7.1.1.1. We say that $Y$ is a colimit of $u$ if there exists a natural transformation $\beta : u \rightarrow \underline{Y}$ which exhibits $Y$ as a colimit of $u$.

Proposition 7.1.1.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then:

  • Suppose that the diagram $u$ has limit $Y \in \operatorname{\mathcal{C}}$. Then an object $X \in \operatorname{\mathcal{C}}$ is a limit of $u$ if and only if it is isomorphic to $Y$.

  • Suppose that the diagram $u$ has colimit $Y \in \operatorname{\mathcal{C}}$. Then an object $X \in \operatorname{\mathcal{C}}$ is a colimit of $u$ if and only if it is isomorphic to $Y$.

Proof. Let $\beta : \underline{Y} \rightarrow u$ be a natural transformation which exhibits $Y$ as a limit of the diagram $u$. For any object $X$ and any natural transformation $\alpha : \underline{X} \rightarrow u$, there exists a morphism $f: X \rightarrow Y$ such that $\alpha $ is a composition of $\beta $ with the constant natural transformation $\underline{f}: \underline{X} \rightarrow \underline{Y}$. If $\alpha $ also exhibits $X$ as a limit of the diagram $u$, then $f$ is an isomorphism (Remark 7.1.1.8); in particular, $X$ is isomorphic to $Y$. Conversely, if $f: X \rightarrow Y$ is an isomorphism, then any composition of $\underline{f}$ with $\beta $ is a natural transformation $\underline{X} \rightarrow u$ which exhibits $X$ as a limit of $u$ (Remark 7.1.1.8), so that $X$ is a limit of $Y$. This proves the first assertion; the proof of the second follows by applying the same argument to the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. $\square$

Notation 7.1.1.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. It follows from Proposition 7.1.1.11 that, if the diagram $f$ admits a limit $Y$, then the isomorphism class of the object $Y$ depends only on the diagram $u$. To emphasize this dependence, we will often denote $Y$ by $\varprojlim (u)$ and refer to it as the limit of the diagram $u$. Similarly, if $u$ admits a colimit $X \in \operatorname{\mathcal{C}}$, we will often denote $X$ by $\varinjlim (u)$ and refer to it as the colimit of the diagram $u$. Beware that this terminology is somewhat abusive, since the objects $\varprojlim (u)$ and $\varinjlim (u)$ are only well-defined up to isomorphism.

In situations where the limit $\varprojlim (u)$ and colimit $\varinjlim (u)$ are defined, they depend functorially on the diagram $u: K \rightarrow \operatorname{\mathcal{C}}$.

Definition 7.1.1.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. We will say that $\operatorname{\mathcal{C}}$ admits $K$-indexed limits if, for every diagram $u: K \rightarrow \operatorname{\mathcal{C}}$, there exists an object $Y \in \operatorname{\mathcal{C}}$ which is a limit of $u$. We will say that $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits if, for every diagram $u: K \rightarrow \operatorname{\mathcal{C}}$, there exists an object $X \in \operatorname{\mathcal{C}}$ which is a colimit of $u$.

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every simplicial set $K$, precomposition with the projection map $K \rightarrow \Delta ^0$ determines a functor

\[ \delta : \operatorname{\mathcal{C}}\simeq \operatorname{Fun}( \Delta ^0, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( K, \operatorname{\mathcal{C}}). \]

We will refer to $\delta $ as the diagonal functor: it carries each object $X \in \operatorname{\mathcal{C}}$ to the constant diagram $\underline{X}: K \rightarrow \operatorname{\mathcal{C}}$ taking the value $X$.

Proposition 7.1.1.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. Then:

  • The $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed limits if and only if the diagonal functor $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ admits a right adjoint $G$. If this condition is satisfied, then the right adjoint $G: \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ carries each diagram $u: K \rightarrow \operatorname{\mathcal{C}}$ to a limit $\varprojlim (u) \in \operatorname{\mathcal{C}}$.

  • The $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits if and only if the diagonal functor $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ admits a left adjoint $F$. In this condition is satisfied, then the left adjoint $F: \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ carries each diagram $u: K \rightarrow \operatorname{\mathcal{C}}$ to a colimit $\varinjlim (u) \in \operatorname{\mathcal{C}}$.

Proof. Apply Corollary 6.2.4.7. $\square$