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7.1.3 Limits and Colimits

We are now ready to introduce the main objects of interest in this chapter.

Definition 7.1.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. A limit of $f$ is an object of $\operatorname{\mathcal{C}}$ which represents the right fibration $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$, in the sense of Definition 7.1.2.1. A colimit of $f$ is an object of $\operatorname{\mathcal{C}}$ which corepresents the left fibration $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}$, in the sense of Variant 7.1.2.2.

Remark 7.1.3.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then an object $X \in \operatorname{\mathcal{C}}$ is a limit of $f$ if and only if it is a colimit of the opposite diagram $f^{\operatorname{op}}: K^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$.

Example 7.1.3.3. Let $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a diagram of ordinary categories, which we regard as an object of the functor category $\operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})$, and let $f = \operatorname{N}_{\bullet }(F)$ be the induced functor from the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})$ to the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Applying Example 4.3.5.7, we obtain isomorphisms of simplicial sets

\[ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/f} \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/F}) \quad \quad \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{f/} \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{F/} ). \]

Combining this observation with Example 7.1.1.3, we see that:

  • An object of $\operatorname{\mathcal{C}}$ is the limit of the diagram of $\infty $-categories $f: \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ if and only if it is the image of a final object of the category $\operatorname{\mathcal{C}}_{/F}$: that is, if and only if it is the limit of the diagram or ordinary categories $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ (see Exercise 7.1.0.4).

  • An object of $\operatorname{\mathcal{C}}$ is the colimit of the diagram of $\infty $-categories $f: \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ if and only if it is the image of a initial object of the category $\operatorname{\mathcal{C}}_{F/}$: that is, if and only if it is the colimit of a diagram of ordinary categories $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$.

In other words, we can view Definition 7.1.3.1 as a generalization of Definition 7.1.0.1.

Example 7.1.3.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then an object $X \in \operatorname{\mathcal{C}}$ is initial (in the sense of Definition 7.1.1.1) if and only if it is a colimit of the empty diagram $\emptyset \hookrightarrow \operatorname{\mathcal{C}}$. Similarly, the object $X \in \operatorname{\mathcal{C}}$ is final if and only if it is a limit of the empty diagram.

Example 7.1.3.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$ and $Y$. The following conditions are equivalent:

  • The objects $X$ and $Y$ are isomorphic.

  • The object $Y$ is a limit of the diagram $\{ X \} \hookrightarrow \operatorname{\mathcal{C}}$.

  • The object $Y$ is a colimit of the diagram $\{ X\} \hookrightarrow \operatorname{\mathcal{C}}$.

See Example 7.1.3.15 for a more precise statement.

Remark 7.1.3.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let $X$ be an object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The object $X$ is a limit of the diagram $f$: that is, $X$ represents the right fibration $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$.

$(2)$

The object $X$ represents the right fibration $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ f \} $.

$(3)$

The object $X$ represents the right fibration $\operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/f}$. Here we abuse notation by identifying $f$ with a single object of the diagram $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$.

See Theorem 4.6.5.16 and Variant 4.6.5.21.

In general, limits and colimits are uniquely determined up to isomorphism when they exist:

Proposition 7.1.3.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then:

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

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

Proof. This is a special case of Proposition 7.1.2.5. $\square$

Notation 7.1.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. It follows from Proposition 7.1.3.7 that, if the diagram $f$ admits a limit $C$, then the isomorphic class of the object $C$ depends only on the diagram $f$. To emphasize this dependence, we will often denote $C$ by $\varprojlim (f)$ and refer to it as the limit of the diagram $f$. Similarly, if $f$ admits a colimit $C \in \operatorname{\mathcal{C}}$, we will often denote $C$ by $\varinjlim (f)$ and refer to it as the colimit of the diagram $f$. Beware that this terminology is somewhat abusive, since the objects $\varprojlim (f)$ and $\varinjlim (f)$ are only well-defined up to isomorphism.

We now show that, in situations where the limit $\varprojlim (f)$ and colimit $\varinjlim (f)$ are defined, they depend functorially on the diagram $f: K \rightarrow \operatorname{\mathcal{C}}$.

Definition 7.1.3.9. 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 $f: K \rightarrow \operatorname{\mathcal{C}}$, the slice $\infty $-category $\operatorname{\mathcal{C}}_{/f}$ has a final object. We will say that $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits if, for every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$, the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ has an initial object.

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 $C \in \operatorname{\mathcal{C}}$ to the constant diagram $\underline{C}: K \rightarrow \operatorname{\mathcal{C}}$ taking the value $\operatorname{\mathcal{C}}$.

Proposition 7.1.3.10. 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 $G$. In this condition is satisfied, then the right adjoint $G: \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ carries each diagram $f: K \rightarrow \operatorname{\mathcal{C}}$ to a limit $\varprojlim (f) \in \operatorname{\mathcal{C}}$.

  • 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 $f: K \rightarrow \operatorname{\mathcal{C}}$ to a colimit $\varinjlim (f) \in \operatorname{\mathcal{C}}$.

For many purposes, the language of Definition 7.1.3.1 is insufficiently precise. If $\operatorname{\mathcal{C}}$ is an $\infty $-category and $f: K \rightarrow \operatorname{\mathcal{C}}$ is a diagram, then the limit $\varprojlim (f)$ should not be viewed merely as an object of $\operatorname{\mathcal{C}}$ but as an object equipped with additional structure which relates it to the diagram $f$.

Definition 7.1.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets carrying the cone point of $K^{\triangleleft }$ to an object $Y \in \operatorname{\mathcal{C}}$. Set $f = \overline{f}|_{K}$, so that the diagram $\overline{f}$ can be identified with an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/f}$. We will say that $\overline{f}$ is a limit diagram if it is a final object of $\operatorname{\mathcal{C}}_{/f}$. If this condition is satisfied, we say that $\overline{f}$ exhibits $Y$ as a limit of the diagram $f$.

Variant 7.1.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets carrying the cone point of $K^{\triangleright }$ to an object $X \in \operatorname{\mathcal{C}}$. Set $f = \overline{f}|_{K}$, so that the diagram $\overline{f}$ can be identified with an object of the coslice $\infty $-category $\operatorname{\mathcal{C}}_{F/}$. We will say that $\overline{f}$ is a colimit diagram if it is an initial object of $\operatorname{\mathcal{C}}_{f/}$. If this condition is satisfied, we say that $\overline{f}$ exhibits $X$ as a colimit of the diagram $f$.

Remark 7.1.3.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then an object $Y \in \operatorname{\mathcal{C}}$ is a limit of $f$ if and only if there exists a diagram $K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ which exhibits $Y$ as a limit of $f$, in the sense of Definition 7.1.3.11. Similarly, an object $X \in \operatorname{\mathcal{C}}$ is a colimit of $f$ if and only if there exists a diagram $K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which exhibits $X$ as a colimit of $f$.

Example 7.1.3.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then an object $X \in \operatorname{\mathcal{C}}$ is initial (in the sense of Definition 7.1.1.1) if and only if the map

\[ (\emptyset )^{\triangleright } \simeq \Delta ^0 \xrightarrow {C} \operatorname{\mathcal{C}} \]

is a colimit diagram in $\operatorname{\mathcal{C}}$. Stated more informally, initial objects of $\operatorname{\mathcal{C}}$ can be identified with colimits of the empty diagram $\emptyset \rightarrow \operatorname{\mathcal{C}}$. Similarly, final objects of $\operatorname{\mathcal{C}}$ can be identified with limits of the empty diagram $\emptyset \rightarrow \operatorname{\mathcal{C}}$.

Example 7.1.3.15. 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).

  • When regarded as a morphism of simplicial sets from $\Delta ^{1} \simeq ( \Delta ^0 )^{\triangleleft }$ to $\operatorname{\mathcal{C}}$, $f$ is a limit diagram. That is, it exhibits $X$ as a limit of the inclusion map $\{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$.

  • When regarded as a morphism of simplicial sets from $\Delta ^{1} \simeq ( \Delta ^0 )^{\triangleright }$ to $\operatorname{\mathcal{C}}$, $f$ is a colimit diagram. That is, it exhibits $Y$ as a colimit of the inclusion map $\{ X\} \hookrightarrow \operatorname{\mathcal{C}}$.

This is a reformulation of Proposition 7.1.2.8.

Remark 7.1.3.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. An extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$ if and only if the opposite map $\overline{f}^{\operatorname{op}}: (K^{\operatorname{op}})^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. In particular, a colimit of the diagram $f: K \rightarrow \operatorname{\mathcal{C}}$ (if it exists) can be identified with a limit of the diagram $f^{\operatorname{op}}: K^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$.

Remark 7.1.3.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $g: B \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and suppose we are given a diagram $\overline{f}: A^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}_{/g}$, which we can identify with a morphism of simplicial sets

\[ \overline{q}: (A \star B)^{\triangleleft } \simeq A^{\triangleleft } \star B \rightarrow \operatorname{\mathcal{C}}. \]

Then $\overline{f}$ is a limit diagram in the slice $\infty $-category $\operatorname{\mathcal{C}}_{/g}$ if and only if $\overline{q}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$.

Proposition 7.1.3.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism with restriction $f = \overline{f}|_{K}$. The following conditions are equivalent:

$(1)$

The morphism $\overline{f}$ is a limit diagram (Definition 7.1.3.11).

$(2)$

The restriction map $\operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \operatorname{\mathcal{C}}_{/f}$ is a trivial Kan fibration.

$(3)$

The restriction map $\operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \operatorname{\mathcal{C}}_{/f}$ is an equivalence of $\infty $-categories.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 7.1.1.9. Note that the restriction map $\operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \operatorname{\mathcal{C}}_{/f}$ is a right fibration of $\infty $-categories (Corollary 4.3.6.9), and therefore an isofibration (Example 4.4.1.10). The equivalence $(2) \Leftrightarrow (3)$ now follows from Proposition 4.5.7.14. $\square$