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7.1.4 Limits and Colimit Diagrams

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}})} \{ u\} $ denote the oriented fiber product of Construction 4.6.4.1. By definition, we can identify objects of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ u\} $ with pairs $(Y, \alpha )$, where $Y$ is an object of $\operatorname{\mathcal{C}}$ and $\alpha : \underline{Y} \rightarrow u$ is a natural transformation (here $\underline{Y}$ denotes the constant diagram $K \rightarrow \{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$). Using Proposition 7.1.3.12, we can reformulate Definition 7.1.1.1 as follows:

Proposition 7.1.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $Y$, let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let $\underline{Y}: K \rightarrow \operatorname{\mathcal{C}}$ denote the constant taking the value. Then:

  • A natural transformation $\alpha : \underline{Y} \rightarrow u$ exhibits $Y$ as a limit of the diagram $u$ if and only if it is final when regarded as an object of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ u\} $.

  • A natural transformation $\beta : u \rightarrow \underline{Y}$ exhibits $Y$ as a colimit of the diagram $u$ if and only if it is initial when regarded as an object of the oriented fiber product $\{ u\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$.

Proof. We will prove the first assertion; the second follows by a similar argument. Projection onto the first factor determines a right fibration $\theta : \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ u\} \rightarrow \operatorname{\mathcal{C}}$. For each object $X \in \operatorname{\mathcal{C}}$, we can identify $\theta ^{-1}(X)$ with the morphism space $\operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}})}( \underline{X}, u )$. Let

\[ \rho _{X}: \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}})}( \underline{Y}, u) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Y) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) }( \underline{X}, u) \]

be the parametrized contravariant transport map of Variant 5.2.7.6. Using Remark 5.2.7.5 and Proposition 5.2.7.7, we see that $\rho _{X}$ factors as a composition

\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}})}( \underline{Y}, u) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Y) & \rightarrow & \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}})}( \underline{Y}, u) \times \operatorname{Hom}_{\operatorname{Fun}(K,\operatorname{\mathcal{C}})}( \underline{X}, \underline{Y} )\\ & \xrightarrow {\circ } & \operatorname{Hom}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) }( \underline{X}, u),\end{eqnarray*}

given on objects by the construction $( \alpha , f) \mapsto \alpha \circ \underline{f}$. It follows that a natural transformation $\alpha : \underline{Y} \rightarrow u$ exhibits $Y$ as a limit of $u$ if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the restriction $\rho _{X}|_{ \{ \alpha \} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) }$ is a homotopy equivalence of Kan complexes. By virtue of Proposition 7.1.3.12, this is equivalent to the requirement that $\alpha $ is final when regarded as an object of the $\infty $-category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ u\} $. $\square$

Corollary 7.1.4.2. 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. The following conditions are equivalent:

$(1)$

The object $Y$ is a limit of the diagram $u$.

$(2)$

The object $Y$ represents the right fibration $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ u\} \rightarrow \operatorname{\mathcal{C}}$ given by projection onto the first factor.

$(3)$

The object $Y$ represents the right fibration $\operatorname{\mathcal{C}}_{/u} \rightarrow \operatorname{\mathcal{C}}$ of Proposition 4.3.6.1.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows immediately from Proposition 7.1.4.1, and the equivalence $(2) \Leftrightarrow (3)$ follows from the observation that the slice diagonal $\operatorname{\mathcal{C}}_{/u} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}})} \{ u\} $ of Construction 4.6.4.12 is an equivalence of $\infty $-categories (Theorem 4.6.4.16). $\square$

Corollary 7.1.4.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. The following conditions are equivalent:

$(1)$

The diagram $u$ has a limit in $\operatorname{\mathcal{C}}$.

$(2)$

The oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ u\} \rightarrow \operatorname{\mathcal{C}}$ has a final object.

$(3)$

The slice $\infty $-category $\operatorname{\mathcal{C}}_{/u}$ has a final object.

Let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram in an $\infty $-category $\operatorname{\mathcal{C}}$. If $Y$ is an object of $\operatorname{\mathcal{C}}$, then supplying a natural transformation of diagrams $\alpha : \underline{Y} \rightarrow u$ is equivalent to giving a morphism of simplicial sets $\overline{u}: \Delta ^0 \diamond K \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{u}|_{ \Delta ^0} = Y$ and $\overline{u}|_{K}$, where

\[ \Delta ^0 \diamond K = \Delta ^0 \coprod _{ (\{ 0\} \times K) } (\Delta ^1 \times K) \]

is the simplicial set introduced in Notation 4.5.5.3. In practice, a datum of this type can be somewhat cumbersome to work with. For example, if $K$ is an $\infty $-category, then $\Delta ^0 \diamond K$ need not be an $\infty $-category. It is therefore often convenient to work with the following variant of Definition 7.1.1.1:

Definition 7.1.4.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets carrying the cone point of $K^{\triangleleft }$ to an object $Y \in \operatorname{\mathcal{C}}$. Set $u = \overline{u}|_{K}$, so that the diagram $\overline{u}$ can be identified with an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/u}$. We will say that $\overline{u}$ is a limit diagram if it is a final object of $\operatorname{\mathcal{C}}_{/u}$. If this condition is satisfied, we say that $\overline{u}$ exhibits $Y$ as a limit of the diagram $u$.

Variant 7.1.4.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $\overline{u}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets carrying the cone point of $K^{\triangleright }$ to an object $Y \in \operatorname{\mathcal{C}}$. Set $u = \overline{u}|_{K}$, so that the diagram $\overline{u}$ can be identified with an object of the coslice $\infty $-category $\operatorname{\mathcal{C}}_{u/}$. We will say that $\overline{u}$ is a colimit diagram if it is an initial object of $\operatorname{\mathcal{C}}_{u/}$. If this condition is satisfied, we say that $\overline{u}$ exhibits $Y$ as a colimit of the diagram $u$.

Remark 7.1.4.6. Let $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be as in Definition 7.1.4.4. Then $\overline{u}$ is a limit diagram if and only if the composite map

\[ \Delta ^1 \times K \simeq K \star _{K} K \rightarrow \Delta ^0 \star _{ \Delta ^0} K = K^{\triangleleft } \xrightarrow {\overline{u}} \operatorname{\mathcal{C}} \]

corresponds to 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. This follows from the characterization of Proposition 7.1.4.1, together with the observation that the slice diagonal $\operatorname{\mathcal{C}}_{/u} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}})} \{ u\} $ of Construction 4.6.4.12 is an equivalence of $\infty $-categories (Theorem 4.6.4.16).

Remark 7.1.4.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then an object $Y \in \operatorname{\mathcal{C}}$ is a limit of $u$ (in the sense of Definition 7.1.1.10) if and only if there exists a diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ which exhibits $Y$ as a limit of $u$. This is a reformulation of Corollary 7.1.4.2. Similarly, $Y$ is a colimit of $u$ if and only if there exists a diagram $\overline{u}': K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which exhibits $Y$ as a colimit of $u$.

Remark 7.1.4.8. 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 $\operatorname{\mathcal{C}}^{\operatorname{op}}$.

Example 7.1.4.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then an object $Y \in \operatorname{\mathcal{C}}$ is final (in the sense of Definition 7.1.2.1) if and only if the map

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

is a limit diagram in $\operatorname{\mathcal{C}}$. Similarly, $Y$ is initial if and only if the map

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

is a colimit diagram in $\operatorname{\mathcal{C}}$.

Example 7.1.4.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

  • The morphism $f$ is an isomorphism.

  • When regarded as a morphism $(\Delta ^0)^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$, $f$ is a limit diagram.

  • When regarded as a morphism $(\Delta ^0)^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$, $f$ is a colimit diagram.

This is a restatement of Proposition 7.1.3.8 (and also of Example 7.1.1.5, by virtue of Remark 7.1.4.6).

Remark 7.1.4.11. 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 7.1.4.12. 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 7.1.4.4).

$(2)$

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 7.1.2.12. Note that the restriction map $\operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \operatorname{\mathcal{C}}_{/f}$ is a right fibration of $\infty $-categories (Corollary 4.3.6.11), and therefore an isofibration (Example 4.4.1.10). The equivalence $(2) \Leftrightarrow (3)$ now follows from Proposition 4.5.7.16. $\square$