7.1.2 Limit 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 5.6.6.21, we can reformulate Definition 7.1.1.1 as follows:
Proposition 7.1.2.1. 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 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 5.2.8.6. Using Remark 5.2.8.5 and Proposition 5.2.8.7, 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 5.6.6.21, 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 7.1.2.2. 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 7.1.2.1, 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 4.6.4.13 is an equivalence of $\infty $-categories (Theorem 4.6.4.17).
$\square$
Corollary 7.1.2.3. 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 $\overline{u}|_{K} = u$, where
\[ \Delta ^0 \diamond K = \Delta ^0 {\coprod }_{ (\{ 0\} \times K) } (\Delta ^1 \times K) \]
is the simplicial set introduced in Notation 4.5.8.3. 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 7.1.2.4. 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 7.1.2.5. 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 $u$.
Example 7.1.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then an object $Y \in \operatorname{\mathcal{C}}$ is final (in the sense of Definition 4.6.7.1) 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 7.1.2.10. 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 4.6.7.22 (and also of Example 7.1.1.5, by virtue of Remark 7.1.2.6).
Proposition 7.1.2.12. 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 7.1.2.4).
- $(2)$
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.
- $(4)$
For every object $X \in \operatorname{\mathcal{C}}$, the restriction map $\{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \{ X\} \times \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/f}$ is a homotopy equivalence of Kan complexes.
Proof.
The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 4.6.7.10. 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 4.4.1.11). The equivalence $(2) \Leftrightarrow (3)$ now follows from Proposition 4.5.5.20, and the equivalence $(3) \Leftrightarrow (4)$ follows from Corollary 5.1.6.4.
$\square$
Proposition 7.1.2.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $\overline{\rho }: \overline{F} \rightarrow \overline{G}$ be a natural transformation between diagrams $\overline{F}, \overline{G}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. Assume that, for every vertex $x \in K$, the morphism $\overline{\rho }_{x}: \overline{F}(x) \rightarrow \overline{G}(x)$ is an isomorphism in $\operatorname{\mathcal{C}}$. Then any two of the following conditions imply the third:
- $(1)$
The morphism of simplicial sets $\overline{F}$ is a limit diagram in $\operatorname{\mathcal{C}}$.
- $(2)$
The morphism of simplicial sets $\overline{G}$ is a limit diagram in $\operatorname{\mathcal{C}}$.
- $(3)$
The natural transformation $\overline{\rho }$ carries the cone point ${\bf 0} \in K^{\triangleleft }$ to an isomorphism $\overline{\rho }_{\bf 0}: \overline{F}( {\bf 0} ) \rightarrow \overline{G}( {\bf 0} )$.
Proof.
Set $F = \overline{F}|_{K}$ and $G = \overline{G}|_{K}$, so that $\overline{\rho }$ restricts to an isomorphism $\rho : F \rightarrow G$ in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ (Theorem 4.4.4.4). Set $X = \overline{F}( {\bf 0} )$ and $Y = \overline{F}( {\bf 0} )$, and let $\underline{X}, \underline{Y}: K \rightarrow \operatorname{\mathcal{C}}$ be the constant maps taking the values $X$ and $Y$, respectively. Let $c$ denote the composition $\Delta ^1 \times K \simeq K \star _{K} K \rightarrow \Delta ^{0} \star _{ \Delta ^{0} } K = K^{\triangleleft }$. Then the composition
\[ \Delta ^1 \times \Delta ^1 \times K \xrightarrow { \operatorname{id}\times c} \Delta ^1 \times K^{\triangleleft } \xrightarrow { \overline{\rho } } \operatorname{\mathcal{C}} \]
can be identified with a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ \underline{X} \ar [d]^{ \alpha } \ar [dr]^{\gamma } \ar [r]^-{ \underline{f} } & \underline{Y} \ar [d]^{\beta } \\ F \ar [r]^-{ \rho }_{\sim } & G } \]
in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Using Remark 7.1.2.6, we can reformulate conditions $(1)$ and $(2)$ as follows:
- $(1')$
The natural transformation $\alpha $ exhibits $X$ as a limit of $F$.
- $(2')$
The natural transformation $\beta $ exhibits $Y$ as a limit of $G$.
Since $\rho $ is an isomorphism, we can use Remark 7.1.1.8 to reformulate $(1')$ as follows:
- $(1'')$
The natural transformation $\gamma $ exhibits $X$ as a limit of $G$.
It will therefore suffice to show that any two of the conditions $(1'')$, $(2')$, and $(3)$ imply the third, which is a special case of Remark 7.1.1.9.
$\square$
Corollary 7.1.2.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. Then:
- $(1)$
Let $\overline{u}, \overline{v}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams which are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{C}})$. Then $\overline{u}$ is a limit diagram if and only if $\overline{v}$ is a limit diagram.
- $(2)$
Let $\overline{u}, \overline{v}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams which are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}})$. Then $\overline{u}$ is a colimit diagram if and only if $\overline{v}$ is a colimit diagram.
Corollary 7.1.2.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and suppose we are given a pair of morphisms $u,v: K \rightarrow \operatorname{\mathcal{C}}$ which are isomorphic as objects of the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$. Then:
- $(1)$
The morphism $u$ can be extended to a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ if and only if $v$ can be extended to a limit diagram $\overline{v}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$.
- $(2)$
The morphism $u$ can be extended to a colimit diagram $\overline{u}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ if and only if $v$ can be extended to a colimit diagram $\overline{v}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.
Proof.
We will prove $(1)$; the proof of $(2)$ is similar. Suppose that $u$ can be extended to a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. Since the diagrams $u$ and $v$ are isomorphic, it follows from Corollary 4.4.5.3 that $\overline{u}$ is isomorphic to a diagram $\overline{v}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{v}|_{K} = v$. Applying Corollary 7.1.2.14, we conclude that $\overline{v}$ is also a limit diagram.
$\square$