We begin by introducing some terminology.
Definition 7.7.1.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathscr {F}, \mathscr {G}: K \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams indexed by the same simplicial set $K$. We say that a natural transformation $\gamma : \mathscr {F} \rightarrow \mathscr {G}$ is cartesian if, for every edge $e: x \rightarrow y$ of the simplicial set $K$, the corresponding diagram
7.79
\begin{equation} \begin{gathered}\label{equation:cartesian-natural-transformation} \xymatrix { \mathscr {F}(x) \ar [r]^{ \mathscr {F}(e) } \ar [d]^{ \gamma _ x } & \mathscr {F}(y) \ar [d]^{ \gamma _{y} } \\ \mathscr {G}(x) \ar [r]^{ \mathscr {F}(e) } & \mathscr {G}(y) } \end{gathered} \end{equation}
is a pullback square in the $\infty $-category $\operatorname{\mathcal{C}}$.
Example 7.7.1.2. In the situation of Definition 7.7.1.1, suppose that the natural transformation $\gamma $ is an isomorphism. Then, for every edge $e: x \rightarrow y$ of $K$, the diagram (7.79) is automatically a pullback square, since the vertical maps are isomorphisms (see Corollary 7.6.2.27). It follows that every isomorphism in the $\infty $-category $\operatorname{Fun}( K, \operatorname{\mathcal{C}})$ is a cartesian natural transformation. In particular, for every diagram $\mathscr {F}: K \rightarrow \operatorname{\mathcal{C}}$, the identity transformation $\operatorname{id}: \mathscr {F} \rightarrow \mathscr {F}$ is cartesian.
Example 7.7.1.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f$ and $g$ be morphisms in $\operatorname{\mathcal{C}}$, and let $\gamma : f \rightarrow g$ be a morphism in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. Then $\gamma $ is cartesian (in the sense of Definition 7.7.1.1) if and only if it corresponds to a pullback diagram $\Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$.
Example 7.7.1.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and suppose we are given a pair of diagrams $\mathscr {F}, \mathscr {G}: K \rightarrow \operatorname{\mathcal{C}}$ indexed by the same simplicial set $K$. If either $K$ or $\operatorname{\mathcal{C}}$ is a Kan complex, then every natural transformation $\gamma : \mathscr {F} \rightarrow \mathscr {G}$ is cartesian.
Example 7.7.1.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: C \rightarrow D$ be a morphism of $\operatorname{\mathcal{C}}$. Then, for every simplicial set $K$, the induced map of constant diagrams $\underline{f}: \underline{C} \rightarrow \underline{D}$ is a cartesian natural transformation.
For diagrams indexed by a cone $K^{\triangleright }$, Definition 7.7.1.1 can be simplified. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and suppose we are given a natural transformation $\overline{\gamma }: \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$ between diagrams $\overline{\mathscr {F}}, \overline{\mathscr {G}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. Set $\mathscr {F} = \overline{\mathscr {F}}|_{K}$ and $\mathscr {G} = \overline{\mathscr {G}}|_{K}$, so that $\overline{\gamma }$ restricts to a natural transformation $\gamma : \mathscr {F} \rightarrow \mathscr {G}$. Let $C$ and $D$ denote the values of $\overline{\mathscr {F}}$ and $\overline{\mathscr {G}}$ at the cone point of $K^{\triangleright }$, so that $\overline{\gamma }$ determines a morphism $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$. Precomposition with the canonical map
\[ \Delta ^1 \times K \rightarrow K \diamond \Delta ^0 \twoheadrightarrow K \star \Delta ^0 = K^{\triangleright } \]
carries $\overline{\gamma }$ to a morphism in the $\infty $-category $\operatorname{Fun}( \Delta ^1 \times K, \operatorname{\mathcal{C}})$, which we can identify with a commutative diagram
7.80
\begin{equation} \begin{gathered}\label{equation:cartesian-transformation-cone-indexed} \xymatrix { \mathscr {F} \ar [r] \ar [d]^{\gamma } & \underline{C} \ar [d]^{ \underline{f} } \\ \mathscr {G} \ar [r] & \underline{D} } \end{gathered} \end{equation}
of $K$-indexed diagrams in $\operatorname{\mathcal{C}}$.
Proposition 7.7.1.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $\overline{\gamma }: \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$ be a natural transformation between diagrams $\overline{\mathscr {F}}, \overline{\mathscr {G}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The natural transformation $\overline{\gamma }$ is cartesian.
- $(2)$
The diagram (7.80) is a levelwise pullback square.
Proof.
For each vertex $x \in K$, let $e_{x}$ denote the edge of $K^{\triangleright }$ given by the map $\Delta ^1 \simeq \{ x\} ^{\triangleright } \hookrightarrow K^{\triangleright }$. Note that $\overline{\gamma }$ carries $e_ x$ to a commutative diagram
\[ \xymatrix { \mathscr {F}(x) \ar [r] \ar [d]^{\gamma _{x}} & C \ar [d]^{f} \\ \mathscr {G}(x) \ar [r] & D } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$, which is obtained from (7.80) by evaluating at $x$. It follows that $(1)$ implies $(2)$. Conversely, suppose that condition $(2)$ is satisfied, and let $e: x \rightarrow y$ be any edge of $K^{\triangleright }$; we wish to show that $\overline{\gamma }$ carries $e$ to a pullback square in $\operatorname{\mathcal{C}}$. Without loss of generality, we may assume that $e$ is nondegenerate (Remark 7.7.1.3). If $y$ is the cone point of $K^{\triangleright }$, then $x$ belongs to $K$ and $e = e_ x$, so the desired result follows from assumption $(2)$. We are therefore reduced to the case where $e$ is an edge of $K$: that is, to showing that the natural transformation $\gamma : \mathscr {F} \rightarrow \mathscr {G}$ is cartesian. This follows from Remark 7.7.1.11, since $\gamma $ is a pullback of the cartesian natural transformation $\underline{f}: \underline{C} \rightarrow \underline{D}$ (Example 7.7.1.6).
$\square$
Variant 7.7.1.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and suppose we are given a natural transformation $\overline{\gamma }: \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$ between diagrams $\overline{\mathscr {F}}, \overline{\mathscr {G}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. If there exists a morphism $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$ and a levelwise pullback square $\sigma :$
7.81
\begin{equation} \begin{gathered}\label{equation:levelwise-pullback-for-cartesian} \xymatrix { \overline{\mathscr {F}} \ar [d]^{ \overline{\gamma } } \ar [r] & \underline{C} \ar [d]^{ \underline{f} } \\ \overline{\mathscr {G}} \ar [r] & \underline{D}, } \end{gathered} \end{equation}
in $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}})$, then $\overline{\gamma }$ is cartesian (see Remark 7.7.1.11 and Example 7.7.1.6). Conversely, suppose that $\overline{\gamma }$ is cartesian. Let $v$ denote the cone point of $K^{\triangleright }$ and set $C = \overline{\mathscr {F}}(v)$ and $D = \overline{\mathscr {G}}(v)$, so that $\overline{\gamma }$ determines a morphism $f: C \rightarrow D$. There is a unique morphism of simplicial sets $h: \Delta ^1 \times K^{\triangleright } \rightarrow K^{\triangleright }$ such that $h|_{ \{ 0\} \times K^{\triangleright } }$ is the identity and $h|_{ \{ 1\} \times K^{\triangleright } }$ is the constant map taking the value $v$. Precomposition with $h$ carries $\overline{\gamma }$ to a morphism in the $\infty $-category $\operatorname{Fun}( \Delta ^1 \times K^{\triangleright }, \operatorname{\mathcal{C}})$, which we can identify with a levelwise pullback square of the form (7.81).
Definition 7.7.1.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. We say that a morphism $\overline{\mathscr {F}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a universal colimit diagram if, for every cartesian natural transformation $\overline{\mathscr {F}}' \rightarrow \overline{\mathscr {F}}$, the morphism $\overline{\mathscr {F}}': K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$.
Example 7.7.1.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K = \emptyset $ be the empty simplicial set. Then we can identify a diagram $\overline{\mathscr {F}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which an object $C \in \operatorname{\mathcal{C}}$. In this case, the following conditions are equivalent:
- $(a)$
The morphism $\overline{\mathscr {F}}$ is a universal colimit diagram.
- $(b)$
For every morphism $u: C' \rightarrow C$ of $\operatorname{\mathcal{C}}$, the object $C' \in \operatorname{\mathcal{C}}$ is initial.
- $(c)$
The object $C \in \operatorname{\mathcal{C}}$ is initial and every morphism $u: C' \rightarrow C$ is an isomorphism.
If these conditions are satisfied, we say that $C$ is a universal initial object of $\operatorname{\mathcal{C}}$.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\overline{ \mathscr {F} }: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a diagram carrying the cone point of $K^{\triangleright }$ to an object $C \in \operatorname{\mathcal{C}}$. Recall that $\overline{ \mathscr {F} }$ determines a natural transformation $\beta $ from $\mathscr {F} = \overline{\mathscr {F}}|_{K}$ to the constant diagram $\underline{C}$, given by the composition
\[ \Delta ^1 \times K \simeq K \star _{K} K \rightarrow K \star _{ \Delta ^0 } \Delta ^0 = K^{\triangleright } \xrightarrow { \overline{\mathscr {F}} } \operatorname{\mathcal{C}}. \]
Moreover, $\overline{\mathscr {F}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$ if and only if the natural transformation $\beta $ exhibits $C$ as a colimit of $\mathscr {F}$ (Remark 7.1.3.6). This observation has a counterpart for universal colimit diagrams:
Proposition 7.7.1.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\overline{\mathscr {F}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism, and define $\beta : \mathscr {F} \rightarrow \underline{C}$ as above. The following conditions are equivalent:
- $(1)$
The morphism $\overline{ \mathscr {F} }$ is a universal colimit diagram, in the sense of Definition 7.7.1.15.
- $(2)$
For every morphism $u: C' \rightarrow C$ in $\operatorname{\mathcal{C}}$ and every levelwise pullback diagram
7.82
\begin{equation} \begin{gathered}\label{equation:universal-colimit-diagram-reformulated} \xymatrix { \mathscr {F}' \ar [r]^{ \beta ' } \ar [d] & \underline{C}' \ar [d]^{ \underline{u} } \\ \mathscr {F} \ar [r]^{\beta } & \underline{C} } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$, the natural transformation $\beta '$ exhibits $C'$ as a colimit of $\mathscr {F}'$
Proof.
The implication $(2) \Rightarrow (1)$ follows from Remark 7.1.3.6 and Proposition 7.7.1.13. To prove the converse, it suffices to observe that every diagram of the form (7.82) is isomorphic to one which is obtained from a natural transformation $\overline{\mathscr {F}}' \rightarrow \overline{\mathscr {F}}$ of $K^{\triangleright }$-indexed diagrams in $\operatorname{\mathcal{C}}$. This follows from the fact that the comparison map $K \diamond \Delta ^0 \twoheadrightarrow K \star \Delta ^0$ of Theorem 4.5.8.8 is a categorical equivalence of simplicial sets.
$\square$
Corollary 7.7.1.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. The following conditions are equivalent:
- $(1)$
Every colimit diagram $\overline{\mathscr {F}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a universal colimit diagram.
- $(2)$
In the $\infty $-category $\operatorname{\mathcal{C}}$, $K$-indexed colimits are universal.
Proof.
The implication $(2) \Rightarrow (1)$ follows from Proposition 7.7.1.19. The reverse implication follows from the observation that, if $\mathscr {F}: K \rightarrow \operatorname{\mathcal{C}}$ is a diagram, then every natural transformation from $\mathscr {F}$ to a constant diagram is homotopic to one which arises from a diagram $\overline{\mathscr {F}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ extending $\mathscr {F}$ (see Theorem 4.6.4.17).
$\square$
Definition 7.7.1.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. We say that a diagram $\mathscr {F}: K \rightarrow \operatorname{\mathcal{C}}$ has a universal colimit if there exists a universal colimit diagram $\overline{\mathscr {F}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{\mathscr {F}}|_{K} = \mathscr {F}$.