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7.1.5 Relative Limits and Colimits

In practice, it will be useful to consider the following relative version of Definition 7.1.3.11.

Definition 7.1.5.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets and let $K$ be a simplicial set, and let $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism with $p = \overline{p}|_{K}$. We will say that $\overline{p}$ is an $F$-limit diagram if the induced map

\[ \operatorname{\mathcal{C}}_{ / \overline{p} } \rightarrow \operatorname{\mathcal{C}}_{/p} \times _{ \operatorname{\mathcal{D}}_{/(F \circ p)}} \operatorname{\mathcal{D}}_{ / (F \circ \overline{p} ) } \]

is a trivial Kan fibration of simplicial sets. We will say that a morphism $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ with restriction $q = \overline{q}|_{K}$ is an $F$-colimit diagram if the induced map

\[ \operatorname{\mathcal{C}}_{ \overline{q} /} \rightarrow \operatorname{\mathcal{C}}_{q/} \times _{ \operatorname{\mathcal{D}}_{(F \circ q)/}} \operatorname{\mathcal{D}}_{ (F \circ \overline{q} )/ } \]

is a trivial Kan fibration of simplicial sets.

Remark 7.1.5.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets. Then a morphism $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is an $F$-limit diagram if and only if the opposite map $\overline{p}^{\operatorname{op}}: (K^{\operatorname{op}})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is an $F^{\operatorname{op}}$-colimit diagram.

Remark 7.1.5.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets and let $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism with restriction $p = \overline{p}|_{K}$. Then the restriction map

\[ U: \operatorname{\mathcal{C}}_{ / \overline{p} } \rightarrow \operatorname{\mathcal{C}}_{/p} \times _{ \operatorname{\mathcal{D}}_{/(F \circ p)}} \operatorname{\mathcal{D}}_{ / (F \circ \overline{p} ) } \]

is automatically a right fibration of simplicial sets (Proposition 4.3.6.6); in particular, it is an isofibration (Corollary 5.6.6.5). Consequently, the following conditions are equivalent:

$(1)$

The morphism $\overline{p}$ is an $F$-limit diagram: that is, $U$ is a trivial Kan fibration of simplicial sets.

$(2)$

Each fiber of $U$ is a contractible Kan complex.

$(3)$

The morphism $U$ is a categorical equivalence of simplicial sets.

The equivalence of $(1)$ and $(2)$ follows from Proposition 4.4.2.13, and the equivalence of $(1)$ and $(3)$ from Proposition 4.5.7.14.

Example 7.1.5.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and $F: \operatorname{\mathcal{C}}\rightarrow \Delta ^{0}$ be the projection map. Then a morphism $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is an $F$-limit diagram (in the sense of Definition 7.1.5.1) if and only if it is a limit diagram (in the sense of Definition 7.1.3.11); this is a reformulation of Proposition 7.1.3.18. Similarly, a morphism $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is an $F$-colimit diagram if and only if it is a colimit diagram.

Example 7.1.5.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets. Then:

  • An edge $e$ of $\operatorname{\mathcal{C}}$ is $F$-cartesian (in the sense of Definition 5.1.1.1) if and only if it is an $F$-limit diagram when viewed as a morphism of simplicial sets $(\Delta ^0)^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$.

  • An edge $e$ of $\operatorname{\mathcal{C}}$ is $F$-cocartesian (in the sense of Definition 5.1.1.1) if and only if it is an $F$-colimit diagram when viewed as a morphism of simplicial sets $(\Delta ^0)^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.

See Proposition 5.1.1.12.

Example 7.1.5.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a trivial Kan fibration of simplicial sets. Then every morphism $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is an $F$-limit diagram, and every morphism $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is an $F$-colimit diagram (see Proposition 4.3.7.15).

Example 7.1.5.7. Let $K$ be a weakly contractible simplicial sets and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a right fibration of simplicial sets. Then every morphism $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is an $F$-limit diagram (see Proposition 4.3.7.2). Similarly, if $F$ is a left fibration, then every morphism $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is an $F$-colimit diagram.

Remark 7.1.5.8 (Base Change). Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r]^-{U} \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}, } \]

where $F$ and $F'$ are inner fibrations. Let $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}'$ be a morphism of simplicial sets. If $U \circ \overline{p}$ is an $F$-limit diagram, then $\overline{p}$ is an $F'$-limit diagram. Similarly, if $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}'$ is a morphism of simplicial sets for which $U \circ \overline{q}$ is an $F$-colimit diagram, then $\overline{q}$ is an $F'$-colimit diagram.

Proposition 7.1.5.9 (Transitivity). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be inner fibrations of simplicial sets. Then:

$(1)$

Let $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets such that $F \circ \overline{p}$ is a $G$-limit diagram. Then $\overline{p}$ is an $F$-limit diagram if and only if it is a $(G \circ F)$-limit diagram.

$(2)$

Let $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets such that $F \circ \overline{q}$ is a $G$-colimit diagram. Then $\overline{q}$ is an $F$-colimit diagram if and only if it is a $(G \circ F)$-colimit diagram.

Proof. We will prove assertion $(1)$; the proof of $(2)$ is similar. Set $p = \overline{p}|_{K}$, and consider the restriction maps

\[ \operatorname{\mathcal{C}}_{ / \overline{p} } \xrightarrow {U} \operatorname{\mathcal{C}}_{ / p } \times _{ \operatorname{\mathcal{D}}_{ /(F \circ p)} } \operatorname{\mathcal{D}}_{ /(F \circ \overline{p} )} \xrightarrow {V} \operatorname{\mathcal{C}}_{/p} \times _{ \operatorname{\mathcal{E}}_{ /(G \circ F \circ p)} } \operatorname{\mathcal{E}}_{ /(G \circ F \circ \overline{p} )}. \]

Assume that $F \circ \overline{p}$ is a $G$-limit diagram. The morphism $V$ is a pullback of the restriction map

\[ \operatorname{\mathcal{D}}_{ / (F \circ \overline{p} )} \rightarrow \operatorname{\mathcal{D}}_{/(F \circ p)} \times _{ \operatorname{\mathcal{E}}_{ /(G \circ F \circ p)} } \operatorname{\mathcal{E}}_{ /(G \circ F \circ \overline{p} )}, \]

and is therefore a trivial Kan fibration. It follows that $U$ is a categorical equivalence of simplicial sets if and only if $(V \circ U)$ is a categorical equivalence of simplicial sets. The desired result now follows from Remark 7.1.5.3. $\square$

Corollary 7.1.5.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:

$(1)$

Let $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets such that $F \circ \overline{p}$ is a limit diagram in $\operatorname{\mathcal{D}}$. Then $\overline{p}$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if it is an $F$-limit diagram.

$(2)$

Let $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets such that $F \circ \overline{q}$ is a colimit diagram in $\operatorname{\mathcal{D}}$. Then $\overline{q}$ is a colimit diagram in $\operatorname{\mathcal{C}}$ if and only if it is an $F$-colimit diagram.

Proof. Apply Proposition 7.1.5.9 in the case $\operatorname{\mathcal{E}}= \Delta ^{0}$ (and use Example 7.1.5.4). $\square$

Corollary 7.1.5.11. Let $K$ be a weakly contractible simplicial set and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. If $F$ is a left fibration, then it creates $K$-indexed colimits. If $F$ is a right fibration, then it creates $K$-indexed limits.

Proof. Assume $F$ is a right fibration; we will show that it creates $K$-indexed limits (the analogous statement for left fibrations follows by a similar argument). Let $q: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram and suppose that $F \circ q$ can be extended to a limit diagram $u: K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$. Since the inclusion $K \hookrightarrow K^{\triangleleft }$ is right anodyne (Example 4.3.7.10), our assumption that $F$ is a right fibration guarantees that the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ K \ar [d] \ar [r]^-{q} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ K^{\triangleleft } \ar [r]^-{u} \ar@ {-->}[ur]^{ \overline{q} } & \operatorname{\mathcal{D}}} \]

has a solution. Since $K$ is weakly contractible, the morphism $\overline{q}$ is automatically an $F$-limit diagram (Example 7.1.5.7). Applying Corollary 7.1.5.10, we see that $\overline{q}$ is a limit diagram. $\square$

Corollary 7.1.5.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $K$ be a weakly contractible simplicial set. Then:

  • If $F$ is a right fibration and the $\infty $-category $\operatorname{\mathcal{D}}$ admits $K$-indexed limits, then $\operatorname{\mathcal{C}}$ also admits $K$-indexed limits and $F$ preserves $K$-indexed limits.

  • If $F$ is a left fibration and the $\infty $-category $\operatorname{\mathcal{D}}$ admits $K$-indexed colimits, then $\operatorname{\mathcal{C}}$ also admits $K$-indexed colimits and $F$ preserves $K$-indexed colimits.

Definition 7.1.5.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets. We say that a vertex $C \in \operatorname{\mathcal{C}}$ is $F$-final if the inclusion map $\{ C\} \hookrightarrow \operatorname{\mathcal{C}}$ is an $F$-limit diagram: that is, the induced map $\operatorname{\mathcal{C}}_{/C} \rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ /F(C)}$ is a trivial Kan fibration. We say that $C$ is $F$-initial if the inclusion map $\{ C\} \hookrightarrow \operatorname{\mathcal{C}}$ is an $F$-colimit diagram.

Remark 7.1.5.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets, let $q: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let

\[ F_{q/}: \operatorname{\mathcal{C}}_{q/} \rightarrow \operatorname{\mathcal{D}}_{(F \circ q)/ } \quad \quad F_{/q}: \operatorname{\mathcal{C}}_{/q} \rightarrow \operatorname{\mathcal{D}}_{/ (F \circ q)} \]

be the induced maps. Then an extension $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ of $q$ is an $F$-limit diagram if and only if it is $F_{/q}$-final when viewed as a vertex of the simplicial set $\operatorname{\mathcal{C}}_{/q}$. Similarly, an extension $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is an $F$-colimit diagram if and only if is $F_{q/}$-initial when viewed as a vertex of the simplicial set $\operatorname{\mathcal{C}}_{q/}$.

Proposition 7.1.5.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of simplicial sets and let $C$ be a vertex of $\operatorname{\mathcal{C}}$ having image $D = F(C)$ in $\operatorname{\mathcal{D}}$. Then $C$ is $F$-initial (in the sense of Definition 7.1.5.13) if and only if is an initial object of the $\infty $-category $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$.

Proof. If the vertex $C$ is $F$-initial, then it is an initial object of $\operatorname{\mathcal{C}}_{D}$ by virtue of Remark 7.1.5.8 and Example 7.1.5.4 (note that this implication does not require the assumption that $F$ is a cartesian fibration). Conversely, suppose that $C$ is an initial object of $\operatorname{\mathcal{C}}_ D$. To show that $C$ is $F$-initial, we must show that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r]^-{ \sigma _0 } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{n} \ar [r]^-{\overline{\sigma } } \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]

admits a solution, provided that $n > 0$ and $\sigma _0(0) = C$. Replacing $F$ by the projection map $\Delta ^ n \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$, we may assume without loss of generality that $\operatorname{\mathcal{D}}= \Delta ^ n$ is a standard simplex and that $\overline{\sigma }$ is the identity map (so that $D = 0$ is the initial vertex of $\operatorname{\mathcal{D}}= \Delta ^ n$). In this case, the simplicial set $\operatorname{\mathcal{C}}$ is an $\infty $-category. We will complete the proof by showing that $C$ is an initial object of $\operatorname{\mathcal{C}}$, so that $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ (note that the equality $F(\sigma ) = \overline{\sigma }$ is then automatically satisfied). Fix an object $Y \in \operatorname{\mathcal{C}}$; we wish to show that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y)$ is contractible. Since $F$ is a cartesian fibration, we can choose an $F$-cartesian morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ satisfying $F(X) = D$. It follows that composition with the homotopy class $[u]$ induces an isomorphism

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{D}}(C,X) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C,Y) \]

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ (Corollary 5.1.2.3). Our assumption that $C$ is an initial object of $\operatorname{\mathcal{C}}_{D}$ guarantees that $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{D}}(C,X)$ is a contractible Kan complex, so that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C,Y)$ is also contractible. $\square$

Corollary 7.1.5.16. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of $\infty $-categories and let $C$ be an object of $\operatorname{\mathcal{C}}$ whose image $D = F(C)$ is an initial object of $\operatorname{\mathcal{D}}$. Then $C$ is an initial object of $\operatorname{\mathcal{C}}$ if and only if it is an initial object of the fiber $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$.

Corollary 7.1.5.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of simplicial sets, let $D \in \operatorname{\mathcal{D}}$ be a vertex, and let

\[ \overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}} \]

be a morphism of simplicial sets. Then $\overline{q}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{D}$ if and only if it is an $F$-colimit diagram in the simplicial set $\operatorname{\mathcal{C}}$.

Proof. Set $q = \overline{q}|_{K}$. By virtue of Proposition 5.1.4.18, the induced map $F_{q/}: \operatorname{\mathcal{C}}_{q/} \rightarrow \operatorname{\mathcal{D}}_{(F \circ q)/}$ is also a cartesian fibration. The desired result now follows by combining Proposition 7.1.5.15 with Remark 7.1.5.14. $\square$

Remark 7.1.5.18. Corollary 7.1.5.17 has an obvious counterpart for $F$-limit diagrams under the assumption that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a cocartesian fibration, which can be proved in the same way. It also has a more subtle counterpart for $F$-colimit diagrams when $F$ is a cocartesian fibration (or $F$-limit diagrams when $F$ is a cartesian fibration), which we will discuss in Chapter (see Proposition ).