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5.6.3 Universal Fibrations

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then $F$ determines a pullback functor $F^{\ast }: (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}} \rightarrow (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$, given on objects by the formula $F^{\ast }( \widetilde{\operatorname{\mathcal{D}}} ) = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}$.

Proposition 5.6.3.1. Let $V: \widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty$-categories, let $\operatorname{\mathcal{C}}$ be a simplicial set, and let $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be morphisms of simplicial sets which are isomorphic when viewed as objects of the diagram $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Then the isofibrations $F^{\ast }( \widetilde{\operatorname{\mathcal{D}}} ) \rightarrow \operatorname{\mathcal{C}}$ and $G^{\ast }( \widetilde{\operatorname{\mathcal{D}}} ) \rightarrow \operatorname{\mathcal{C}}$ are equivalent (in the sense of Definition 5.6.2.1).

Warning 5.6.3.2. The conclusion of Proposition 5.6.3.1 does not necessarily hold if $U: \widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$ is assumed only to be an inner fibration of simplicial sets. See Warning 5.6.2.6.

Proof of Proposition 5.6.3.1. Since $F$ and $G$ are isomorphism as objects of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, there exists a contractible Kan complex $\operatorname{\mathcal{Q}}$ containing vertices $f$ and $g$ and a functor $H: \operatorname{\mathcal{Q}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ satisfying $H(f) = F$ and $H(g) = G$. Let us identify $H$ with a morphism of simplicial sets $\operatorname{\mathcal{Q}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, and let $\widetilde{\operatorname{\mathcal{C}}}$ denote the fiber product $(\operatorname{\mathcal{Q}}\times \operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}$. We will show that the inclusion maps

$F^{\ast }( \widetilde{\operatorname{\mathcal{D}}} ) = \{ f\} \times _{\operatorname{\mathcal{Q}}} \widetilde{\operatorname{\mathcal{C}}} \hookrightarrow \widetilde{\operatorname{\mathcal{C}}} \hookleftarrow \{ g\} \times _{\operatorname{\mathcal{Q}}} \widetilde{\operatorname{\mathcal{C}}} = G^{\ast }( \widetilde{\operatorname{\mathcal{D}}})$

are equivalences of inner fibrations over $\operatorname{\mathcal{C}}$. To prove this, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex (Proposition 5.6.2.9); in this case, we wish to show that both inclusion maps are equivalences of $\infty$-categories (Corollary 5.6.2.8). This follows by applying Corollary 4.5.4.5 to the diagram of pullback squares

$\xymatrix@R =50pt@C=50pt{ F^{\ast }(\widetilde{\operatorname{\mathcal{D}}}) \ar [r] \ar [d] & \widetilde{\operatorname{\mathcal{C}}} \ar [d] & G^{\ast }(\widetilde{\operatorname{\mathcal{D}}}) \ar [l] \ar [d] \\ \{ f\} \times \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{Q}}\times \operatorname{\mathcal{C}}& \{ g\} \times \operatorname{\mathcal{C}}; \ar [l] }$

here the vertical maps are isofibrations (since they are pullbacks of $V$) and the lower horizontal maps are equivalences of $\infty$-categories (since $\operatorname{\mathcal{Q}}$ is a contractible Kan complex). $\square$

Corollary 5.6.3.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $\mathscr {F}, \mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be diagrams, and let $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ and $U': \int _{\operatorname{\mathcal{C}}} \mathscr {F}' \rightarrow \operatorname{\mathcal{C}}$ be the projection maps. If $\mathscr {F}$ and $\mathscr {F}'$ are isomorphic as objects of the diagram $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$, then $U$ and $U'$ are equivalent as cocartesian fibrations over $\operatorname{\mathcal{C}}$.

Proof. Apply Proposition 5.6.3.1 to the cocartesian fibration $\operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}$ of Proposition 5.4.6.11. $\square$

Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. It follows from Corollary 5.6.3.3 that the $\infty$-category of elements construction induces a map of sets

$\{ \textnormal{Functors \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}} \} / \textnormal{Isomorphism} \rightarrow \{ \textnormal{Cocartesian fibrations \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}} \} / \textnormal{Equivalence}.$

Our main result is that, modulo set-theoretic difficulties, this map is bijective (see Corollary 5.6.3.12). To address the relevant technicalities, it will be convenient to introduce some terminology.

Definition 5.6.3.4. Let $\operatorname{\mathcal{QC}}$ denote the $\infty$-category of small $\infty$-categories (Construction 5.4.4.1), and let $\operatorname{\mathcal{Q}}\subseteq \operatorname{\mathcal{QC}}$ be a full subcategory. We will say that a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is $\operatorname{\mathcal{Q}}$-small if, for every object $C \in \operatorname{\mathcal{D}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is equivalent to an $\infty$-category which belongs to $\operatorname{\mathcal{Q}}$.

Example 5.6.3.5. Let $\operatorname{\mathcal{Q}}$ be a full subcategory of $\operatorname{\mathcal{QC}}$, and let $\widetilde{\operatorname{\mathcal{Q}}}$ denote the fiber product $\operatorname{\mathcal{Q}}\times _{ \operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}}$: that is, the $\infty$-category of elements of the inclusion map $\operatorname{\mathcal{Q}}\hookrightarrow \operatorname{\mathcal{QC}}$. Then the projection map $V: \widetilde{\operatorname{\mathcal{Q}}} \rightarrow \operatorname{\mathcal{Q}}$ is a $\operatorname{\mathcal{Q}}$-small cocartesian fibration of $\infty$-categories. This follows from Example 5.5.4.19: for every object $Q \in \operatorname{\mathcal{Q}}$, the fiber $\{ Q\} \times _{ \operatorname{\mathcal{Q}}} \widetilde{\operatorname{\mathcal{Q}}}$ is an $\infty$-category equivalent to $Q$.

Theorem 5.6.0.2 is a consequence of the following more precise assertion:

Theorem 5.6.3.6 (Universality Theorem). Let $\operatorname{\mathcal{Q}}$ be a full subcategory of $\operatorname{\mathcal{QC}}$. For every simplicial set $\operatorname{\mathcal{C}}$, the construction

$( \mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{Q}}) \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F}$

induces a bijection from $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{Q}})^{\simeq } )$ to the set of equivalence classes of $\operatorname{\mathcal{Q}}$-small cocartesian fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$.

We will deduce Theorem 5.6.3.6 from a more precise statement (Theorem 5.6.4.9), which we formulate in §5.6.4 and prove in §5.6.7.

Remark 5.6.3.7. In the statement of Theorem 5.6.3.6, it is not necessary to assume that the simplicial set $\operatorname{\mathcal{C}}$ is an $\infty$-category, or that it is small.

Remark 5.6.3.8. Let $\operatorname{\mathcal{Q}}$ be a full subcategory of $\operatorname{\mathcal{QC}}$. We can summarize Theorem 5.6.3.6 more informally by saying that the projection map $V: \widetilde{\operatorname{\mathcal{Q}}} \rightarrow \operatorname{\mathcal{Q}}$ of Example 5.6.3.5 is a universal $\operatorname{\mathcal{Q}}$-small cocartesian fibration. That is, for every simplicial set $\operatorname{\mathcal{C}}$, the construction

$(\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{Q}}) \mapsto \mathscr {F}^{\ast }( \widetilde{\operatorname{\mathcal{Q}}} ) = \int _{\operatorname{\mathcal{C}}} \mathscr {F}$

induces a bijection from the set of isomorphism classes $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{Q}})^{\simeq } )$ to the set of equivalence classes of $\operatorname{\mathcal{Q}}$-small cocartesian fibrations over $\operatorname{\mathcal{C}}$. Note that this property characterizes the $\infty$-category $\operatorname{\mathcal{Q}}$ up to equivalence.

Remark 5.6.3.9. We will later show that the bijection of Theorem 5.6.3.6 can be upgraded to an equivalence of $\infty$-categories; see Theorem .

Warning 5.6.3.10. The statement of Theorem 5.6.3.6 assumes that $\operatorname{\mathcal{Q}}$ is a full subcategory of $\operatorname{\mathcal{QC}}$. However, it will sometimes be convenient to apply Theorem 5.6.3.6 when $\operatorname{\mathcal{Q}}$ is an enlargement of $\operatorname{\mathcal{QC}}$, whose objects include $\infty$-categories which are not necessarily small.

Definition 5.6.3.11. Let $\operatorname{\mathcal{E}}$ be an $\infty$-category. We say that $\operatorname{\mathcal{E}}$ is essentially small if there exists an equivalence of $\infty$-categories $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$, where $\operatorname{\mathcal{E}}'$ is a small $\infty$-category.

Corollary 5.6.3.12 (The Universal Cocartesian Fibration). For every simplicial set $\operatorname{\mathcal{C}}$, the construction

$\mathscr {F} \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F} = \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}}$

induces a bijection from $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})^{\simeq } )$ to the collection of equivalence classes of cocartesian fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ having essentially small fibers.

Proof. Apply Theorem 5.6.3.6 in the case $\operatorname{\mathcal{Q}}= \operatorname{\mathcal{QC}}$. $\square$

Let $\operatorname{\mathcal{S}}$ denote the $\infty$-category of spaces (Construction 5.4.1.1) and let $\operatorname{\mathcal{S}}_{\ast } = \operatorname{\mathcal{S}}_{\Delta ^0/}$ denote the $\infty$-category of pointed spaces (Construction 5.4.3.1).

Corollary 5.6.3.14 (The Universal Left Fibration). For every simplicial set $\operatorname{\mathcal{C}}$, the construction

$\mathscr {F} \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F} = \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}_{\ast }$

induces a bijection from $\pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})^{\simeq } )$ to the set of equivalence classes of left fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ having essentially small fibers.

Proof. Combine Theorem 5.6.3.6 (applied to the full subcategory $\operatorname{\mathcal{S}}\subset \operatorname{\mathcal{QC}}$) with with Proposition 5.1.4.14. $\square$