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7.3.1 Limits of $\infty $-Categories

Let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of (small) $\infty $-categories (Construction 5.4.4.1). Our goal in this section (and ยง7.3.3) is to show that the $\infty $-category $\operatorname{\mathcal{QC}}$ admits small limits (Corollary 7.3.1.20). In fact, we will prove something more precise: if $\operatorname{\mathcal{C}}$ is a small $\infty $-category, then the limit of any diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ can be realized as explicitly as a full subcategory of the $\infty $-category of sections of the cocartesian fibration $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ of Proposition 5.5.4.2 (Corollary 7.3.1.19).

Notation 7.3.1.1 (Cocartesian Sections). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets. Then the simplicial set

\[ \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}}) = \{ U' \} \times _{ \operatorname{Fun}(\operatorname{\mathcal{E}}',\operatorname{\mathcal{C}}) } \operatorname{Fun}(\operatorname{\mathcal{E}}', \operatorname{\mathcal{E}}) \]

is an $\infty $-category (see Corollary 4.1.4.7). We let $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})$ denote the full subcategory of $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})$ whose objects are morphisms $F: \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}$ which satisfy the identity $U \circ F = U'$ and carry $U'$-cocartesian edges of $\operatorname{\mathcal{E}}'$ to $U$-cocartesian edges of $\operatorname{\mathcal{E}}$. We will be particularly interested in the special case where $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}$ (and $U'$ is the identity map); in this case, we will refer to $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ as the $\infty $-category of cocartesian sections of $U$.

Variant 7.3.1.2 (Cartesian Sections). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be cartesian fibrations of simplicial sets. We let $\operatorname{Fun}^{\operatorname{Cart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})$ denote the full subcategory of $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})$ whose objects are morphisms $F: \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}$ which satisfy the identity $U \circ F = U'$ and carry $U'$-cartesian edges of $\operatorname{\mathcal{E}}'$ to $U$-cartesian edges of $\operatorname{\mathcal{E}}$. Note that we have a canonical isomorphism of simplicial sets

\[ \operatorname{Fun}^{\operatorname{Cart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})^{\operatorname{op}} = \operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}^{\operatorname{op}} }( \operatorname{\mathcal{E}}'^{\operatorname{op}}, \operatorname{\mathcal{E}}^{\operatorname{op}} ). \]

In the special case $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}$, we will refer to $\operatorname{Fun}^{\operatorname{Cart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ as the $\infty $-category of cartesian sections of $U$.

Remark 7.3.1.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets. Then the full subcategory $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}}) \subseteq \operatorname{Fun}_{/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})$ is replete (Example 4.4.1.11). That is, if $F$ and $G$ are isomorphic objects of $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})$, then $F$ carries $U'$-cocartesian edges of $\operatorname{\mathcal{E}}'$ to $U$-cocartesian edges of $\operatorname{\mathcal{E}}$ if and only if $G$ has the same property. In fact, we can be more precise: for every particular edge $e$ of $\operatorname{\mathcal{E}}'$, the image $F(e)$ is $U$-cocartesian if and only if $G(e)$ is $U$-cocartesian. To prove this, we can assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^1$, in which case it follows from Corollary 5.1.2.5.

Remark 7.3.1.4 (Functoriality). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets. Suppose that we are given a morphism of simplicial sets $F: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$, and set $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}'_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$. Then pullback along $F$ determines a morphism of simplicial sets

\[ F^{\ast }: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0 }^{\operatorname{CCart}}( \operatorname{\mathcal{E}}'_0, \operatorname{\mathcal{E}}_0 ), \]

which we will refer to as the restriction map.

Remark 7.3.1.5. In the situation of Remark 7.3.1.4, suppose that $F: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ is a monomorphism of simplicial sets. Then the restriction map $F^{\ast }: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0 }^{\operatorname{CCart}}( \operatorname{\mathcal{E}}'_0, \operatorname{\mathcal{E}}_0 )$ is an isofibration. To see this, we first observe that $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})$ can be regarded as a replete subcategory of the fiber product

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0 }^{\operatorname{CCart}}( \operatorname{\mathcal{E}}'_0, \operatorname{\mathcal{E}}_0 ) \times _{ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0 }( \operatorname{\mathcal{E}}'_0, \operatorname{\mathcal{E}}_0 ) } \operatorname{Fun}_{ /\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}}) \]

(Remark 7.3.1.3). It will therefore suffice to show that the restriction map

\[ \operatorname{Fun}_{ /\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0 }( \operatorname{\mathcal{E}}'_0, \operatorname{\mathcal{E}}_0 ) \simeq \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}'_0, \operatorname{\mathcal{E}}) \]

is an isofibration, which follows from Proposition 4.5.7.10.

Remark 7.3.1.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets, and let $K$ be an arbitrary simplicial set. Then:

  • The projection map $\operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ is also a cocartesian fibration.

  • The canonical isomorphism

    \[ \operatorname{Fun}(K, \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}}) ) \simeq \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{E}}) ) \]

    restricts to an isomorphism of full subcategories

    \[ \operatorname{Fun}(K, \operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}}) ) \simeq \operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{E}}) ). \]

Both assertions follow immediately from Theorem 5.2.1.1.

Remark 7.3.1.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Let $\operatorname{\mathcal{E}}^{\circ } \subseteq \operatorname{\mathcal{E}}$ be the simplicial subset whose $n$-simplices are maps $\Delta ^{n} \rightarrow \operatorname{\mathcal{E}}$ which carry each edge of $\Delta ^ n$ to a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$, so that $U$ restricts to a left fibration $U^{\circ }: \operatorname{\mathcal{E}}^{\circ } \rightarrow \operatorname{\mathcal{C}}$ (see Corollary 5.1.4.15). Then $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}^{\circ } )$ can be identified with the core of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.

We can now state our main result.

Theorem 7.3.1.8 (Diffraction Criterion). Suppose we are given a pullback diagram of small simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleleft }, } \]

where $U$ and $\overline{U}$ are cocartesian fibrations. The following conditions are equivalent:

$(1)$

The restriction map

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]

is an equivalence of $\infty $-categories.

$(2)$

The covariant transport representation

\[ \operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleleft } }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{QC}} \]

of Notation 5.6.2.12 is a limit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$.

Remark 7.3.1.9. In the situation of Theorem 7.3.1.8, the restriction map $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is automatically an isofibration of $\infty $-categories (Remark 7.3.1.5). Using Proposition 4.5.7.16, we see that condition $(1)$ of Theorem 7.3.1.8 is equivalent to the following a priori stronger condition:

$(1')$

The restriction map

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]

is a trivial Kan fibration of simplicial sets.

In the situation of Theorem 7.3.1.8, the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} )$ admits a more concrete description: it can be identified with the fiber of $\overline{U}$ over the cone point of the simpicial set $\operatorname{\mathcal{C}}^{\triangleleft }$. This is a consequence of the following more general observation:

Proposition 7.3.1.10. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $F: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ be a left cofinal morphism of simplicial sets, and set $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Then the restriction map

\[ F^{\ast }: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0 }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{E}}_0 ) \]

of Remark 7.3.1.4 is an equivalence of $\infty $-categories.

Proof. By virtue of Theorem 4.5.4.1, it will suffice to show that for every simplicial set $K$, the induced map

\[ \operatorname{Fun}(K, \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}))^{\simeq } \rightarrow \operatorname{Fun}(K, \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0 }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{E}}_0 ))^{\simeq } \]

is a homotopy equivalence of Kan complexes (in fact, it suffices to verify this for $K = \Delta ^1$). Replacing $\operatorname{\mathcal{E}}$ by the fiber product $\operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{E}})$ and using Remark 7.3.1.6, we are reduced to proving that $F^{\ast }$ restricts to a homotopy equivalence $\theta : F^{\ast }: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})^{\simeq } \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{E}})^{\simeq }$. Let $U^{\circ }: \operatorname{\mathcal{E}}^{\circ } \rightarrow \operatorname{\mathcal{E}}$ denote the underlying left fibration of $U$. Using Remark 7.3.1.7, we can identify $\theta $ with the map

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}^{\circ } ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0 }( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}^{\circ } ) \simeq \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{E}}^{\circ } ), \]

given by precomposition with $F$, which is a homotopy equivalence by virtue of our assumption that $F$ is left cofinal. $\square$

Corollary 7.3.1.11. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $F: \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ be a left anodyne morphism of simplicial sets, and set $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Then the restriction map

\[ F^{\ast }: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0 }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{E}}_0 ) \]

of Remark 7.3.1.4 is a trivial Kan fibration.

Proof. Combining Proposition 7.2.1.3 with Proposition 7.3.1.10, we see that $F^{\ast }$ is an equivalence of $\infty $-categories. Since it is also an isofibration (Remark 7.3.1.5), it is a trivial Kan fibration (Proposition 4.5.7.16). $\square$

Corollary 7.3.1.12. Let $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleleft }$ be a cocartesian fibration of simplicial sets. Let $\overline{\operatorname{\mathcal{E}}}_{ {\bf 0} }$ denote the fiber of $\overline{U}$ over the cone point ${\bf 0} \in \operatorname{\mathcal{C}}^{\triangleleft }$. Then evaluation at ${\bf 0}$ induces a trivial Kan fibration of simplicial sets

\[ \operatorname{Fun}_{/ \operatorname{\mathcal{C}}^{\triangleleft } }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \{ {\bf 0} \} \times _{ \operatorname{\mathcal{C}}^{\triangleleft } } \overline{\operatorname{\mathcal{E}}}. \]

Construction 7.3.1.13 (Covariant Diffraction). Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleleft }, } \]

where $U$ and $\overline{U}$ are cocartesian fibrations. Let $\overline{\operatorname{\mathcal{E}}}_{ {\bf 0} }$ denote the fiber of $\overline{U}$ over the cone point ${\bf 0} \in \operatorname{\mathcal{C}}^{\triangleleft }$. We then have restriction maps

\[ \overline{\operatorname{\mathcal{E}}}_{ {\bf 0} } \xleftarrow {\operatorname{ev}} \operatorname{Fun}_{/ \operatorname{\mathcal{C}}^{\triangleleft } }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \xrightarrow {\theta } \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}), \]

where $\operatorname{ev}$ is a trivial Kan fibration (Corollary 7.3.1.12). Composing $\theta $ with a section of $\operatorname{ev}$, we obtain a functor of $\infty $-categories $\mathrm{Df}: \overline{\operatorname{\mathcal{E}}}_{ {\bf 0} } \rightarrow \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ which is well-defined up to isomorphism. We will refer to $\mathrm{Df}$ as the covariant diffraction functor associated to the cocartesian fibration $\overline{U}$.

Remark 7.3.1.14. In the situation of Construction 7.3.1.13, let $C \in \operatorname{\mathcal{C}}$ be a vertex and let $\operatorname{ev}_{C}: \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}_{C}$ be the evaluation functor, given on objects by $\operatorname{ev}_{C}(F) = F(C)$. Then the composition

\[ \overline{\operatorname{\mathcal{E}}}_{ {\bf 0} } \xrightarrow {\mathrm{Df}} \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \xrightarrow {\operatorname{ev}_ C} \operatorname{\mathcal{E}}_{C} \]

is given by covariant transport along the unique edge ${\bf 0} \rightarrow C$ of $\operatorname{\mathcal{C}}^{\triangleleft }$.

Remark 7.3.1.15. Suppose we are given a pullback diagram of small simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleleft }. } \]

Then the covariant diffraction functor $\mathrm{Df}: \overline{\operatorname{\mathcal{E}}}_{ {\bf 0} } \rightarrow \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of Construction 7.3.1.13 is an equivalence of $\infty $-categories if and only if the covariant transport representation $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleleft } }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{QC}}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$ (this is a restatement of Theorem 7.3.1.8).

We now show that there exists a good supply of cocartesian fibrations which satisfy the hypotheses of Theorem 7.3.1.8.

Proposition 7.3.1.16. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then there exists a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleleft }, } \]

where $\overline{U}$ is a cocartesian fibration and the restriction map

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]

is an equivalence of $\infty $-categories.

Proof. Let $\operatorname{ev}: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ denote the evaluation morphism (given on vertices by the formula $\operatorname{ev}( F, C) = F(C)$), and let

\[ \operatorname{\mathcal{E}}' = (\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}) \star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}} \]

denote the relative join of Construction 5.2.4.1. Note that we have a canonical map

\[ U': \operatorname{\mathcal{E}}' = (\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}) \star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\simeq \Delta ^{1} \times \operatorname{\mathcal{C}}. \]

Let $\pi : \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ be given by projection onto the second factor. Note that $\pi $ is a cocartesian fibration, and that an edge of the product $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}$ is $\pi $-cocartesian if and only if its image in $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is an isomorphism. It follows that the $\operatorname{ev}$ carries $\pi $-cocartesian edges of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}$ to $U$-cocartesian edges of $\operatorname{\mathcal{E}}$. Applying Lemma 5.2.4.17, we deduce that $U'$ is a cocartesian fibration. By construction, we can identify $\operatorname{\mathcal{E}}$ with the inverse image of $\{ 1\} \times \operatorname{\mathcal{C}}$ under $U'$.

Let $\operatorname{\mathcal{E}}''$ denote the pushout

\[ ( \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}^{\triangleleft } ) \coprod _{ ( \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}^{\triangleleft } ) } \operatorname{\mathcal{E}}'. \]

Amalgamating $U'$ with the projection map $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}^{\triangleleft }$, we obtain a morphism of simplicial sets $U'': \operatorname{\mathcal{E}}'' \rightarrow K$, where $K$ denotes the pushout $( \{ 0 \} \times \operatorname{\mathcal{C}})^{\triangleleft } \coprod _{ ( \{ 0\} \times \operatorname{\mathcal{C}}) } ( \Delta ^1 \times \operatorname{\mathcal{C}})$. It follows from Proposition 5.1.4.7 that $U''$ is also a cocartesian fibration.

Let us abuse notation by identifying $K$ with its image in the simplicial set $( \Delta ^1 \times \operatorname{\mathcal{C}})^{\triangleleft }$. Since the inclusion map $\{ 0\} \times \operatorname{\mathcal{C}}\hookrightarrow \Delta ^1 \times \operatorname{\mathcal{C}}$ is left anodyne (Proposition 4.2.5.3), the inclusion $K \hookrightarrow (\Delta ^1 \times \operatorname{\mathcal{C}})^{\triangleleft }$ is inner anodyne (Example 4.3.6.5). Applying Proposition 5.6.4.2, we can write $U''$ as the pullback of a cocartesian fibration $U''': \operatorname{\mathcal{E}}''' \rightarrow (\Delta ^1 \times \operatorname{\mathcal{C}})^{\triangleleft }$. We then have a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \operatorname{\mathcal{E}}' \ar [r] \ar [d]^{U'} & \operatorname{\mathcal{E}}'' \ar [r] \ar [d]^{U''} & \operatorname{\mathcal{E}}''' \ar [d]^{U'''} \\ \{ 1\} \times \operatorname{\mathcal{C}}\ar [r] & \Delta ^1 \times \operatorname{\mathcal{C}}\ar [r] & K \ar [r] & ( \Delta ^1 \times \operatorname{\mathcal{C}})^{\triangleleft }, } \]

where each square is a pullback and each vertical map is a cocartesian fibration. Let $\overline{\operatorname{\mathcal{E}}}$ denote the pullback $( \{ 1\} \times \operatorname{\mathcal{C}})^{\triangleleft } \times _{ (\Delta ^1 \times \operatorname{\mathcal{C}})^{\triangleleft } } \operatorname{\mathcal{E}}'''$, so that $U'''$ restricts to a cocartesian fibration $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow ( \{ 1\} \times \operatorname{\mathcal{C}})^{\triangleleft }$. We will complete the proof by showing that the commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \{ 1\} \times \operatorname{\mathcal{C}}\ar [r] & (\{ 1\} \times \operatorname{\mathcal{C}})^{\triangleleft } } \]

satisfies the requirements of Proposition 7.3.1.16.

For every simplicial subset $A \subseteq (\Delta ^1 \times \operatorname{\mathcal{C}})^{\triangleleft }$, let $\operatorname{\mathcal{D}}(A)$ denote the $\infty $-category

\[ \operatorname{Fun}_{ / A}^{\operatorname{CCart}}( A, A \times _{ ( \Delta ^1 \times \operatorname{\mathcal{C}})^{\triangleleft } } \operatorname{\mathcal{E}}'''). \]

Let ${\bf 0}$ denote the cone point of $(\Delta ^1 \times \operatorname{\mathcal{C}})^{\triangleleft }$. Note that we have a commutative diagram of restriction functors

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}( ( \Delta ^1 \times \operatorname{\mathcal{C}})^{\triangleleft } ) \ar [r]^-{\alpha '} \ar [d]^{\beta } & \operatorname{\mathcal{D}}( (\{ 1\} \times \operatorname{\mathcal{C}})^{\triangleleft } ) \ar [d]^{\alpha } \\ \operatorname{\mathcal{D}}(K) \ar [r]^-{\beta '} \ar [d]^{\gamma } & \operatorname{\mathcal{D}}( \{ 1\} \times \operatorname{\mathcal{C}}) \\ \operatorname{\mathcal{D}}( \{ {\bf 0} \} ). & } \]

We wish to show that $\alpha $ is an equivalence of $\infty $-categories. Since the inclusion $K \hookrightarrow ( \Delta ^1 \times \operatorname{\mathcal{C}})^{\triangleleft }$ is inner anodyne (as noted above) and the inclusion $( \{ 1\} \times \operatorname{\mathcal{C}})^{\triangleleft } \hookrightarrow ( \Delta ^1 \times \operatorname{\mathcal{C}})^{\triangleleft }$ is left anodyne (Lemma 4.3.7.8), the morphisms $\alpha '$ and $\beta $ are trivial Kan fibrations (Corollary 7.3.1.11). It will therefore suffice to show that $\beta '$ is an equivalence of $\infty $-categories.

Amalgamating the map

\begin{eqnarray*} \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \Delta ^1 \times \operatorname{\mathcal{C}}& \simeq & (\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}) \star _{ (\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}) } (\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}) \\ & \rightarrow & (\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}) \star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}\\ & = & \operatorname{\mathcal{E}}' \end{eqnarray*}

with the identity on $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}^{\triangleleft }$, we obtain a morphism of simplicial sets $F: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times K \rightarrow \operatorname{\mathcal{E}}''$. If $e$ is an edge of the product $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times K$ whose image in $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is an isomorphism, then $F(e)$ is a $U''$-cocartesian edge of $\operatorname{\mathcal{E}}''$. We can therefore identify $F$ with a morphism of simplicial sets $f: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{D}}(K)$. Unwinding the definitions, we see that $\beta ' \circ f$ is an isomorphism of simplicial sets. Consequently, to show that $\beta '$ is an equivalence of $\infty $-categories, it will suffice to show that $f$ is an equivalence of $\infty $-categories. Similarly, the composite map $\gamma \circ f$ is an isomorphism, so we are reduced to proving that $\gamma $ is an equivalence of $\infty $-categories. Since $\beta $ is a trivial Kan fibration, this is equivalent to the assertion that $\gamma \circ \beta $ is an equivalence of $\infty $-categories, which is a special case of Corollary 7.3.1.12. $\square$

Remark 7.3.1.17. If $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of small simplicial sets, then the simplicial set $\overline{\operatorname{\mathcal{E}}}$ constructed in the proof of Proposition 7.3.1.16 will also be small.

Corollary 7.3.1.18. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of small simplicial sets and let $\operatorname{Tr}_{ \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a covariant transport representation for $U$. Then the diagram $\operatorname{Tr}_{ \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}$ has a limit in the $\infty $-category $\operatorname{\mathcal{QC}}$, given by the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of cocartesian sections of $U$.

Proof. Using Proposition 7.3.1.16 (and Remark 7.3.1.17), we see that there exists a pullback diagram of small simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleleft }, } \]

where $\overline{U}$ is a cocartesian fibration and the restriction map $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is a trivial Kan fibration. Using Corollary 5.6.2.9, we can extend $\operatorname{Tr}_{ \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}$ to a diagram $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleleft } }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{QC}}$ which is a covariant transport representation for $\overline{U}$. Let ${\bf 0}$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleleft }$. It follows from Theorem 7.3.1.8 that $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleleft } }$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$, and therefore exhibits the $\infty $-category $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleleft } }( {\bf 0} ) \simeq \overline{\operatorname{\mathcal{E}}}_{ {\bf 0} }$ as a limit of the diagram $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$. Using Remark 7.3.1.15, we see that covariant diffraction supplies an equivalence of $\infty $-categories $\overline{\operatorname{\mathcal{E}}}_{ {\bf 0} } \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$, so that $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is also a limit of the diagram $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ (Proposition 7.1.1.11). $\square$

Corollary 7.3.1.19. Let $\operatorname{\mathcal{C}}$ be a small simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$. Then the $\infty $-category of cocartesian sections $\operatorname{Fun}_{ /\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \int _{\operatorname{\mathcal{C}}} \mathscr {F} )$ is a limit of the diagram $\mathscr {F}$.

Proof. Apply Corollary 7.3.1.18 to the cocartesian fibration $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$. $\square$

Corollary 7.3.1.20. The $\infty $-category $\operatorname{\mathcal{QC}}$ admits small limits.