Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

5.4.9 Smallness of Fibrations

It will sometimes be convenient to work with a relative version of Definition 5.4.5.1.

Definition 5.4.9.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets and let $\kappa $ be an uncountable regular cardinal. We say that $U$ is essentially $\kappa $-small if, for every simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small. We say that $U$ is locally $\kappa $-small if, for every simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is locally $\kappa $-small.

Variant 5.4.9.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. We say that $U$ is essentially small if, for every simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially small. We say that $U$ is locally small if, for every $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is locally small.

Remark 5.4.9.3. Let $\kappa $ be an uncountable regular cardinal and suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix { \operatorname{\mathcal{E}}' \ar [d]^{U'} \ar [r] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}, } \]

where $U$ and $U'$ are inner fibrations. If $U$ is essentially $\kappa $-small, then $U'$ is essentially $\kappa $-small. If $U$ is locally $\kappa $-small, then $U'$ is locally $\kappa $-small.

Remark 5.4.9.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets and let $\kappa $ be an uncountable regular cardinal. Then $U$ is essentially $\kappa $-small if and only if it is locally $\kappa $-small and, for each vertex $C \in \operatorname{\mathcal{C}}$, the set of isomorphism classes $\pi _0( \operatorname{\mathcal{E}}_{C}^{\simeq } )$ is $\kappa $-small. See Proposition 5.4.8.7.

Proposition 5.4.9.5. Let $\kappa $ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\kappa $-small, and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration. The following conditions are equivalent:

$(1)$

The inner fibration $U$ is locally $\kappa $-small.

$(2)$

For every edge $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is locally $\kappa $-small.

$(3)$

The $\infty $-category $\operatorname{\mathcal{E}}$ is locally $\kappa $-small.

Proof. The implication $(1) \Rightarrow (2)$ is immediate from the definitions. Assume next that $(2)$ is satisfied; we will prove $(3)$. Let $X$ and $Y$ be objects of $\operatorname{\mathcal{E}}$, and set $\overline{X} = U(X)$ and $\overline{Y} = U(Y)$. We wish to show that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)$ is essentially $\kappa $-small. By virtue of Proposition 4.6.1.21, the functor $U$ induces a Kan fibration $\theta : \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} )$. Our assumption that $\operatorname{\mathcal{C}}$ is locally $\kappa $-small guarantees that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} )$ is essentially $\kappa $-small. By virtue of Corollary 5.4.7.2, it will suffice to show that for every morphism $\overline{e}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\overline{e}} = \{ \overline{e} \} \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} ) } \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)$ is essentially $\kappa $-small. This follows immediately from assumption $(2)$.

We now complete the proof by showing that $(3)$ implies $(1)$. Assume that $\operatorname{\mathcal{E}}$ is locally $\kappa $-small and choose a simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$; we will show that the $\infty $-category $\operatorname{\mathcal{E}}_{\sigma } = \Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is locally $\kappa $-small. Fix a pair of objects $\widetilde{X}, \widetilde{Y} \in \operatorname{\mathcal{E}}_{\sigma }$; we wish to show that the Kan complex $\operatorname{Hom}_{ \operatorname{\mathcal{E}}_{\sigma } }( \widetilde{X}, \widetilde{Y} )$ is essentially $\kappa $-small. Let $X$ and $Y$ denote the images of $\widetilde{X}$ and $\widetilde{Y}$ in the $\infty $-category $\operatorname{\mathcal{E}}$, and set $\overline{X} = U(X)$ and $\overline{Y} = U(Y)$ as above. If the Kan complex $\operatorname{Hom}_{ \operatorname{\mathcal{E}}_{\sigma } }( \widetilde{X}, \widetilde{Y} )$ is nonempty, then it can be identified with a fiber of the Kan fibration $\theta : \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} )$, which is essentially $\kappa $-small by virtue of Corollary 5.4.7.2. $\square$

Corollary 5.4.9.6. Let $\kappa $ be an uncountable regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small, and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration. If the $\infty $-category $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small, then then $U$ is essentially $\kappa $-small. The converse holds if $U$ is an isofibration.

Proof. Assume first that $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small. Applying Proposition 5.4.9.5, we deduce that $U$ is locally $\kappa $-small. It will therefore suffice to show that, for each object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small (Remark 5.4.9.4). Using Corollary 5.3.7.7, we can factor $U$ as a composition $\operatorname{\mathcal{E}}\xrightarrow {\iota } \operatorname{\mathcal{E}}' \xrightarrow {U'} \operatorname{\mathcal{C}}$, where $U'$ is an isofibration and $\iota $ is an equivalence of $\infty $-categories. Then $\operatorname{\mathcal{E}}'$ is essentially $\kappa $-small, so Corollary 5.4.5.16 guarantees that the fiber $\operatorname{\mathcal{E}}'_{C}$ is essentially $\kappa $-small. Remark 5.3.7.8 guarantees that the map of fibers $\operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$, so $\operatorname{\mathcal{E}}_{C}$ is also essentially $\kappa $-small (Corollary 5.4.5.13).

Now suppose that $U$ is an isofibration which is essentially $\kappa $-small; we wish to show that the $\infty $-category $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small. Proposition 5.4.9.5 guarantees that $U$ is locally $\kappa $-small. It will therefore suffice to show that the set of isomorphism classes $\pi _0( \operatorname{\mathcal{E}}^{\simeq } )$ is $\kappa $-small. In fact, we will show that the core $\operatorname{\mathcal{E}}^{\simeq }$ is an essentially $\kappa $-small Kan complex. This is a special case of Corollary 5.4.7.2, since $U$ induces a Kan fibration $U^{\simeq }: \operatorname{\mathcal{E}}^{\simeq } \rightarrow \operatorname{\mathcal{C}}^{\simeq }$ whose fibers are essentially $\kappa $-small (see Proposition 4.4.3.7). $\square$

Warning 5.4.9.7. If $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is not assumed to be an isofibration, then the converse assertion of Corollary 5.4.9.6 does not necessarily hold. For example, suppose that $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$ is the nerve of an essentially $\kappa $-small category $\operatorname{\mathcal{C}}_0$, and let $\operatorname{\mathcal{E}}= \operatorname{sk}_0( \operatorname{\mathcal{C}})$ be the constant simplicial set associated to the collection of objects of $\operatorname{\mathcal{C}}_0$. Then the inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{C}}$ is an essentially $\kappa $-small inner fibration (which is usually not an isofibration). However, the $\infty $-category $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small if and only if the set of objects of $\operatorname{\mathcal{C}}_0$ is $\kappa $-small.

Corollary 5.4.9.8. Let $\kappa $ be an uncountable regular cardinal. Then an $\infty $-category $\operatorname{\mathcal{C}}$ is locally $\kappa $-small (in the sense of Definition 5.4.8.1) if and only if the inner fibration $U: \operatorname{\mathcal{C}}\rightarrow \Delta ^0$ is locally $\kappa $-small (in the sense of Definition 5.4.9.1). Similarly, $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small (in the sense of Definition 5.4.5.1) if and only if $U$ is essentially $\kappa $-small.

Corollary 5.4.9.9 (Transitivity of Local Smallness). Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be inner fibrations of simplicial sets, let $\kappa $ be an uncountable regular cardinal, and suppose that $U$ is locally $\kappa $-small. Then $V$ is locally $\kappa $-small if and only if $U \circ V$ is locally $\kappa $-small.

Proof. Suppose first that $V$ is locally $\kappa $-small. Choose an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$; we wish to show that the $\infty $-category $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is locally $\kappa $-small. This follows by applying Proposition 5.4.9.5 to the inner fibration of $\infty $-categories $(\operatorname{id}\times V): \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, which is locally $\kappa $-small by virtue of Remark 5.4.9.3.

Now suppose that $U \circ V$ is locally $\kappa $-small, and choose an $n$-simplex $\widetilde{\sigma }: \Delta ^ n \rightarrow \operatorname{\mathcal{D}}$; we wish to show that the fiber product $\Delta ^ n \times _{ \operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}$ is locally $\kappa $-small. To prove this, we are free to replace $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ by $\Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ and $\Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, respectively, and thereby reduce to the case where $\operatorname{\mathcal{C}}= \Delta ^ n$ is a locally $\kappa $-small $\infty $-category. In this case, our assumptions on $U$ and $U \circ V$ guarantee that the $\infty $-categories $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ are also locally $\kappa $-small (Proposition 5.4.9.5), so that $V$ is automatically locally $\kappa $-small (by Proposition 5.4.9.5 again). $\square$

Variant 5.4.9.10 (Transitivity of Essential Smallness). Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be inner fibrations of simplicial sets, let $\kappa $ be an uncountable regular cardinal, and suppose that $U$ is essentially $\kappa $-small. Then:

  • If $V$ is an essentially $\kappa $-small isofibration, then $U \circ V$ is essentially $\kappa $-small.

  • If $U \circ V$ is essentially $\kappa $-small, then $V$ is essentially $\kappa $-small.

Proof. We proceed as in the proof of Corollary 5.4.9.9. Assume first that $V$ is an essentially $\kappa $-small isofibration, and choose an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$; we wish to show that the $\infty $-category $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small. This follows by applying Corollary 5.4.9.6 to the isofibration of $\infty $-categories $(\operatorname{id}\times V): \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, which is essentially $\kappa $-small by virtue of Remark 5.4.9.3.

Now suppose that $U \circ V$ is essentially $\kappa $-small, and choose an $n$-simplex $\widetilde{\sigma }: \Delta ^ n \rightarrow \operatorname{\mathcal{D}}$; we wish to show that the fiber product $\Delta ^ n \times _{ \operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small. To prove this, we are free to replace $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ by $\Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ and $\Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, respectively, and thereby reduce to the case where $\operatorname{\mathcal{C}}= \Delta ^ n$. In particular, we may assume that $\operatorname{\mathcal{C}}$ is an essentially $\kappa $-small $\infty $-category and that the functors $U$ and $U \circ V$ are isofibrations (Example 4.4.1.6). Applying Corollary 5.4.9.6, we deduce that the $\infty $-categories $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ are essentially $\kappa $-small, so that $V$ is also automatically essentially $\kappa $-small (by Corollary 5.4.9.6 again). $\square$

Corollary 5.4.9.11. Let $\kappa $ be an uncountable regular cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories. Then $U$ is locally $\kappa $-small if and only if, for every edge $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is locally $\kappa $-small.

Proof. The “only if” direction is immediate from the definitions. To prove the converse, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex, in which case the desired result follows from Proposition 5.4.9.5. $\square$

Corollary 5.4.9.12. Let $\kappa $ be an uncountable regular cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories. Then $U$ is essentially $\kappa $-small if and only if, for every edge $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small.

Warning 5.4.9.13. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories. If $U$ is essentially $\kappa $-small, then the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small for each object $C \in \operatorname{\mathcal{C}}$. Beware that the converse is false in general. For example, let $S$ be a set and let $\operatorname{\mathcal{E}}$ be the category containing a pair of objects $X$ and $Y$, with morphisms given by

\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( X, X) = \{ \operatorname{id}_{X} \} \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{E}}}( Y, Y ) = \{ \operatorname{id}_{Y} \} \]
\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y) = S \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{E}}}(Y,X) = \emptyset . \]

Then there is a unique isofibration $U: \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}) \rightarrow \Delta ^1$ satisfying $U(X) = 0$ and $U(Y) = 1$. The fibers $U^{-1} \{ 0\} $ and $U^{-1} \{ 1\} $ are isomorphic to $\Delta ^0$ (and are therefore essentially $\kappa $-small for every uncountable cardinal $\kappa $). However, the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{E}})$ is essentially $\kappa $-small if and only if the set $S$ is $\kappa $-small.

The behavior described in Warning 5.4.9.13 can be avoided if we restrict our attention to cartesian (or cocartesian) fibrations.

Proposition 5.4.9.14. Let $\kappa $ be an uncountable regular cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets. Then $U$ is locally $\kappa $-small if and only if, for every object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is locally $\kappa $-small.

Proof. Assume that, for every object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ is locally $\kappa $-small; we will show that $U$ is locally $\kappa $-small (the reverse implication is immediate from the definitions). By virtue of Corollary 5.4.9.11, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^1$; in this case, we wish to show that $\operatorname{\mathcal{E}}$ is locally $\kappa $-small. Fix a pair of objects $X,Z \in \operatorname{\mathcal{E}}$; we wish to show that the Kan complex $\operatorname{Hom}_{ \operatorname{\mathcal{E}}}(X, Z )$ is essentially $\kappa $-small. We may assume without loss of generality that $U(X) \leq U(Z)$ (otherwise, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z)$ is empty). If $U(X) = U(Z)$, then the desired result follows from our hypothesis. It will therefore suffice to treat the case where $U(X) = 0$ and $U(Z) = 1$. Since $U$ is a cartesian fibration, we can choose a $U$-cartesian morphism $f: Y \rightarrow Z$ of $\operatorname{\mathcal{E}}$, where $U(Y) = 0$. In this case, composition with the homotopy class $[f]$ induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z)$ (Corollary 5.1.2.3), so the desired result follows from the local $\kappa $-smallness of the fiber $\{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. $\square$

Corollary 5.4.9.15. Let $\kappa $ be an uncountable regular cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets. Then $U$ is essentially $\kappa $-small if and only if, for every object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small.

Corollary 5.4.9.16. Let $\kappa $ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and suppose that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. The following conditions are equivalent:

$(1)$

The $\infty $-category $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small.

$(2)$

For every vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small.

Proof. Using Corollary 4.1.3.3, we can choose an inner anodyne morphism $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}'$, where $\operatorname{\mathcal{C}}'$ is an $\infty $-category. Using Proposition 5.7.7.2, we can write $U$ as the pullback of a cocartesian fibration $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}'$. Proposition 5.3.6.1 then guarantees that the inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{E}}'$ is a categorical equivalence of simplicial sets. Since the inclusion $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}'$ is bijective on vertices, every fiber of $U'$ can also be regarded as a fiber of $U$. We can therefore replace $U$ by $U'$, and thereby reduce to proving Corollary 5.4.9.16 in the special case where $U$ is a cocartesian fibration of $\infty $-categories. The desired result now follows by combining Corollaries 5.4.9.15 and 5.4.9.6. $\square$