7.6.6 Small Limits
We now study limits and colimits indexed by diagrams of bounded size.
Definition 7.6.6.1. Let $\kappa $ be an infinite cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ is $\kappa $-complete if admits $K$-indexed limits, for every $\kappa $-small simplicial set $K$.
We say that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $\kappa $-small limits if it preserves $K$-indexed limits, for every $\kappa $-small simplicial set $K$ (Definition 7.1.4.4).
Example 7.6.6.3. An $\infty $-category $\operatorname{\mathcal{C}}$ is $\aleph _0$-complete if and only if it admits finite limits. Similarly, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $\aleph _0$-small limits if and only if it preserves finite limits.
Example 7.6.6.4. Let $\lambda $ be an uncountable cardinal and let $\kappa = \mathrm{ecf}(\lambda )$ be its exponential cofinality (Definition 4.7.3.16). Let $\operatorname{\mathcal{S}}^{< \lambda }$ denote the $\infty $-category of $\lambda $-small spaces (Variant 5.5.4.13) and let $\operatorname{\mathcal{QC}}^{ < \lambda }$ denote the $\infty $-category of $\lambda $-small $\infty $-categories (Variant 5.5.4.11). Then the $\infty $-categories $\operatorname{\mathcal{S}}^{< \lambda }$ and $\operatorname{\mathcal{QC}}^{< \lambda }$ are $\kappa $-complete. Moreover, the inclusion maps
\[ \operatorname{\mathcal{S}}^{ < \lambda } \hookrightarrow \operatorname{\mathcal{S}}\quad \quad \operatorname{\mathcal{QC}}^{< \lambda } \hookrightarrow \operatorname{\mathcal{QC}} \]
preserve $\kappa $-small limits. See Variants 7.4.1.15 and 7.4.4.14. In particular, if $\kappa = \lambda $ is a strongly inaccessible cardinal, then the $\infty $-categories $\operatorname{\mathcal{S}}^{< \kappa }$ and $\operatorname{\mathcal{QC}}^{< \kappa }$ are $\kappa $-complete.
Moreover, in either case, it suffices to consider the case where $K$ is an $\infty $-category. See Remark 7.1.1.15 and Proposition 4.7.5.5. Similarly, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $\kappa $-small limits if and only if it preserves $K$-indexed limits, for every simplicial set $K$ which is essentially $\kappa $-small (and it again suffices to consider the case where $K$ is an $\infty $-category).
Variant 7.6.6.7. Let $\kappa $ be an infinite cardinal. We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete if it admits $K$-indexed colimits, for every $\kappa $-small simplicial set $K$. Equivalently, the $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete if the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is $\kappa $-complete.
We say that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $\kappa $-small colimits if it preserves $K$-indexed colimits, for every $\kappa $-small simplicial set $K$. Equivalently, $F$ preserves $\kappa $-small colimits if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ preserves $\kappa $-small limits.
Example 7.6.6.8. Let $\lambda $ be an uncountable cardinal and let $\kappa = \mathrm{cf}(\lambda )$ denote its cofinality. Then the $\infty $-categories $\operatorname{\mathcal{S}}^{< \lambda }$ and $\operatorname{\mathcal{QC}}^{< \lambda }$ are $\kappa $-cocomplete. Moreover, the inclusion maps
\[ \operatorname{\mathcal{S}}^{ < \lambda } \hookrightarrow \operatorname{\mathcal{S}}\quad \quad \operatorname{\mathcal{QC}}^{< \lambda } \hookrightarrow \operatorname{\mathcal{QC}} \]
preserve $\kappa $-small colimits. See Corollaries 7.4.5.20 and 7.4.3.8. In particular, if $\kappa = \lambda $ is an uncountable regular cardinal, then the $\infty $-categories $\operatorname{\mathcal{S}}^{< \kappa }$ and $\operatorname{\mathcal{QC}}^{< \kappa }$ are $\kappa $-cocomplete.
Proposition 7.6.6.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa $ be an infinite cardinal. Then $\operatorname{\mathcal{C}}$ is $\kappa $-complete if and only if it satisfies the following conditions:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\kappa $-small products. That is, every collection of objects $\{ X_ j \} _{j \in J}$ indexed by a $\kappa $-small set $J$ admits a product in $\operatorname{\mathcal{C}}$.
- $(2)$
The $\infty $-category $\operatorname{\mathcal{C}}$ admits finite limits.
Proof.
Assume that $\operatorname{\mathcal{C}}$ satisfies conditions $(1)$ and $(2)$; we wish to show that $\operatorname{\mathcal{C}}$ is $\kappa $-complete (the converse is immediate from the definitions). Let $S$ be a $\kappa $-small simplicial set; we wish to show that $\operatorname{\mathcal{C}}$ admits $S$-indexed limits. If $\kappa = \aleph _0$, this follows immediately from assumption $(2)$ (Example 7.6.6.3). We may therefore assume that $\kappa $ is uncountable, so that $\operatorname{\mathcal{C}}$ admits countable products.
For each $n \geq 0$, let $\operatorname{sk}_{n}(S)$ denote the $n$-skeleton of $S$ (Construction 1.1.4.1), so that $S = \bigcup _{n} \operatorname{sk}_ n(S)$. It follows from Proposition 7.6.5.16 that $\operatorname{\mathcal{C}}$ admits sequential limits. Consequently, to show that $\operatorname{\mathcal{C}}$ admits $S$-indexed limits, it will suffice to show that it admits $\operatorname{sk}_ n(S)$-indexed limits, for each $n \geq 0$ (Corollary 7.6.5.14). We may therefore assume without loss of generality that the simplicial set $S$ has finite dimension. We proceed by induction on the dimension $n$ of $S$. If $n=-1$, then $S$ is empty and the desired result is immediate. Assume that $n \geq 0$ and let $\{ \sigma _ j \} _{j \in J}$ denote the collection of nondegenerate $n$-simplices of $S$, so that Proposition 1.1.4.12 supplies a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \underset {j \in J}{\coprod } \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \underset {j \in J}{\coprod } \Delta ^{n} \ar [d] \\ \operatorname{sk}_{n-1}(S) \ar [r] & \operatorname{sk}_{n}(S). } \]
Since the horizontal maps in this diagram are monomorphisms, it is also a categorical pushout square (Example 4.5.4.12). By virtue of Proposition 7.6.2.29, it will suffice to show that $\operatorname{\mathcal{C}}$ admits limits indexed by the simplicial sets $\operatorname{sk}_{n-1}(S)$, $J \times \operatorname{\partial \Delta }^{n}$, and $J \times \Delta ^ n$. In the first two cases, this follows from our inductive hypothesis. To handle the third case, we can use assumption $(1)$ and Corollary 7.6.1.22 to reduce to showing that the $\infty $-category $\operatorname{\mathcal{C}}$ admits $\Delta ^{n}$-indexed limits. This is clear, since the simplicial set $\Delta ^ n$ is an $\infty $-category containing an initial object (see Corollary 7.2.2.12).
$\square$
See Corollary 7.6.2.30 and 7.6.4.25.
Exercise 7.6.6.11. Let $\kappa $ be an infinite cardinal, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and suppose that $\operatorname{\mathcal{C}}$ is $\kappa $-complete. Show that $F$ preserves $\kappa $-small limits if and only if it preserves finite limits and $\kappa $-small products.
Corollary 7.6.6.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\lambda $ be an infinite cardinal which is not regular. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\lambda $-complete.
- $(2)$
For every infinite cardinal $\kappa < \lambda $, the $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-complete.
- $(3)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\lambda ^{+}$-complete, where $\lambda ^{+}$ denotes the successor cardinal of $\lambda $.
By virtue of Corollary 7.6.6.12, little information is lost by restricting the use of Definition 7.6.6.1 to the case where $\kappa $ is a regular cardinal.
Proof of Corollary 7.6.6.12.
The equivalence $(1) \Leftrightarrow (2)$ and the implication $(3) \Rightarrow (1)$ follow from Remark 7.6.6.2. We will complete the proof by showing that $(1)$ implies $(3)$. Assume that $\operatorname{\mathcal{C}}$ is $\lambda $-complete; we wish to show that it is $\lambda ^{+}$-complete. By virtue of Proposition 7.6.6.9, it will suffice to show that every collection of objects $\{ X_ i \} _{i \in I}$ admits a product in $\operatorname{\mathcal{C}}$, provided that the index set $I$ has cardinality $\leq \lambda $. Our assumption that $\lambda $ is not regular guarantees that we can decompose $I$ as a disjoint union of $\lambda $-small subsets $\{ I_ j \subseteq I \} _{j \in J}$, where the index set $J$ is $\lambda $-small. It follows from $(1)$ that $\operatorname{\mathcal{C}}$ admits $J$-indexed products and also that it admits $I_{j}$-indexed products for each $j \in J$, and therefore admits $I$-indexed products by virtue of Corollary 7.6.1.22.
$\square$
The existence of $\kappa $-small limits can be used to prove the existence of a large class of Kan extensions.
Proposition 7.6.6.13. Let $\kappa $ be an uncountable regular cardinal and let $K$ be a simplicial set which is essentially $\kappa $-small. Suppose we are given a pair of $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, together with diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$. Suppose that $\operatorname{\mathcal{C}}$ is locally $\kappa $-small and that $\operatorname{\mathcal{D}}$ is $\kappa $-complete. Then $F_0$ admits a right Kan extension along $\delta $.
Proof.
By virtue of Proposition 7.3.5.1, it will suffice to show that for every object $C \in \operatorname{\mathcal{C}}$, the composite map
\[ K \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{C/} \rightarrow K \xrightarrow { F_0 } \operatorname{\mathcal{D}} \]
admits a limit in the $\infty $-category $\operatorname{\mathcal{D}}$. Note that the projection map $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{C/}$ is a left fibration of simplicial sets (Proposition 4.3.6.1), whose fiber over each vertex $x \in K$ can be identified with the Kan complex $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( C,\delta (x) )$. Invoking Proposition 4.6.5.10, we see that $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( C,\delta (x) $ is homotopy equivalent to the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, \delta (x))$, and is therefore essentially $\kappa $-small (by virtue of our assumption that $\operatorname{\mathcal{C}}$ is locally $\kappa $-small). Since $K$ is essentially $\kappa $-small, Corollary 5.6.7.7 implies that the simplicial set $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{C/}$ is essentially $\kappa $-small. The desired result now follows from our assumption that $\operatorname{\mathcal{D}}$ is $\kappa $-complete (Remark 7.6.6.6).
$\square$