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7.2.4 Filtered $\infty $-Categories

We begin by recalling the classical notion of a filtered category.

Definition 7.2.4.1. Let $\operatorname{\mathcal{C}}$ be a category. We say that $\operatorname{\mathcal{C}}$ is filtered if it satisfies the following conditions:

  • The category $\operatorname{\mathcal{C}}$ is nonempty.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, there exists an object $Z \in \operatorname{\mathcal{C}}$ and a pair of morphisms $u: X \rightarrow Z$ and $v: Y \rightarrow Z$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$ and every pair of morphisms $f_0, f_1: X \rightarrow Y$, there exists a morphism $v: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$ satisfying $v \circ f_0 = v \circ f_1$.

Exercise 7.2.4.2. We say that a partially ordered set $(A, \leq )$ is directed if every finite subset $A_0 \subseteq A$ has an upper bound. Show that $(A, \leq )$ is directed if and only if it is filtered, when regarded as a category.

Our goal in this section is to introduce an $\infty $-categorical counterpart of Definition 7.2.4.1:

Definition 7.2.4.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ is filtered if, for every finite simplicial set $K$, every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$ admits an extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.

In ยง7.2.5, we will show that Definition 7.2.4.3 is a generalization of Definition 7.2.4.1: that is, a category $\operatorname{\mathcal{C}}$ is filtered if and only if the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is filtered (Corollary 7.2.5.8).

Variant 7.2.4.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ is cofiltered if, for every finite simplicial set $K$, every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$ admits an extension $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. Equivalently, $\operatorname{\mathcal{C}}$ is cofiltered if the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is filtered.

Example 7.2.4.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which contains a final object $X$. Then every morphism of simplicial sets $f: K \rightarrow \operatorname{\mathcal{C}}$ can be extended to a morphism $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which carries the cone point of $K^{\triangleright }$ to the object $X$. In particular, the $\infty $-category $\operatorname{\mathcal{C}}$ is filtered. For a more general statement, see Proposition 7.2.7.1.

Remark 7.2.4.6. Let $\{ \operatorname{\mathcal{C}}_{\alpha } \} $ be a filtered diagram of simplicial sets, where each $\operatorname{\mathcal{C}}_{\alpha }$ is a filtered $\infty $-category. Then the colimit $\operatorname{\mathcal{C}}= \varinjlim _{\alpha } \operatorname{\mathcal{C}}_{\alpha }$ is also a filtered $\infty $-category. To prove this, we first observe that $\operatorname{\mathcal{C}}$ is an $\infty $-category (Remark 1.3.0.9). If $K$ is a finite simplicial set, then any morphism $f: K \rightarrow \operatorname{\mathcal{C}}$ factors through $f_{\alpha }: K \rightarrow \operatorname{\mathcal{C}}_{\alpha }$ for some index $\alpha $ (see Proposition 3.5.1.9). Our assumption that $\operatorname{\mathcal{C}}_{\alpha }$ is filtered guarantees that $f_{\alpha }$ extends to a diagram $\overline{f}_{\alpha }: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{\alpha }$, from which it follows that $f$ extends to a diagram $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.

Remark 7.2.4.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:

$(1)$

The $\infty $-category $\operatorname{\mathcal{C}}$ is filtered.

$(2)$

For every finite simplicial set $K$ and every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$, the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is nonempty.

$(3)$

For every finite simplicial set $K$ and every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$, the oriented fiber product $\{ f\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$ is nonempty.

$(4)$

For every finite simplicial set $K$ and every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$, there exists a morphism $f \rightarrow f'$ in the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$, where $f': K \rightarrow \operatorname{\mathcal{C}}$ is a constant diagram.

The equivalences $(1) \Leftrightarrow (2)$ and $(3) \Leftrightarrow (4)$ follow immediately from the definitions, and the equivalence $(2) \Leftrightarrow (3)$ follows from Theorem 4.6.4.16.

Proposition 7.2.4.8. Let $\operatorname{\mathcal{C}}$ be a filtered $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, where $K$ is a finite simplicial set. Then the $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is also filtered.

Proof. By virtue of Remark 7.2.4.7, it will suffice to show that for every finite simplicial set $L$ and every morphism $g: L \rightarrow \operatorname{\mathcal{C}}_{f/}$, the $\infty $-category $(\operatorname{\mathcal{C}}_{f/} )_{g/}$ is nonempty. Unwinding the definitions, we can identify $g$ with a morphism of simplicial sets $\overline{f}: K \star L \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{f}|_{K} = f$. This identification supplies an isomorphism $(\operatorname{\mathcal{C}}_{f/} )_{g/} \simeq \operatorname{\mathcal{C}}_{ \overline{f} / }$. We are therefore reduced to showing that the coslice $\infty $-category $\operatorname{\mathcal{C}}_{ \overline{f} / }$ is nonempty. This follows from Remark 7.2.4.7, since the simplicial set $K \star L$ is finite (Remark 4.3.3.16). $\square$

Proposition 7.2.4.9. Let $\operatorname{\mathcal{C}}$ be a filtered $\infty $-category. Then $\operatorname{\mathcal{C}}$ is weakly contractible.

Proof. By virtue of Proposition 3.1.7.1, there exists a functor $Q: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ and a natural transformation $u: \operatorname{id}_{\operatorname{Set_{\Delta }}} \rightarrow Q$ with the following properties:

  • The functor $Q$ commutes with filtered colimits.

  • For every simplicial set $X$, the simplicial set $Q(X)$ is a Kan complex.

  • For every simplicial set $X$, the morphism $u_{X}: X \rightarrow Q(X)$ is a weak homotopy equivalence.

To show that $\operatorname{\mathcal{C}}$ is weakly contractible, it will suffice to show that the Kan complex $Q(\operatorname{\mathcal{C}})$ is contractible. Note that $\operatorname{\mathcal{C}}$ is nonempty, so that $Q(\operatorname{\mathcal{C}})$ is also nonempty. It will therefore suffice to show that for every integer $n \geq 0$, every morphism of simplicial sets $\sigma : \Delta ^ n / \operatorname{\partial \Delta }^ n \rightarrow Q(\operatorname{\mathcal{C}})$ is nullhomotopic (Proposition 3.2.6.14). Since the simplicial set $\Delta ^ n / \operatorname{\partial \Delta }^ n$ is finite and the functor $Q$ commutes with filtered colimits, the morphism $\sigma $ factors as a composition $\Delta ^{n} / \operatorname{\partial \Delta }^{n} \rightarrow Q(K) \xrightarrow {Q(\iota )} Q(\operatorname{\mathcal{C}})$, where $K$ is a finite simplicial subset of $\operatorname{\mathcal{C}}$ and $\iota : K \hookrightarrow \operatorname{\mathcal{C}}$ denotes the inclusion map. We will complete the proof by showing that $Q(\iota )$ is nullhomotopic. Since $u_{K}: K \rightarrow Q(K)$ is a weak homotopy equivalence, this is equivalent to assertion that the composite morphism $Q(\iota ) \circ u_{K} = u_{\operatorname{\mathcal{C}}} \circ \iota $ is nullhomotopic. This is clear: our assumption that $\operatorname{\mathcal{C}}$ is filtered guarantees that there exists a natural transformation from $\iota $ to a constant diagram $K \rightarrow \operatorname{\mathcal{C}}$ (Remark 7.2.4.7). $\square$

Proposition 7.2.4.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:

$(1)$

The $\infty $-category $\operatorname{\mathcal{C}}$ is filtered.

$(2)$

For every finite simplicial set $K$ and every morphism $f: K \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is filtered.

$(3)$

For every finite simplicial set $K$ and every morphism $f: K \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is weakly contractible.

$(4)$

For every finite simplicial set $K$, the diagonal map $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}})$ is right cofinal.

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 7.2.4.8, the implication $(2) \Rightarrow (3)$ from Proposition 7.2.4.9, and the implication $(3) \Rightarrow (1)$ is immediate from the definitions (Remark 7.2.4.7). The equivalence $(3) \Leftrightarrow (4)$ is a special case of Corollary 7.2.3.11. $\square$

Corollary 7.2.4.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories. Then $\operatorname{\mathcal{C}}$ is filtered if and only if $\operatorname{\mathcal{D}}$ is filtered.

Proof. By virtue of Proposition 7.2.4.10, it will suffice to show that for every (finite) simplicial set $K$, the diagonal map $\delta _{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}})$ is right cofinal if and only if the diagonal map $\delta _{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{D}})$ is right cofinal. This follows by applying Corollary 7.2.1.20 to the commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{\delta _{\operatorname{\mathcal{C}}} } \ar [d]^{F} & \operatorname{Fun}(K,\operatorname{\mathcal{C}}) \ar [d]^{ F \circ } \\ \operatorname{\mathcal{D}}\ar [r]^-{ \delta _{\operatorname{\mathcal{D}}} } & \operatorname{Fun}(K, \operatorname{\mathcal{D}}). } \]
$\square$

Corollary 7.2.4.12. Let $\operatorname{\mathcal{C}}$ be a Kan complex. Then $\operatorname{\mathcal{C}}$ is filtered if and only if it is contractible.

Proof. If $\operatorname{\mathcal{C}}$ is a contractible Kan complex, then there exists a categorical equivalence $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$, so that $\operatorname{\mathcal{C}}$ is filtered by virtue of Corollary 7.2.4.11. The converse is a special case of Proposition 7.2.4.9. $\square$