9.1 Filtered $\infty $-Categories
We begin by reviewing the classical notion of a filtered category.
Definition 9.1.0.1. Let $\operatorname{\mathcal{C}}$ be a category. We say that $\operatorname{\mathcal{C}}$ is filtered if it satisfies the following conditions:
- $(\ast _0)$
The category $\operatorname{\mathcal{C}}$ is nonempty.
- $(\ast _1)$
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 $v: Y \rightarrow Z$.
- $(\ast _2)$
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 $v \circ f_0 = v \circ f_1$.
Our goal in this section is to formulate an $\infty $-categorical counterpart of Definition 9.1.0.1. We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is filtered if, for every finite simplicial set $K$, every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$ can be extended to the cone $K^{\triangleright }$ (Definition 9.1.1.1): that is, the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is nonempty. In §9.1.1, we show that if this property is satisfied for every finite diagram in $\operatorname{\mathcal{C}}$, then we can say more: every such coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is weakly contractible. It follows that $\operatorname{\mathcal{C}}$ is filtered if and only if the diagonal map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is right cofinal for every finite simplicial set $K$ (Proposition 9.1.1.14).
To verify that an $\infty $-category $\operatorname{\mathcal{C}}$ is filtered, it is not necessary to show that the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is nonempty for every finite diagram $f: K \rightarrow \operatorname{\mathcal{C}}$. In §9.1.2, we show that it suffices to verify this condition in the special case where $K = \operatorname{\partial \Delta }^ n$ is the boundary of a standard simplex, for each $n \geq 0$ (Lemma 9.1.2.12). When $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0)$ is (the nerve of) an ordinary category $\operatorname{\mathcal{C}}_0$ , this condition is vacuous for $n > 2$, and reduces to condition $(\ast _ n)$ of Definition 9.1.0.1 for $n \leq 2$ (see Example 9.1.2.3 and Exercise 9.1.2.15). It follows that $\operatorname{\mathcal{C}}_0$ is a filtered category (in the sense of Definition 9.1.0.1) if and only if the nerve $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$ is a filtered $\infty $-category.
Our primary interest in the notion of filtered $\infty $-category stems from the exactness properties enjoyed by filtered colimits. Let $\operatorname{\mathcal{C}}$ be a small $\infty $-category, and let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.5.1.1). In §9.1.5, we show that $\operatorname{\mathcal{C}}$ is filtered if and only if the colimit functor $\varinjlim : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}$ preserves finite limits (Theorem 9.1.5.1). One implication is fairly straightforward: if the functor $\varinjlim $ preserves $K$-indexed limits, then the diagonal map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K^{\operatorname{op}}, \operatorname{\mathcal{C}})$ is right cofinal (Corollary 9.1.4.5). Allowing $K$ to vary over all finite simplicial sets, we conclude that $\operatorname{\mathcal{C}}$ is filtered. To prove the converse, we will need to work a bit harder. Assume that the $\infty $-category $\operatorname{\mathcal{C}}$ is filtered; we wish to show that the colimit functor $\varinjlim $ preserves finite limits. In §9.1.4, we show that this is equivalent to the following a priori weaker assertion (see Proposition 9.1.4.3):
- $(\star )$
Let $\operatorname{Fun}'( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$ spanned by those diagrams $\mathscr {F}$ for which the colimit $\varinjlim (\mathscr {F} )$ is contractible. Then $\operatorname{Fun}'( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$ is closed under finite limits.
To prove $(\star )$, it will be convenient to work with another description of the subcategory $\operatorname{Fun}'( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$. Recall that for any small diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$, the colimit $\varinjlim (\mathscr {F} )$ can be regarded as a fibrant replacement for the $\infty $-category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ (Corollary 7.4.3.3). In particular, the colimit $\varinjlim (\mathscr {F} )$ is contractible if and only if the $\infty $-category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is weakly contractible. When $\operatorname{\mathcal{C}}$ is filtered, we show in §9.1.3 that this is equivalent to the a priori stronger condition that the $\infty $-category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is also filtered (Corollary 9.1.3.4). In §9.1.5, we use this characterization to deduce $(\star )$ (see Lemma 9.1.5.4) and thereby establish Theorem 9.1.5.1.
Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. If $U$ is right cofinal, then the colimit of any diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ can be identified with the colimit of the restriction $\mathscr {F} \circ U$. It follows that if the $\infty $-category $\operatorname{\mathcal{D}}$ is filtered, then the $\infty $-category $\operatorname{\mathcal{C}}$ is also filtered (Proposition 9.1.6.1). In §9.1.6, we establish a converse of this result: if $\operatorname{\mathcal{C}}$ is a filtered $\infty $-category, then there exists a filtered category $\operatorname{\mathcal{C}}_0$ and a right cofinal functor $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0) \rightarrow \operatorname{\mathcal{C}}$. Moreover, we can even arrange that $\operatorname{\mathcal{C}}_0$ is a partially ordered set (Corollary 9.1.6.3).
This book has already made extensive use of the theory of filtered colimits in the setting of ordinary categories, such as the category $\operatorname{Set_{\Delta }}$ of simplicial sets. In this context, an essential feature of filtered colimits (not shared by colimits in general) is homotopy invariance: if $f: \{ X_{\alpha } \} \rightarrow \{ Y_{\alpha } \} $ is a levelwise homotopy equivalence between filtered diagrams of Kan complexes, then the colimit $\varinjlim (X_{\alpha } ) \rightarrow \varinjlim ( Y_{\alpha } )$ is also a homotopy equivalence (Proposition 3.2.8.3). In §9.1.7, we apply this observation to show that the inclusion functor $\operatorname{N}_{\bullet }( \operatorname{Kan}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}) = \operatorname{\mathcal{S}}$ preserves filtered colimits (Variant 9.1.7.4). Consequently, questions about filtered colimits in the $\infty $-category $\operatorname{\mathcal{S}}$ can often be translated to concrete assertions about simplicial sets.
Structure
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Subsection 9.1.1: Filtered $\infty $-Categories
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Subsection 9.1.2: Local Characterization of Filtered $\infty $-Categories
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Subsection 9.1.3: Fibrations over Filtered $\infty $-Categories
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Subsection 9.1.4: Digression: Commutation of Limits and Colimits
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Subsection 9.1.5: Filtered Colimits of Spaces
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Subsection 9.1.6: Cofinal Approximation
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Subsection 9.1.7: Filtered Colimits of Simplicial Sets