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7.2 Cofinality

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: B \rightarrow \operatorname{\mathcal{C}}$ be a diagram in $\operatorname{\mathcal{C}}$ indexed by a simplicial set $B$. In §7.1, we introduced the definition of a colimit $\varprojlim (f)$ and colimit $\varinjlim (f)$ of the diagram $f$ (Definition 7.1.3.1). In practice, it is often convenient to replace $f$ by a simpler diagram having the same limit (or colimit). The primary goal of this section is to introduce a general formalism which will allow us to make replacements of this sort.

We begin in §7.2.1 by introducing the notions of left cofinal and right cofinal morphisms of simplicial sets (Definition 7.2.1.1. Roughly speaking, one can regard left cofinality as a homotopy-invariant replacement for the notion of left anodyne morphism introduced in Definition 4.2.4.1. More precisely, the collection of left cofinal morphisms of simplicial sets is uniquely determined by the following assertions:

  • A monomorphism of simplicial sets $f: A \hookrightarrow B$ is left cofinal if and only if it is left anodyne.

  • Suppose we are given a commutative diagram of simplicial sets

    \[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ A' \ar [r]^-{f'} & B', } \]

    where the vertical maps are categorical equivalences. Then $f$ is left cofinal if and only if $f'$ is left cofinal (Corollary 7.2.1.20).

In §7.2.2, we connect the notion of cofinality with the theory of limits and colimits developed in §7.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $g: B \rightarrow \operatorname{\mathcal{C}}$ be a diagram in $\operatorname{\mathcal{C}}$. Our main result (Corollary 7.2.2.2) asserts that if $f: A \rightarrow B$ is a left cofinal morphism of simplicial sets, then the limit of the diagram $g$ (if it exists) can be identified with with the limit of the composite diagram $(g \circ f): A \rightarrow \operatorname{\mathcal{C}}$. Similarly, if $f$ is right cofinal, then the colimit of $g$ can be identified with the colimit of $g \circ f$. Consequently, cofinality is a very useful tool for computing (or verifying the existence of) limits and colimits.

In §7.2.3, we specialize to the study of cofinal functors between $\infty $-categories. Our main result asserts that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is right cofinal if and only if, for every object $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is weakly contractible (Theorem 7.2.3.1). This has several useful consequences:

  • If $Y$ is an object of an $\infty $-category $\operatorname{\mathcal{C}}$, then the inclusion map $\{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$ is left cofinal if and only if $Y$ is an initial object of $\operatorname{\mathcal{C}}$, and right cofinal if and only if $Y$ is a final object of $\operatorname{\mathcal{C}}$ (Corollary 7.2.3.4).

  • If $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories having the property that each fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/D}$ is weakly contractible, then $F$ is a weak homotopy equivalence. This is an $\infty $-categorical generalization of Quillen's “Theorem A” (Example 7.2.3.3).

  • If a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a left adjoint, then it is left cofinal; if it is a right adjoint, then it is right cofinal (Corollary 7.2.3.6).

We will deduce Theorem 7.2.3.1 from a general fact about the stability of right cofinality with respect to pullback along cocartesian fibrations (Proposition 7.2.3.9), which is of independent interest.

We devote the second half of this section to studying properties of $\infty $-categories which are closely related to the notion of cofinality. We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is filtered if, 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 (Definition 7.2.4.3). In §7.2.4, we show that if this property is satisfied for every finite simplicial set $K$, then one 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 7.2.4.10).

To show 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 §7.2.5, we show that it suffices to verify this condition in the case where $K = \operatorname{\partial \Delta }^ n$ is the boundary of a standard simplex, for each $n \geq 0$ (Lemma 7.2.5.13). Using this observation, we show that the condition that an $\infty $-category $\operatorname{\mathcal{C}}$ is filtered can be formulated entirely at the level of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, viewed as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category (Theorem 7.2.5.5). As an application, we show that our notion of filtered $\infty $-category generalizes the classical notion of a filtered category: that is, an ordinary category $\operatorname{\mathcal{C}}$ is filtered if and only if the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a filtered $\infty $-category (Corollary 7.2.5.8). We also formulate a counterpart of this result for the homotopy coherent nerve of a locally Kan simplicial category (Corollary 7.2.5.10).

Our primary interest in the notion of filtered $\infty $-category stems from the exactness properties enjoyed by filtered colimits. We will see later than a small $\infty $-category $\operatorname{\mathcal{C}}$ 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 ). In §7.2.6 we establish a version of this statement, which reformulates the condition that $\operatorname{\mathcal{C}}$ is filtered in terms of fiber products of $\infty $-categories which are left-fibered over $\operatorname{\mathcal{C}}$ (Corollary 7.2.6.3). As a consequence, we show that if $F: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ is a right cofinal functor of $\infty $-categories where $\operatorname{\mathcal{C}}'$ is filtered, then $\operatorname{\mathcal{C}}$ is also filtered (Proposition 7.2.7.1). In §7.2.7, we establish a partial converse to this assertion: if $\operatorname{\mathcal{C}}$ is a filtered $\infty $-category, then there exists a directed partially ordered set $(A, \leq )$ and a right cofinal functor $\operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{C}}$ (Theorem 7.2.7.2).

For many applications, it will be useful to consider a generalization of the notion of filtered $\infty $-category. In §7.2.8, we introduce the larger class of sifted simplicial sets. We say that a simplicial set $K$ is sifted if, for every finite set $I$, the diagonal map $\delta : K \rightarrow K^{I}$ is right cofinal (Definition 7.2.8.1). Equivalently, a simplicial set $K$ is sifted if it is weakly contractible and the diagonal $K \hookrightarrow K \times K$ is right cofinal (Proposition 7.2.8.8). Every filtered $\infty $-category is sifted (Example 7.2.8.4), but the converse is false: for example, the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})^{\operatorname{op}}$ is sifted (Proposition 7.2.8.10), but is not filtered.

Structure

  • Subsection 7.2.1: Cofinal Morphisms of Simplicial Sets
  • Subsection 7.2.2: Cofinality and Limits
  • Subsection 7.2.3: Quillen's Theorem A for $\infty $-Categories
  • Subsection 7.2.4: Filtered $\infty $-Categories
  • Subsection 7.2.5: Local Characterization of Filtered $\infty $-Categories
  • Subsection 7.2.6: Left Fibrations over Filtered $\infty $-Categories
  • Subsection 7.2.7: Cofinal Approximation
  • Subsection 7.2.8: Sifted Simplicial Sets