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4 The Homotopy Theory of $\infty $-Categories

Let $q: X \rightarrow S$ be a morphism of simplicial sets. Recall that $q$ is a Kan fibration if and only if it has the right lifting property with respect to every horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for $n > 0$ and $0 \leq i \leq n$. The theory of Kan fibrations can be viewed as a relativization of the theory of Kan complexes, which plays an essential role in the classical homotopy theory of simplicial sets (as in Chapter 3). In this chapter, we study several weaker notions of fibration, which will play an analogous role in the study of $\infty $-categories:

  • We say that a morphism of simplicial sets $q: X \rightarrow S$ is an inner fibration if it has the right lifting property with respect to every inner horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$, $0 < i < n$ (Definition 4.1.1.1). If this condition is satisfied, then for each vertex $s \in S$, the fiber $X_{s} = \{ s\} \times _{S} X$ an $\infty $-category (Remark 4.1.1.6). Consequently, the theory of inner fibrations can be regarded as a relative version of the theory of $\infty $-categories, which we study in §4.1.

  • We say that a morphism of simplicial sets $q: X \rightarrow S$ is a left fibration if it has the right lifting property with respect to the horn inclusions $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for $0 \leq i < n$, and a right fibration if it has the right lifting property with respect to the horn inclusions $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for $0 < i \leq n$ (Definition 4.2.1.1). If either of these conditions are satisfied, then the fiber $X_{s} = \{ s\} \times _{S} X$ is a Kan complex for each vertex $s \in S$ (Corollary 4.4.2.2). We will see later that the construction $s \mapsto X_{s}$ is covariantly functorial when $q$ is a left fibration, and contravariantly functorial when $q$ is a right fibration (see §5.1.1). In §4.2, we develop some basic formal properties of left and right fibrations; we will carry out a more detailed analysis in Chapter 5.

  • We say that a morphism of simplicial sets $q: X \rightarrow S$ is an isofibration if it has the right lifting property with respect to every inclusion of simplicial sets $A \hookrightarrow B$ which is a categorical equivalence (Definition 4.5.7.1). This condition is primarily useful in the case where $X$ and $S$ are $\infty $-categories, in which case it is equivalent to the requirement that $q$ is an inner fibration which satisfies a lifting property with respect to isomorphisms (Proposition 4.5.6.1). We study isofibrations between $\infty $-categories in §4.4, and between general simplicial sets in §4.5.7).

If $q: X \rightarrow S$ is a morphism of simplicial sets, we have the following diagram of implications:

\[ \xymatrix { & \txt { $q$ is a trivial Kan fibration} \ar@ {=>}[d] & \\ & \txt { $q$ is a Kan fibration} \ar@ {=>}[dr] \ar@ {=>}[dl] & \\ \txt { $q$ is a left fibration} \ar@ {=>}[dr] & & \txt { $q$ is a right fibration} \ar@ {=>}[dl] \\ & \txt { $q$ is an isofibration } \ar@ {=>}[d] & \\ & \txt { $q$ is an inner fibration. } & } \]

Beware that, in general, none of these implications is reversible.

In §4.3, we consider some prototypical examples of left and right fibrations which arise frequently in practice. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. To each object $X \in \operatorname{\mathcal{C}}$, one can associate a simplicial set $\operatorname{\mathcal{C}}_{/X}$, whose $n$-simplices are given by maps $\sigma : \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$ which carry the final vertex of $\Delta ^{n+1}$ to the object $X \in \operatorname{\mathcal{C}}$. In particular, vertices of $\operatorname{\mathcal{C}}_{/X}$ can be identified with morphisms $f: Y \rightarrow X$ in $\operatorname{\mathcal{C}}$ having target $X$, and edges of $\operatorname{\mathcal{C}}_{/X}$ can be identified with commutative diagrams

\[ \xymatrix { Z \ar [rr] \ar [dr] & & Y \ar [dl] \\ & X. & } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$ (see Notation 4.3.5.6 for a precise definition). The simplicial set $\operatorname{\mathcal{C}}_{/X}$ is itself an $\infty $-category, which we we will refer to as the slice $\infty $-category of $\operatorname{\mathcal{C}}$ over the object $X$. Moreover, the evident forgetful functor $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ (given on objects by the construction $(f: Y \rightarrow X) \mapsto Y$) is a right fibration (Proposition 4.3.6.1). A dual version of this construction produces another $\infty $-category $\operatorname{\mathcal{C}}_{X/}$ whose objects are morphisms $f: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$, which we refer to as the coslice $\infty $-category of $\operatorname{\mathcal{C}}$ under the object $X$. The slice and coslice constructions (and generalizations thereof) provide a rich supply of right and left fibrations between simplicial sets, and will play an essential role in the theory developed in this book.

Our final goal in this chapter is to generalize the notion of equivalence to the $\infty $-categorical setting. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We will say that $F$ is an equivalence of $\infty $-categories if there exists a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ such that $G \circ F$ and $F \circ G$ are isomorphic to $\operatorname{id}_{\operatorname{\mathcal{C}}}$ and $\operatorname{id}_{\operatorname{\mathcal{D}}}$ as objects of the $\infty $-categories $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ and $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$, respectively (Definition 4.5.1.9). In §4.5, we study equivalences of $\infty $-categories in detail, as well as the more general notion of categorical equivalence between arbitrary simplicial sets (Definition 4.5.2.1).

Structure

  • Section 4.1: Inner Fibrations
    • Subsection 4.1.1: Inner Fibrations of Simplicial Sets
    • Subsection 4.1.2: Subcategories of $\infty $-Categories
    • Subsection 4.1.3: Inner Anodyne Morphisms
    • Subsection 4.1.4: Exponentiation for Inner Fibrations
  • Section 4.2: Left and Right Fibrations
    • Subsection 4.2.1: Left and Right Fibrations of Simplicial Sets
    • Subsection 4.2.2: Left Anodyne and Right Anodyne Morphisms
    • Subsection 4.2.3: Exponentiation for Left and Right Fibrations
    • Subsection 4.2.4: The Homotopy Extension Lifting Property
    • Subsection 4.2.5: Example: Fibrations in Groupoids
  • Section 4.3: The Slice and Join Constructions
    • Subsection 4.3.1: Slices of Categories
    • Subsection 4.3.2: Joins of Categories
    • Subsection 4.3.3: Joins of Simplicial Sets
    • Subsection 4.3.4: Joins of Topological Spaces
    • Subsection 4.3.5: Slices of Simplicial Sets
    • Subsection 4.3.6: Slices of $\infty $-Categories
    • Subsection 4.3.7: Slices of Left and Right Fibrations
  • Section 4.4: Isomorphisms and Isofibrations
    • Subsection 4.4.1: Isofibrations of $\infty $-Categories
    • Subsection 4.4.2: Isomorphisms and Lifting Properties
    • Subsection 4.4.3: The Core of an $\infty $-Category
    • Subsection 4.4.4: Natural Isomorphisms
    • Subsection 4.4.5: Exponentiation for Isofibrations
  • Section 4.5: Equivalence
    • Subsection 4.5.1: Equivalences of $\infty $-Categories
    • Subsection 4.5.2: Categorical Equivalences of Simplicial Sets
    • Subsection 4.5.3: Categorical Pushout Diagrams
    • Subsection 4.5.4: Detecting Equivalences of $\infty $-Categories
    • Subsection 4.5.5: Application: Universal Property of the Join
    • Subsection 4.5.6: Lifting Property of Isofibrations
    • Subsection 4.5.7: Isofibrations of Simplicial Sets