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1.4 Functors of $\infty $-Categories

Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories, and let $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ and $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ denote the corresponding $\infty $-categories. According to Proposition 1.2.2.1, the nerve functor $\operatorname{N}_{\bullet }$ induces a bijection

\[ \{ \text{Functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$} \} \simeq \{ \text{Maps of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$} \} . \]

Consequently, the notion of functor admits an obvious generalization to the setting of $\infty $-categories:

Definition 1.4.0.1. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. A functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a map of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.

This section is devoted to the study of functors between $\infty $-categories, in the sense of Definition 1.4.0.1. We begin in §1.4.1 with some simple examples, which illustrate the meaning of Definition 1.4.0.1 in the case of $\infty $-categories which arise from ordinary categories (via the construction $\operatorname{\mathcal{E}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}})$) or topological spaces (via the construction $X \mapsto \operatorname{Sing}_{\bullet }(X)$).

In ordinary category theory, one can think of a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ as a kind of commutative diagram in $\operatorname{\mathcal{D}}$, having vertices indexed by the objects of $\operatorname{\mathcal{C}}$ and arrows indexed by the morphisms of $\operatorname{\mathcal{C}}$. This perspective is quite useful: if the category $\operatorname{\mathcal{C}}$ is sufficiently small, one can communicate the datum of a functor by drawing a graphical representation of the corresponding diagram. In §1.4.2, we discuss the notion of commutative diagram in an $\infty $-category (Convention 1.4.2.11) and describe some dangers associated with diagrammatic reasoning in the higher-categorical setting (Remark 1.4.2.12).

If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are ordinary categories, then the collection of all functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ can itself be organized into a category, which we denote by $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. In §1.4.3, we describe a counterpart of this construction in the setting of $\infty $-categories. For every pair of simplicial sets $S_{\bullet }$ and $T_{\bullet }$, one can form a new simplicial set $\operatorname{Fun}( S_{\bullet }, T_{\bullet } )$ whose vertices are maps from $S_{\bullet }$ to $T_{\bullet }$ (Construction 1.4.3.1). The main result of this section asserts that if $T_{\bullet }$ is an $\infty $-category, then $\operatorname{Fun}( S_{\bullet }, T_{\bullet } )$ is also an $\infty $-category (Theorem 1.4.3.7). Moreover, our notation is consistent: in the case where $S_{\bullet }$ and $T_{\bullet }$ are isomorphic to the nerves of categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{Fun}( S_{\bullet }, T_{\bullet } )$ is isomorphic to the nerve of the functor category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ (Proposition 1.4.3.3).

In order to prove Theorem 1.4.3.7, we will need to introduce some auxiliary ideas. Recall that if $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are composable morphisms in an $\infty $-category $\operatorname{\mathcal{C}}$, then we can form a composition of $f$ and $g$ by choosing a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ which satisfies $d_0( \sigma ) = g$ and $d_2(\sigma ) = f$, as indicated in the diagram

\[ \xymatrix { & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{g \circ f} & & Z. } \]

We proved in §1.3.4 that the resulting morphism $g \circ f$ is well-defined up to homotopy (Proposition 1.3.4.2). In §1.4.6, we prove a variant of this assertion which asserts that the $2$-simplex $\sigma $ is “unique up to a contractible space of choices” (see Corollary 1.4.6.2 for a precise statement). Moreover, we show that a strong version of this uniqueness result is equivalent to the assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category (Theorem 1.4.6.1), and deduce the existence of functor $\infty $-categories $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ as a consequence (Theorem 1.4.3.7). The precise formulation and proof of Theorem 1.4.6.1 will require some general ideas about categorical lifting properties and the homotopy theory of simplicial sets, which we develop in §1.4.4 and §1.4.5, respectively.

Structure

  • Subsection 1.4.1: Examples of Functors
  • Subsection 1.4.2: Commutative Diagrams
  • Subsection 1.4.3: The $\infty $-Category of Functors
  • Subsection 1.4.4: Digression: Lifting Properties
  • Subsection 1.4.5: Trivial Kan Fibrations
  • Subsection 1.4.6: Uniqueness of Composition