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4.5 Equivalence

Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an isomorphism of categories if there exists a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ satisfying the identities $G \circ F = \operatorname{id}_{\operatorname{\mathcal{C}}}$ and $F \circ G = \operatorname{id}_{\operatorname{\mathcal{D}}}$. This condition is somewhat unnatural, since it refers to equalities between objects of the functor categories $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ and $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$. For most purposes, it is better to adopt a looser definition. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of categories if there exists a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ for which the composite functors $G \circ F$ and $F \circ G$ are isomorphic to the identity functors $\operatorname{id}_{\operatorname{\mathcal{C}}}$ and $\operatorname{id}_{\operatorname{\mathcal{D}}}$, respectively. In category theory, the notion of equivalence between categories plays a much more central role than the notion of isomorphism between categories, and virtually all important concepts are invariant under equivalence.

In §4.5.1, we extend the notion of equivalence to the $\infty $-categorical setting. If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, we will say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories if there exists a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ for which the composite maps $G \circ F$ and $F \circ G$ are isomorphic to $\operatorname{id}_{\operatorname{\mathcal{C}}}$ and $\operatorname{id}_{\operatorname{\mathcal{D}}}$, when viewed 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 4.5.1.10). Phrased differently, a functor $F$ is an equivalence of $\infty $-categories if it is an isomorphism when viewed as a morphism of the category $\mathrm{h} \mathit{\operatorname{QCat}}$, whose objects are $\infty $-categories and whose morphisms are isomorphism classes of functors (Construction 4.5.1.1).

In the study of $\infty $-categories, it can be technically convenient to work with simplicial sets which do not satisfy the weak Kan extension condition. For example, it is often harmless to replace the standard $n$-simplex $\Delta ^ n$ by its spine $\operatorname{Spine}[n] \subseteq \Delta ^ n$: for any $\infty $-category $\operatorname{\mathcal{C}}$, the restriction map $\operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{Spine}[n], \operatorname{\mathcal{C}})$ is a trivial Kan fibration (see Example 1.4.7.7). In §4.5.2, we formalize this observation by introducing the notion of categorical equivalence between simplicial sets. By definition, a morphism of simplicial sets $f: X \rightarrow Y$ is a categorical equivalence if, for every $\infty $-category $\operatorname{\mathcal{C}}$, the induced functor of $\infty $-categories $\operatorname{Fun}(Y, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{C}})$ is bijective on isomorphism classes of objects (Definition 4.5.2.1). If $X$ and $Y$ are $\infty $-categories, this reduces to the condition that $f$ is an equivalence of $\infty $-categories in the sense of §4.5.1 (Example 4.5.2.3). However, we will encounter many other examples of categorical equivalences between simplicial sets which are not $\infty $-categories: for example, every inner anodyne morphism of simplicial sets is a categorical equivalence (Corollary 4.5.2.11).

Recall that every $\infty $-category $\operatorname{\mathcal{C}}$ contains a largest Kan complex, which we denote by $\operatorname{\mathcal{C}}^{\simeq }$ and refer to as the core of $\operatorname{\mathcal{C}}$ (Construction 4.4.3.1). The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\simeq }$ can often be used to reformulate questions about $\infty $-categories in terms of the classical homotopy theory of Kan complexes. In §4.5.4, we illustrate this principle by showing that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence if and only if, for every simplicial set $K$, the induced map $\operatorname{Fun}( K, \operatorname{\mathcal{C}})^{\simeq } \xrightarrow {F \circ } \operatorname{Fun}(K, \operatorname{\mathcal{D}})^{\simeq }$ is a homotopy equivalence of Kan complexes. Moreover, it suffices to check this condition in the special case $K = \Delta ^1$ (Theorem 4.5.4.1). As an application, we show that the collection of categorical equivalences is stable under the formation of filtered colimits (Corollary 4.5.4.2), and under pullback along isofibrations of $\infty $-categories (Corollary 4.5.4.3).

In §4.5.5, we study an important class of categorical equivalences emerging from the theory of joins developed in §4.3. Recall that, if $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are categories, then the join $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is isomorphic to the iterated pushout

\[ \operatorname{\mathcal{C}}\coprod _{ (\operatorname{\mathcal{C}}\times \{ 0\} \times \operatorname{\mathcal{D}})} (\operatorname{\mathcal{C}}\times [1] \times \operatorname{\mathcal{D}}) \coprod _{ (\operatorname{\mathcal{C}}\times \{ 1\} \times \operatorname{\mathcal{D}}) } \operatorname{\mathcal{D}}, \]

formed in the category $\operatorname{Cat}$ of (small) categories (Remark 4.3.2.14). In the setting of $\infty $-categories, the situation is more subtle (Warning 4.3.3.28). For any simplicial sets $X$ and $Y$, there is a natural comparison map

\[ c_{X,Y}: X \coprod _{ (X \times \{ 0\} \times Y)} (X \times \Delta ^1 \times Y) \coprod _{ (X \times \{ 1\} \times Y)} Y \rightarrow X \star Y \]

(Notation 4.5.5.3), which is almost never an isomorphism. Nevertheless, we show in §4.5.5 that $c_{X,Y}$ is always a categorical equivalence of simplicial sets (Theorem 4.5.5.7). To carry out the proof, it will be useful to use the language of categorical pushout diagrams, which we explain in §4.5.3.

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Recall that $F$ is an inner fibration if and only if every lifting problem

4.16
\begin{equation} \begin{gathered}\label{equation:lifting-problem-char-isofibration} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ B \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{D}}} \end{gathered} \end{equation}

admits a solution, provided that the morphism $i: A \hookrightarrow B$ is inner anodyne (Proposition 4.1.3.1). In §4.5.6, we show that $F$ is an isofibration if and only if the following stronger condition holds: the lifting problem (4.16) admits a solution whenever the map $i: A \hookrightarrow B$ is both a monomorphism and a categorical equivalence (Proposition 4.5.6.1). In §4.5.7, we apply this result to extend the notion of isofibration to simplicial sets which are not necessarily $\infty $-categories (Definition 4.5.7.1).

Structure

  • 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