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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 4.5.1.9). Phrased differently, a functor $F$ is an equivalence of $\infty $-categories if it is an isomorphism when viewed as an object of the category $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$, 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 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 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 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 employ 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

4.15
\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: the lifting problem (4.15) 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