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 1.5.7.7). In §4.5.3, 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 4.5.3.1). 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 4.5.3.3). 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 4.5.3.14).
Throughout this book, we will generally emphasize concepts which are invariant under categorical equivalence. In practice, this requires us to take some care when manipulating elementary constructions, such as fiber products. If $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ are functors of $\infty $-categories, then the fiber product $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ (formed in the category of simplicial sets) need not be an $\infty $-category. Moreover, the construction $(F_0, F_1) \mapsto \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ does not preserve categorical equivalence in general. In §4.5.2, we remedy the situation by enlarging the fiber product $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ to the homotopy fiber product $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_1$, given by the formula
(see Construction 4.5.2.1). The homotopy fiber product $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_1$ is always an $\infty $-category (Remark 4.5.2.2), and the construction $(F_0, F_1) \mapsto \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_1$ is invariant under equivalence (Corollary 4.5.2.20). We will say that a commutative diagram of $\infty $-categories
is a categorical pullback square if it induces an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ (Definition 4.5.2.8). This is closely related to the notion of homotopy pullback diagram introduced in §3.4.1:
A commutative diagram of Kan complexes is a homotopy pullback square if and only if it is a categorical pullback square (Proposition 4.5.2.10).
The diagram of $\infty $-categories (4.19) is a categorical pullback square if and only if, for every simplicial set $X$, the induced diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(X,\operatorname{\mathcal{C}}_{01})^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}(X,\operatorname{\mathcal{C}}_0)^{\simeq } \ar [d] \\ \operatorname{Fun}(X,\operatorname{\mathcal{C}}_1)^{\simeq } \ar [r] & \operatorname{Fun}(X,\operatorname{\mathcal{C}})^{\simeq } } \]is a homotopy pullback square (Proposition 4.5.2.14).
In §4.5.4 we study the dual notion of categorical pushout square (Definition 4.5.4.1), which is an $\infty $-categorical counterpart of the theory of homotopy pushout squares developed in §3.4.2.
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. It is not difficult to show 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 $X$, the induced map $\operatorname{Fun}( X, \operatorname{\mathcal{C}})^{\simeq } \xrightarrow {F \circ } \operatorname{Fun}(X, \operatorname{\mathcal{D}})^{\simeq }$ is a homotopy equivalence of Kan complexes (Proposition 4.5.1.22). In §4.5.7, we show that it suffices to verify this condition in the special case $X = \Delta ^1$ (Theorem 4.5.7.1). As an application, we show that the collection of categorical equivalences is stable under the formation of filtered colimits (Corollary 4.5.7.2).
In §4.5.8, 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
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 4.3.3.33). For any simplicial sets $X$ and $Y$, there is a natural comparison map
(Notation 4.5.8.3), which is almost never an isomorphism. Nevertheless, we show in §4.5.8 that $c_{X,Y}$ is always a categorical equivalence of simplicial sets (Theorem 4.5.8.8).
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
admits a solution, provided that the morphism $i: A \hookrightarrow B$ is inner anodyne (Proposition 4.1.3.1). In §4.5.5, we show that $F$ is an isofibration if and only if the following stronger condition holds: the lifting problem (4.20) admits a solution whenever the map $i: A \hookrightarrow B$ is both a monomorphism and a categorical equivalence (Proposition 4.5.5.1). Using this characterization, we extend the notion of isofibration to simplicial sets which are not necessarily $\infty $-categories (Definition 4.5.5.5).
Structure
- Subsection 4.5.1: Equivalences of $\infty $-Categories
- Subsection 4.5.2: Categorical Pullback Squares
- Subsection 4.5.3: Categorical Equivalence
- Subsection 4.5.4: Categorical Pushout Squares
- Subsection 4.5.5: Isofibrations of Simplicial Sets
- Subsection 4.5.6: Isofibrant Diagrams
- Subsection 4.5.7: Detecting Equivalences of $\infty $-Categories
- Subsection 4.5.8: Application: Universal Property of the Join
- Subsection 4.5.9: Relative Exponentiation