Definition 4.5.3.1. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. We say that $f$ is a categorical equivalence if, for every $\infty $-category $\operatorname{\mathcal{C}}$, the induced functor $\operatorname{Fun}( Y, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{C}})$ induces a bijection on isomorphism classes $\pi _0( \operatorname{Fun}(Y,\operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } )$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
4.5.3 Categorical Equivalence
Recall that a morphism of simplicial sets $f: X \rightarrow Y$ is a weak homotopy equivalence if, for every Kan complex $Z$, precomposition with $f$ induces a bijection $\pi _0( \operatorname{Fun}(Y,Z) ) \rightarrow \pi _0( \operatorname{Fun}(X,Z) )$ (Definition 3.1.6.12). If this condition is satisfied, then one should regard $X$ and $Y$ as indistinguishable from the perspective of classical homotopy theory. However, from the $\infty $-categorical perspective, the relation of weak homotopy equivalence is somewhat too coarse: it is possible for a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ to be a weak homotopy equivalence (or even a homotopy equivalence) without being an equivalence of $\infty $-categories (Warning 4.5.1.14). For this reason, it will be convenient to introduce a finer notion of equivalence.
Example 4.5.3.2. Every isomorphism of simplicial sets is a categorical equivalence.
Example 4.5.3.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is a categorical equivalence (in the sense of Definition 4.5.3.1) if and only if it is an equivalence of $\infty $-categories (in the sense of Definition 4.5.1.10). Both conditions are equivalent to the assertion that for every $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a bijection $ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}} }( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.
Remark 4.5.3.4. Let $f: X \rightarrow Y$ be a categorical equivalence of simplicial sets. Then $f$ is a weak homotopy equivalence (since every Kan complex is an $\infty $-category). Beware that the converse is generally false.
Remark 4.5.3.5 (Two-out-of-Three). Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be morphisms of simplicial sets. If any two of the morphisms $f$, $g$, and $g \circ f$ is a categorical equivalence, then so is the third. In particular, the collection of categorical equivalences is closed under composition.
Remark 4.5.3.6. The collection of categorical equivalences is closed under retracts. That is, if there exists a commutative diagram of simplicial sets where the horizontal compositions are the identity and $f'$ is a categorical equivalence, then $f$ is also a categorical equivalence.
\[ \xymatrix@C =40pt@R=40pt{ X \ar [r] \ar [d]^{f} & X' \ar [d]^{f'} \ar [r] & X \ar [d]^{f} \\ Y \ar [r] & Y' \ar [r] & Y } \]
Remark 4.5.3.7. Let $f: X \rightarrow Y$ be a categorical equivalence of simplicial sets. Then, for any simplicial set $K$, the induced map $f_ K: X \times K \rightarrow Y \times K$ is also a categorical equivalence of simplicial sets. To prove this, we must show that for every $\infty $-category $\operatorname{\mathcal{C}}$, the restriction map $\theta : \operatorname{Fun}( Y \times K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(X \times K, \operatorname{\mathcal{C}})$ induces a bijection on isomorphism classes of objects. This follows from our assumption that $f$ is a categorical equivalence, since $\theta $ can be identified with the map $\operatorname{Fun}(Y, \operatorname{Fun}(K, \operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Fun}(X, \operatorname{Fun}(K, \operatorname{\mathcal{C}}) )$ given by precomposition with $f$.
Proposition 4.5.3.8. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. The following conditions are equivalent:
The morphism $f: X \rightarrow Y$ is a categorical equivalence. That is, for every $\infty $-category $\operatorname{\mathcal{C}}$, precomposition with $f$ induces a bijection
For every $\infty $-category $\operatorname{\mathcal{C}}$, precomposition with $f$ induces a homotopy equivalence of Kan complexes $\operatorname{Fun}(Y, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq }$.
For every $\infty $-category $\operatorname{\mathcal{C}}$, precomposition with $f$ induces an equivalence of $\infty $-categories $\operatorname{Fun}(Y, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{C}})$.
\[ \pi _0( \operatorname{Fun}(Y, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } ). \]
Proof. The implication $(2) \Rightarrow (1)$ follows from Remark 3.1.6.5, and the implication $(3) \Rightarrow (2)$ follows from Remark 4.5.1.19. We will complete the proof by showing that $(1)$ implies $(3)$. Assume that $f$ is a categorical equivalence of simplicial sets, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $f^{\ast }: \operatorname{Fun}(Y, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{C}})$ denote the functor given by precomposition with $f$. We wish to show that $[f^{\ast }]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$. For this, it will suffice to show that for any $\infty $-category $\operatorname{\mathcal{D}}$, the induced map
\[ \theta : \pi _0( \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{Fun}(Y, \operatorname{\mathcal{C}}) )^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{Fun}(X, \operatorname{\mathcal{C}}) )^{\simeq } ) \]
is bijective. We conclude by observing that $\theta $ can be identified with the map
\[ \pi _0( \operatorname{Fun}(Y, \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}) )^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(X, \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}) )^{\simeq } ) \]
given by precomposition with $f$. $\square$
Corollary 4.5.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $f,f': K \rightarrow \operatorname{\mathcal{C}}$ be diagrams which are isomorphic (when viewed as objects of the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$). Then $f$ is a categorical equivalence if and only if $f'$ is a categorical equivalence.
Corollary 4.5.3.10. Let $\{ f_ i: X_{i} \rightarrow Y_ i \} _{i \in I}$ be a collection of categorical equivalences indexed by a set $I$. Then the coproduct map is also a categorical equivalence.
\[ f: \coprod _{i \in I} X_{i} \rightarrow \coprod _{i \in I} Y_ i \]
Proof. By virtue of Proposition 4.5.3.8, it will suffice to show that for every $\infty $-category $\operatorname{\mathcal{C}}$, precomposition with $f$ induces an equivalence of $\infty $-categories
\[ F: \operatorname{Fun}( \coprod _{i \in I} Y_ i, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \coprod _{i \in I} X_ i, \operatorname{\mathcal{C}}). \]
Note that $F$ factors as a product of functors $F_ i: \operatorname{Fun}(Y_ i, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(X_ i, \operatorname{\mathcal{C}})$, each of which is induced by precomposition with $f_{i}$. Since each $f_{i}$ is a categorical equivalence, Proposition 4.5.3.8 guarantees that each $F_{i}$ is an equivalence of $\infty $-categories. Applying Remark 4.5.1.17, we conclude that $F$ is an equivalence of $\infty $-categories. $\square$
Proposition 4.5.3.11. Let $f: X \rightarrow Y$ be a trivial Kan fibration of simplicial sets. Then $f$ is a categorical equivalence.
Proof. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We wish to show that precomposition with $f$ induces a bijection
\[ f^{\ast }: \pi _0( \operatorname{Fun}(Y,\operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(X,\operatorname{\mathcal{C}})^{\simeq } ). \]
Let $s: Y \rightarrow X$ be a section of $f$ (so that $f \circ s = \operatorname{id}_{Y}$). Then precomposition with $s$ induces a function $s^{\ast }: \pi _0(\operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq }) \rightarrow \pi _0( \operatorname{Fun}(Y, \operatorname{\mathcal{C}})^{\simeq } )$ for which the composition $s^{\ast } \circ f^{\ast }$ is equal to the identity on the set $\pi _0( \operatorname{Fun}(Y, \operatorname{\mathcal{C}})^{\simeq } )$. We will complete the proof by showing that the composition $f^{\ast } \circ s^{\ast }$ is isomorphic to the identity on $\pi _0( \operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } )$. Fix a map of simplicial sets $g: X \rightarrow \operatorname{\mathcal{C}}$; we wish to show that $g$ is isomorphic to the composite map
\[ X \xrightarrow {f} Y \xrightarrow {s} X \xrightarrow {g} \operatorname{\mathcal{C}} \]
as an object of the $\infty $-category $\operatorname{Fun}(X, \operatorname{\mathcal{C}})$.
Since $f$ is a trivial Kan fibration, the composition $s \circ f$ is fiberwise homotopic to the identity map $\operatorname{id}_{X}$: that is, we can choose a morphism of simplicial sets $h: \Delta ^1 \times X \rightarrow X$ which is compatible with the projection to $Y$ and which satisfies $h|_{ \{ 0\} \times X } = s \circ f$ and $h|_{\{ 1\} \times X} = \operatorname{id}_{X}$. The composition $g \circ h$ can then be regarded as a natural transformation $u: (g \circ s \circ f) \rightarrow g$. We will complete the proof by showing that $u$ is an isomorphism in the $\infty $-category $\operatorname{Fun}(X, \operatorname{\mathcal{C}})$. By virtue of Theorem 4.4.4.4, it will suffice to prove that for each vertex $x \in X$, the composite map
\[ \Delta ^1 \simeq \Delta ^1 \times \{ x\} \hookrightarrow \Delta ^1 \times X \xrightarrow {h} X \xrightarrow {g} \operatorname{\mathcal{C}} \]
describes an invertible morphism in $\operatorname{\mathcal{C}}$. Setting $y = f(x)$, we note that this composite map factors through the (contractible) Kan complex $X_{y}$, so the desired result follows from Proposition 1.4.6.10. $\square$
Corollary 4.5.3.12. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a trivial Kan fibration. Then $F$ is an equivalence of $\infty $-categories.
Proof. Combine Proposition 4.5.3.11 with Example 4.5.3.3. $\square$
Corollary 4.5.3.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $\operatorname{Isom}(\operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms of $\operatorname{\mathcal{C}}$ (Example 4.4.1.14). Then the diagonal embedding is an equivalence of $\infty $-categories.
\[ \delta : \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Isom}(\operatorname{\mathcal{C}}) \quad \quad C \mapsto \operatorname{id}_{C} \]
Proof. Let $\operatorname{ev}_{0}: \operatorname{Isom}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ denote the evaluation map
\[ \operatorname{Isom}(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}}. \]
Then $\operatorname{ev}_0 \circ \delta $ is the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$. Corollary 4.4.5.10 guarantees that $\operatorname{ev}_0$ is a trivial Kan fibration, and therefore an equivalence of $\infty $-categories (Corollary 4.5.3.12). Applying the two-out-of-three property (Remark 4.5.1.18), we conclude that $\delta $ is also an equivalence of $\infty $-categories. $\square$
Corollary 4.5.3.14. Let $f: A \hookrightarrow B$ be an inner anodyne morphism of simplicial sets. Then $f$ is a categorical equivalence.
Proof. By virtue of Proposition 4.5.3.8, it will suffice to show that for every $\infty $-category $\operatorname{\mathcal{C}}$, the restriction map $f^{\ast }: \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{C}})$ is an equivalence of $\infty $-categories. This follows from Corollary 4.5.3.12, since $f^{\ast }$ is a trivial Kan fibration (Proposition 1.5.7.6). $\square$
Warning 4.5.3.15. Let $f: A \rightarrow B$ be a morphism of simplicial sets. By virtue of Corollary 3.3.7.7, the morphism $f$ is anodyne if and only if it is both a monomorphism and a weak homotopy equivalence. Beware that the analogous assertion for inner anodyne morphisms is false. If $f$ is inner anodyne, then it is both a monomorphism (Remark 1.5.6.5) and a categorical equivalence (Corollary 4.5.3.14). However, the converse fails: a monomorphism $A \hookrightarrow B$ which is a categorical equivalence need not be inner anodyne. For example, an inner anodyne morphism of simplicial sets is automatically bijective on vertices (Exercise 1.5.6.6). However, there can be other obstructions as well: see Example 4.5.3.16.
Example 4.5.3.16 ([MR4042827]). Let $X = \Delta ^{2} \coprod _{ \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) } \Delta ^0$ be the simplicial set obtained from the standard $2$-simplex by collapsing the final edge to a point, which we represent by the diagram Then $X$ has exactly two nondegenerate edges $e,e': \Delta ^1 \rightarrow X$, as indicated in the diagram. We now make the following observations:
There is a pushout diagram of simplicial sets
Consequently, the morphism $e': \Delta ^1 \rightarrow X$ is inner anodyne, and therefore a categorical equivalence (Corollary 4.5.3.14).
There is a unique morphism of simplicial sets $r: X \rightarrow \Delta ^1$ satisfying $r \circ e' = \operatorname{id}_{\Delta ^1}$; the composite map $\Delta ^2 \twoheadrightarrow X \xrightarrow {r} \Delta ^1$ is given on vertices by $0 \mapsto 0$, $1 \mapsto 1$, and $2 \mapsto 1$. Since $e'$ is a categorical equivalence, it follows that $r$ is also a categorical equivalence (Remark 4.5.3.5).
The composite map $\Delta ^1 \xrightarrow {e} X \xrightarrow {r} \Delta ^1$ is equal to the identity map $\operatorname{id}_{\Delta ^1}$. Since $r$ is a categorical equivalence, it follows that $e$ is also a categorical equivalence. Moreover, $e$ is also a monomorphism of simplicial sets which is bijective on vertices.
The morphism $e: \Delta ^1 \hookrightarrow X$ is an inner fibration. This follows from Remark 4.1.1.5, since we have a pullback diagram of simplicial sets
where the horizontal maps are surjective and the inclusion $\Lambda ^2_{2} \hookrightarrow \Delta ^2$ is an inner fibration (since can be realized as the nerve of a morphism between partially ordered sets).
The morphism $e$ is not inner anodyne, since the lifting problem
has no solution.
\[ \xymatrix@R =50pt@C=50pt{ & \bullet \ar@ {=}[dr] \\ \bullet \ar [ur]^{e'} \ar [rr]^{e} & & \bullet . } \]
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{2}_{1} \ar [r] \ar [d] & \Delta ^2 \ar [d] \\ \Delta ^{1} \ar [r]^-{e'} & X. } \]
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{2}_{2} \ar [r] \ar [d] & \Delta ^1 \ar [d]^{e} \\ \Delta ^2 \ar [r] & X, } \]
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{1} \ar [d]^{e} \ar [r]^-{\operatorname{id}} & \Delta ^1 \ar [d]^{e} \\ X \ar [r]^-{\operatorname{id}} \ar@ {-->}[ur] & X } \]
Remark 4.5.3.17 (Axioms for Categorical Equivalence). The collection of categorical equivalences of simplicial sets has the following properties:
If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a categorical equivalence if and only if it is an equivalence of $\infty $-categories (Example 4.5.3.3).
Every inner anodyne morphism of simplicial sets is a categorical equivalence (Corollary 4.5.3.14).
If $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ have the property that two of the morphisms $f$, $g$, and $g \circ f$ are categorical equivalences, then so is the third (Remark 4.5.3.5).
In fact, the collection of categorical equivalences is characterized by assertions $(A)$, $(B)$ and $(C)$. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. Invoking Proposition 4.1.3.1 twice, we can construct a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{u} \ar [d]^{f} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ Y \ar [r]^-{v} & \operatorname{\mathcal{D}}, } \]
where $u$ and $v$ are inner anodyne and $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories. It follows from $(A)$, $(B)$ and $(C)$ that the morphism $f$ is a categorical equivalence if and only if the functor $F$ is an equivalence of $\infty $-categories.