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4.3.1 Fibrations over a Point

Recall that a morphism of simplicial sets $X \rightarrow S$ is a Kan fibration if and only if it is both a left fibration and a right fibration (Example 4.1.0.5). We now show that, in the special case $S = \Delta ^0$, either one of these conditions is individually sufficient.

Proposition 4.3.1.1 (Joyal [MR1935979]). Let $X$ be a simplicial set. The following conditions are equivalent:

$(a)$

The projection map $X \rightarrow \Delta ^0$ is a Kan fibration.

$(b)$

The simplicial set $X$ is a Kan complex.

$(c)$

The simplicial set $X$ is an $\infty $-category and the homotopy category $\mathrm{h} \mathit{X}$ is a groupoid.

$(d)$

The simplicial set $X$ is an $\infty $-category and every morphism in $X$ is an isomorphism.

$(e)$

The projection map $X \rightarrow \Delta ^0$ is a left fibration.

$(f)$

The projection map $X \rightarrow \Delta ^0$ is a right fibration.

Corollary 4.3.1.2. Let $q: X \rightarrow S$ be morphism of simplicial sets which is either a left or a right fibration. Then, for every vertex $s \in S$, the fiber $X_{s} = \{ s\} \times _{S} X$ is a Kan complex.

Our proof of Proposition 4.3.1.1 is based on the following characterization of isomorphisms in an $\infty $-category $\operatorname{\mathcal{C}}$:

Theorem 4.3.1.3 (Joyal). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $u: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The morphism $u$ is an isomorphism.

$(2)$

Let $n \geq 2$ and let $\sigma _0: \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets for which the initial edge

\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0,1\} ) \hookrightarrow \Lambda ^ n_0 \xrightarrow {\sigma _0} \operatorname{\mathcal{C}} \]

is equal to $u$. Then $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$.

$(3)$

Let $n \geq 2$ and let $\sigma _0: \Lambda ^{n}_{n} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets for which the final edge

\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ n-1,n\} ) \hookrightarrow \Lambda ^ n_ n \xrightarrow {\sigma _0} \operatorname{\mathcal{C}} \]

is equal to $u$. Then $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$.

Proof of Proposition 4.3.1.1 from Theorem 4.3.1.3. Let $X$ be a simplicial set. By definition, the projection map $X \rightarrow \Delta ^{0}$ is a left fibration if and only if, for every pair of integers $0 < i \leq n$, every morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow X$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow X$. This condition is automatically satisfied when $n=1$ (we can identify $\sigma _0$ with a vertex $x \in X$, and take $\sigma $ to be the degenerate edge $\operatorname{id}_{x}$), and is satisfied for $0 < i < n$ if and only if $X$ is an $\infty $-category. Assuming that $X$ is an $\infty $-category, it is satisfied for $i = n$ if and only if every morphism in $X$ is an isomorphism (by virtue of Theorem 4.3.1.3). This proves the equivalence $(d) \Leftrightarrow (e)$, and the equivalence $(d) \Leftrightarrow (f)$ follows by applying the same reasoning to the opposite simplicial set $X^{\operatorname{op}}$. In particular, $(e)$ and $(f)$ are equivalent to one another, and therefore equivalent to $(a)$ (see Example 4.1.0.5). The equivalences $(a) \Leftrightarrow (b)$ and $(c) \Leftrightarrow (d)$ are immediate from the definitions. $\square$

The proof of Theorem 4.3.1.3 will require some preliminaries.

Definition 4.3.1.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. We will say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is conservative if it satisfies the following condition:

  • Let $u: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. If $F(u): F(X) \rightarrow F(Y)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$, then $u$ is an isomorphism.

Example 4.3.1.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the canonical map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ is conservative.

Remark 4.3.1.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors between $\infty $-categories, where $G$ is conservative. Then $F$ is conservative if and only if the composition $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is conservative.

Proposition 4.3.1.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. If $F$ is a left or a right fibration, then $F$ is conservative.

Proof. Without loss of generality, we may assume that $F$ is a left fibration. Let $u: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, and suppose that $F(u)$ is an isomorphism in $\operatorname{\mathcal{D}}$. Let $\overline{v}: F(Y) \rightarrow F(X)$ is a homotopy inverse to $F(u)$, so that there exists a $2$-simplex $\overline{\sigma }$ of $\operatorname{\mathcal{D}}$ as depicted in the following diagram:

\[ \xymatrix { & F(Y) \ar [dr]^{\overline{v}} & \\ F(X) \ar [ur]^{F(u)} \ar [rr]^{F( \operatorname{id}_ X) } & & F(X). } \]

Invoking our assumption that $F$ is a left fibration, we can lift $\overline{\sigma }$ to a diagram

\[ \xymatrix { & Y \ar [dr]^{v} & \\ X \ar [ur]^{u} \ar [rr]^{\operatorname{id}_{X}} & & X } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$. This lift supplies a morphism $v: Y \rightarrow X$ and witnesses that $\operatorname{id}_{X}$ as a composition of $v$ with $u$, so that $v$ is a left homotopy inverse to $u$. Moreover, the image $F(v) = \overline{v}$ is an isomorphism in $\operatorname{\mathcal{D}}$. Repeating the preceding argument (with $u: X \rightarrow Y$ replaced by $v: Y \rightarrow X$), we deduce that there exists a morphism $w: X \rightarrow Y$ which is left homotopy inverse to $v$. It follows that $u$ and $w$ are homotopic, so that $v$ is a homotopy inverse to $u$ (Remark 1.3.6.6). In particular, $u$ is an isomorphism. $\square$

Lemma 4.3.1.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a left fibration of $\infty $-categories, let $Y$ be an object of $\operatorname{\mathcal{C}}$, and let $\overline{u}: \overline{X} \rightarrow F(Y)$ be an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$. Then there exists a morphism $u: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ such that $F(u) = \overline{u}$ (note that $u$ is automatically an isomorphism, by virtue of Proposition 4.3.1.7).

Proof. Let $\overline{v}: F(Y) \rightarrow \overline{X}$ be a homotopy inverse to $\overline{u}$ in the $\infty $-category $\operatorname{\mathcal{D}}$. Since $F$ is a left fibration, there exists a morphism $v: Y \rightarrow X$ in the $\infty $-category $\operatorname{\mathcal{C}}$ satisfying $F(v) = \overline{v}$. Choose a $2$-simplex $\overline{\sigma }$ in $\operatorname{\mathcal{C}}$ which witnesses $\operatorname{id}_{F(Y)} = F( \operatorname{id}_ Y )$ as a composition of $\overline{u}$ with $\overline{v} = F(v)$, as indicated in the diagram

\[ \xymatrix { & F(X) \ar [dr]^{\overline{u}} & \\ F(Y) \ar [ur]^{ F(v) } \ar [rr]^{ F( \operatorname{id}_ Y) } & & F(Y). } \]

Applying our assumption that $F$ is a left fibration, we can lift $\overline{\sigma }$ to a $2$-simplex

\[ \xymatrix { & X \ar [dr]^{u} & \\ Y \ar [ur]^{v} \ar [rr]^{\operatorname{id}_{Y}} & & Y, } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$. In particular, the morphism $u: X \rightarrow Y$ satisfies $F(u) = \overline{u}$. $\square$

Proof of Theorem 4.3.1.3. We first prove the implication $(1) \Rightarrow (3)$. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $u$ be a morphism in $\operatorname{\mathcal{C}}$, and let $\sigma _0: \Lambda ^{n}_{n} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets for some $n \geq 2$ with the property that the composition

\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ n-1,n\} ) \hookrightarrow \Lambda ^{n}_{n} \xrightarrow {\sigma _0} \operatorname{\mathcal{C}} \]

is equal to $u$. We wish to show that $\sigma _0$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$. Using Lemma 4.2.6.8, we can identify the horn $\Lambda ^{n}_{0}$ with the pushout

\[ (\Delta ^{n-2} \star \{ 1\} ) \coprod _{ ( \operatorname{\partial \Delta }^{n-2} \star \{ 1\} ) } ( \operatorname{\partial \Delta }^{n-2} \star \Delta ^1) \subseteq \Delta ^{n-2} \star \Delta ^1 \simeq \Delta ^{n}. \]

Set $f= \sigma _0|_{ \Delta ^{n-2} }$ and $f_0 = \sigma _0|_{ \operatorname{\partial \Delta }^{n-2} }$. Then the restriction of $\sigma _0$ to $\Delta ^{n-2} \star \{ 1\} $ can be identified with an object $Y$ of the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$. Let $\overline{Y}$ denote the image of $Y$ in the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f_0/}$. Then the restriction of $\sigma _0$ to $\operatorname{\partial \Delta }^{n-2} \star \Delta ^1$ can be identified with an edge $\overline{v}: \overline{X} \rightarrow \overline{Y}$ in the $\infty $-category $\operatorname{\mathcal{C}}_{f_0/}$, fitting into a commutative diagram

\[ \xymatrix { \{ 1\} \ar [d] \ar [r]^-{Y} & \operatorname{\mathcal{C}}_{f/} \ar [d]^{q} \\ \Delta ^1 \ar [r]^-{\overline{v}} \ar@ {-->}[ur]^{v} & \operatorname{\mathcal{C}}_{f_0/}. } \]

To prove the existence of $\sigma $, it will suffice to show that there exists a dotted arrow as indicated, rendering the diagram commutative. In other words, we must show that $\overline{v}$ can be lifted to a morphism $v: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}_{/f}$. It follows from Corollary 4.2.6.5 that the morphism $q$ is a left fibration. By virtue of Lemma 4.3.1.8, to prove the existence of $v$, it will suffice to show that $\overline{v}$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}_{f_0/}$. Proposition 4.2.6.1 guarantees that the projection map $p: \operatorname{\mathcal{C}}_{f_0/} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration of $\infty $-categories, and therefore conservative (Proposition 4.3.1.7). We conclude by observing that $p( \overline{v} ) = u$ is an isomorphism by assumption.

We now show that $(3) \Rightarrow (1)$. Let $u: X \rightarrow Y$ be a morphism in the $\infty $-category $\operatorname{\mathcal{C}}$, and consider the map $\sigma _0: \Lambda ^{2}_{2} \rightarrow \operatorname{\mathcal{C}}$ depicted in the diagram

\[ \xymatrix { & X \ar [dr]^{u} & \\ Y \ar@ {-->}[ur]^{v} \ar [rr]^{ \operatorname{id}_ Y} & & Y. } \]

If $u$ satisfies condition $(3)$, then we can complete $\sigma _0$ to a $2$-simplex of $\operatorname{\mathcal{C}}$, which witnesses the morphism $v = d_2(\sigma )$ as a right homotopy inverse of $u$. The tuple $(\sigma , s_0(u), s_1(u), \bullet )$ then determines a morphism of simplicial sets $\tau _0: \Lambda ^3_3 \rightarrow \operatorname{\mathcal{C}}$ (see Exercise 1.1.2.14). Invoking assumption $(3)$ again, we can extend $\tau _0$ to a $3$-simplex $\tau $ of $\operatorname{\mathcal{C}}$. The face $d_3(\tau )$ then witnesses that $v$ is also a left homotopy inverse to $u$, so that $u$ is an isomorphism as desired. This completes the proof of the equivalence $(1) \Leftrightarrow (3)$. The equivalence $(1) \Leftrightarrow (2)$ follows by applying the same arguments to the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. $\square$