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5.1.7 Example: Path Fibrations

Recall that every morphism of Kan complexes $f: X \rightarrow Y$ admits a canonical factorization

\[ X \xrightarrow {\delta } P(f) \xrightarrow {\pi } Y, \]

where $\delta $ is a homotopy equivalence and $\pi $ is the path fibration

\[ P(f) = X \times _{ \operatorname{Fun}( \{ 0\} , Y)} \operatorname{Fun}( \Delta ^1, Y) \rightarrow \operatorname{Fun}( \{ 1\} , Y) \simeq Y \]

of Example 3.1.7.10. Note that the simplicial set $P(f) = X \operatorname{\vec{\times }}_{Y} Y$ is an example of an oriented fiber product (Definition 4.6.4.1), which is defined for any morphism of simplicial sets $f: X \rightarrow Y$. Beware that if $X$ and $Y$ are not Kan complexes, then $\delta $ need not be a homotopy equivalence and $\pi $ need not be a Kan fibration. However, if $X = \operatorname{\mathcal{C}}$ and $Y = \operatorname{\mathcal{D}}$ are $\infty $-categories, then we have the following weaker statements:

$(a)$

The functor $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is fully faithful (Proposition 5.1.7.4).

$(b)$

The functor $\pi : \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ is a cocartesian fibration of $\infty $-categories (Proposition 5.1.7.1).

Because of these features, the factorization

\[ \operatorname{\mathcal{C}}\xrightarrow { \delta } \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\xrightarrow {\pi } \operatorname{\mathcal{D}} \]

remains quite useful in the $\infty $-categorical setting.

Proposition 5.1.7.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets and let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ denote the oriented fiber product of Definition 4.6.4.1, so that evaluation at the vertices $0,1 \in \Delta ^1$ determines maps

\[ \operatorname{\mathcal{C}}\xleftarrow {\pi '} \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\xrightarrow {\pi } \operatorname{\mathcal{D}}. \]

Then:

$(1)$

If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$ are $\infty $-categories, then the evaluation map $\pi : \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ is a cocartesian fibration of simplicial sets. Moreover, an edge $e$ of the simplicial set $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is $\pi $-cocartesian if and only if $\pi '(e)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.

$(2)$

If $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ are $\infty $-categories, then the evaluation map $\pi ': \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is a cartesian fibration of simplicial sets. Moreover, an edge $e$ of the simplicial set $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is $\pi '$-cartesian if and only if $\pi (e)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.

Example 5.1.7.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Applying Proposition 5.1.7.1 in the case where both $F$ and $G$ are the identity functor $\operatorname{id}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$, we deduce that the evaluation functor

\[ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}} \]

is a cartesian fibration of $\infty $-categories, and the evaluation functor

\[ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}} \]

is a cocartesian fibration of $\infty $-categories.

Proof of Proposition 5.1.7.1. We will prove assertion $(1)$; the proof of $(2)$ is similar. Note that the oriented fiber product $(\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ fits into a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\ar [d]^-{\pi } \ar [r] & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}\ar [d] \\ \operatorname{\mathcal{D}}\ar [r]^-{G} & \operatorname{\mathcal{E}}. } \]

By virtue of Remark 5.1.4.6, we can replace $\operatorname{\mathcal{D}}$ by $\operatorname{\mathcal{E}}$ and thereby reduce to the case where $\operatorname{\mathcal{D}}= \operatorname{\mathcal{E}}$ and $G$ is the identity functor $\operatorname{id}_{\operatorname{\mathcal{D}}}$. It follows from Proposition 4.6.4.2 that the map $(\pi ', \pi ): \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is an isofibration of $\infty $-categories, so that $\pi : \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ is also an isofibration. Let us say that a morphism $e$ of the $\infty $-category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is special if $\pi '(e)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$. Proposition 5.1.7.1 is an immediate consequence of the following three assertions:

$(a)$

For object $x$ in the $\infty $-category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ and every morphism $\overline{e}: \pi (x) \rightarrow \overline{y}$ in the $\infty $-category $\operatorname{\mathcal{D}}$, there exists a special morphism $e: x \rightarrow y$ in $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ satisfying $\pi (e) = \overline{e}$.

$(b)$

Every special morphism of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is $\pi $-cocartesian.

$(c)$

Every $\pi $-cocartesian morphism of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is special.

We begin with the proof of $(a)$. Let $x$ be an object of the $\infty $-category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$, which we identify with a triple $(\overline{x}', \overline{x}, u)$ where $\overline{x}' = \pi '(x)$ is an object of $\operatorname{\mathcal{C}}$, $\overline{x} = \pi (x)$ is an object of $\operatorname{\mathcal{D}}$, and $u: F( \overline{x}' ) \rightarrow \overline{x}$ is a morphism of $\operatorname{\mathcal{D}}$. Since $\operatorname{\mathcal{D}}$ is an $\infty $-category, we can choose a $2$-simplex $\sigma $ of $\operatorname{\mathcal{E}}$ satisfying $d_0(\sigma ) = \overline{e}$ and $d_2(\sigma ) = f$. Set $g = d_1(\sigma )$. Then the $2$-simplices $\sigma $ and $s_0(g)$ together determine a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ F( \overline{x}' ) \ar [d]_-{ \operatorname{id}_{ F( \overline{x}' ) }} \ar [r]^-{f} \ar [dr]_{g} & \overline{x} \ar [d]^-{ \overline{e} } \\ F( \overline{x}' ) \ar [r]^-{g} & \overline{y} } \]

in the $\infty $-category $\operatorname{\mathcal{D}}$, which we can identify with an edge $e: x \rightarrow y$ of the $\infty $-category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ satisfying $\pi '(e) = \operatorname{id}_{ \overline{x}'}$ and $\pi (e) = \overline{e}$.

We now prove $(b)$. Let $n \geq 2$ and suppose that we are given a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{0} \ar [r]^-{ \tau _0 } \ar [r] \ar [d] & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\ar [d]^-{ \pi } \\ \Delta ^{n} \ar [r]^-{ \overline{\tau } } \ar@ {-->}[ur]^{\tau } & \operatorname{\mathcal{D}}} \]

for which the composite map

\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \hookrightarrow \Lambda ^{n}_0 \xrightarrow { \tau _0 } \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}} \]

is a special edge of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$. Then $\pi ' \circ \tau _0$ carries the initial edge of $\Lambda ^{n}_0$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$, and can therefore be extended to an $n$-simplex $\rho : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$. Let $X(0)$ denote the simplicial subset of $\Delta ^1 \times \Delta ^ n$ given by the union of $\{ 0 \} \times \Delta ^ n$, $\{ 1\} \times \Delta ^ n$, and $\Delta ^1 \times \Lambda ^{n}_0$. The morphisms $\rho $, $\overline{\tau }$, and $\tau _0$ can then be amalgamated to a morphism of simplicial sets $h_0: X(0) \rightarrow \operatorname{\mathcal{D}}$. We wish to show that $h_0$ can be extended to a map $h: \Delta ^1 \times \Delta ^ n \rightarrow \operatorname{\mathcal{D}}$. Choose a filtration

\[ X(0) \subset X(1) \subset X(2) \subset \cdots \subset X(t) = \Delta ^{1} \times \Delta ^{n} \]

satisfying the requirements of Lemma 4.4.4.7. We will complete the proof of $(b)$ by showing that, for each $s \leq t$, the morphism $h_0$ admits an extension $h_ s: X(s) \rightarrow \operatorname{\mathcal{D}}$. The proof proceeds by induction on $s$, the case $s = 0$ being vacuous. Let us therefore assume that $0 < s \leq t$ and that $h_0$ has already been extended to a morphism of simplicial sets $h_{s-1}: X(s-1) \rightarrow \operatorname{\mathcal{D}}$. By construction, we have a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{q}_{p} \ar [r]^-{\varphi } \ar [d] & X(s-1) \ar [d] \\ \Delta ^{q} \ar [r] & X(s) } \]

for some $q \geq 2$ and $0 \leq p < q$. Consequently, to prove the existence of $h_ s$, it will suffice to show that $h_{s-1} \circ \varphi $ can be extended to a $q$-simplex of $\operatorname{\mathcal{D}}$. For $p \neq 0$, the existence of this extension follows from our assumption that $\operatorname{\mathcal{D}}$ is an $\infty $-category. In the case $p = 0$, Lemma 4.4.4.7 guarantees that the morphism $\varphi $ carries the initial edge of $\Delta ^{q}$ to the edge $(0,0) \rightarrow (0,1)$ of $\Delta ^{1} \times \Delta ^ n$, so that $h_{s-1} \circ \varphi $ carries the initial edge of $\Delta ^ q$ to an isomorphism in $\operatorname{\mathcal{D}}$. In this case, the existence of the desired extension follows from Theorem 4.4.2.6.

We now prove $(c)$. Let $e: x \rightarrow z$ be a $\pi $-cocartesian morphism in the $\infty $-category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$; we wish to show that $e$ is special. By virtue of $(a)$, there exists a special morphism $e': x \rightarrow y$ of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ satisfying $\pi (e) = \pi (e')$. It follows from $(b)$ that $e'$ is also $\pi $-cocartesian. Applying Remark 5.1.3.8, we deduce that there exists a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ & y \ar [dr]^{ e'' } & \\ x \ar [ur]^{e'} \ar [rr]^{e} & & z } \]

in the $\infty $-category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$, where $e''$ is an isomorphism. Then $\pi '(e')$ and $\pi '(e'')$ are isomorphisms in the $\infty $-category $\operatorname{\mathcal{C}}$, so that $\pi '(e)$ is also an isomorphism in $\operatorname{\mathcal{C}}$ (Remark 1.3.6.3). It follows that $e$ is a special morphism of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$, as desired. $\square$

Corollary 5.1.7.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. Then:

$(1)$

The restriction map $U: \operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is a cocartesian fibration. Moreover, a morphism $e$ of $\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}})$ is $U$-cocartesian if and only if it carries the cone point ${\bf 0} \in K^{\triangleleft }$ to an isomorphism in $\operatorname{\mathcal{C}}$.

$(2)$

The restriction map $V: \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is a cartesian fibration. Moreover, a morphism $e$ of $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$ is $U$-cartesian if and only if it carries the cone point ${\bf 1} \in K^{\triangleright }$ to an isomorphism in $\operatorname{\mathcal{C}}$.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Let $\Delta ^0 \diamond K$ be the simplicial set introduced in Notation 4.5.5.3, and let $c: \Delta ^0 \diamond K \rightarrow \Delta ^{0} \star K = K^{\triangleleft }$ be the categorical equivalence of Theorem 4.5.5.8. We have a commutative diagram of $\infty $-categories

\[ \xymatrix { \operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{C}}) \ar [dr]^{U} \ar [rr]^{ \circ c} & & \operatorname{Fun}( \Delta ^0 \diamond K, \operatorname{\mathcal{C}}) \ar [dl]^{U'} \\ & \operatorname{Fun}(K, \operatorname{\mathcal{C}}) & } \]

where the horizontal map is an equivalence of $\infty $-categories (Proposition 4.5.2.8) and the vertical maps are isofibrations (Corollary 4.4.5.3). Unwinding the definitions, we can identify $\operatorname{Fun}( \Delta ^0 \diamond K, \operatorname{\mathcal{C}})$ with the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Under this identification, the functor $U'$ is given by projection onto the second factor, and is therefore a cocartesian fibration (Proposition 5.1.7.1). Applying Corollary 5.1.5.2, we deduce that $U$ is also a cocartesian fibration. Moreover, a morphism $e$ of $\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}})$ is $U$-cocartesian if and only if its image in $\operatorname{Fun}( \Delta ^0 \diamond K, \operatorname{\mathcal{C}})$ is $U'$-cocartesian (Proposition 5.1.5.6). Using the criterion of Proposition 5.1.7.1, we see that this is equivalent to the requirement that $e$ carries the cone point ${\bf 0} \in K^{\triangleleft }$ to an isomorphism in $\operatorname{\mathcal{C}}$. $\square$

Proposition 5.1.7.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let

\[ \delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(\{ 0\} , \operatorname{\mathcal{D}})} \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}}) \]

be map induced by the diagonal embedding $c: \operatorname{\mathcal{D}}\hookrightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}})$. Then $\delta $ is fully faithful.

Proof. The functor $\delta $ fits into a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{\delta } \ar [d]^-{F} & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\ar [d] \ar [r]^-{\pi '} & \operatorname{\mathcal{C}}\ar [d]^-{F} \\ \operatorname{\mathcal{D}}\ar [r]^-{c} & \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}}) \ar [r] & \operatorname{\mathcal{D}}, } \]

where the horizontal compositions are the identity. Consequently, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we obtain a commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=25pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}}(\delta (X), \delta (Y) ) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(X), F(Y) ) \ar [r] & \operatorname{Fun}(\Delta ^1, \operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(X), F(Y) )) \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(X), F(Y) ).} \]

Note that the right half of the diagram is a pullback square in which the horizontal maps are Kan fibrations, and therefore a homotopy pullback square (Example 3.4.1.5). Since the outer rectangle is a homotopy pullback square, it follows that the left half of the diagram is also a homotopy pullback square (Proposition 3.4.1.9). Since the simplicial set $\Delta ^1$ is weakly contractible, the lower left horizontal map $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(X), F(Y) ) \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(X), F(Y) ))$ is a homotopy equivalence. Applying Corollary 3.4.1.3), we deduce that the upper left horizontal map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}}(\delta (X), \delta (Y) )$ is also a homotopy equivalence. $\square$