# Kerodon

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### 5.2.2 Covariant Transport Functors

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration between categories (Definition 5.0.0.3) and let $f: C \rightarrow D$ be a morphism in the category $\operatorname{\mathcal{C}}$. If $X$ is an object of the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, then our assumption that $U$ is a cocartesian fibration guarantees that we can choose an object $f_{!}(X)$ of the fiber $\operatorname{\mathcal{E}}_{D} = \{ D\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ together with a $U$-cocartesian morphism $\widetilde{f}_{X}: X \rightarrow f_{!}(X)$ satisfying $U( \widetilde{f}_ X) = f$. In this case, we can view the construction $X \mapsto f_{!}(X)$ as a functor from the category $\operatorname{\mathcal{E}}_{C}$ to the category $\operatorname{\mathcal{E}}_{D}$:

Proposition 5.2.2.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of categories and let $f: C \rightarrow D$ be a morphism of $\operatorname{\mathcal{C}}$. For each object $X \in \operatorname{\mathcal{E}}_{C}$, let $\widetilde{f}_{X}$ be a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$ having source $X$ and satisfying $U( \widetilde{f}_{X} ) = f$. Then there is a unique functor $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ with the following properties:

• For each object $X \in \operatorname{\mathcal{E}}_{C}$, the object $f_{!}(X) \in \operatorname{\mathcal{E}}_{D}$ is the target of the morphism $\widetilde{f}_{X}$.

• The construction $X \mapsto \widetilde{f}_{X}$ determines a natural transformation from the inclusion functor $\operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}$ to the functor $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D} \subseteq \operatorname{\mathcal{E}}$.

Proof. For each object $X \in \operatorname{\mathcal{E}}_{C}$, let $f_{!}(X)$ denote the target of the morphism $\widetilde{f}_{X}$. Let $u: X \rightarrow Y$ be a morphism in the category $\operatorname{\mathcal{E}}_{C}$. Invoking our assumption that $\widetilde{f}_{X}$ is $U$-cocartesian, we see that there is a unique morphism $f_{!}(u): f_{!}(X) \rightarrow f_{!}(Y)$ for which the diagram

5.12
$$\begin{gathered}\label{equation:defining-contravariant-transport} \xymatrix@R =50pt@C=50pt{ X \ar [d]^{u} \ar [r]^-{ \widetilde{f}_{X} } & f_{!}(X) \ar [d]^{ f_{!}(u) } \\ Y \ar [r]^-{ \widetilde{f}_{Y} } & f_{!}(Y) } \end{gathered}$$

is commutative (in the category $\operatorname{\mathcal{E}}$). Note that if $v: Y \rightarrow Z$ is another morphism in the category $\operatorname{\mathcal{E}}_{C}$, then the calculation

$f_{!}(v) \circ f_{!}(u) \circ \widetilde{f}_{X} = f_{!}(v) \circ \widetilde{f}_{Y} \circ u = \widetilde{f}_{Z} \circ v \circ u$

shows that $f_{!}( v \circ u) = f_{!}(v) \circ f_{!}(u)$. Similarly, for each object $X \in \operatorname{\mathcal{E}}_{C}$, the calculation $\widetilde{f}_{X} \circ \operatorname{id}_{ f_{!}(X) } = \widetilde{f}_{X} = \operatorname{id}_{X} \circ \widetilde{f}_{X}$ shows that $f_{!}( \operatorname{id}_{X} ) = \operatorname{id}_{ f_{!}(X)}$. We can therefore regard $f_{!}$ as a functor from the category $\operatorname{\mathcal{E}}_{C}$ to $\operatorname{\mathcal{E}}_{D}$, and the commutativity of (5.12) guarantees that the construction $X \mapsto \widetilde{f}_{X}$ determines a natural transformation from the inclusion $\operatorname{\mathcal{E}}_{C} \hookrightarrow \operatorname{\mathcal{E}}$ to the functor $f_{!}$. $\square$

Construction 5.2.2.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of categories, let $f: C \rightarrow D$ be a morphism of the category $\operatorname{\mathcal{C}}$, and let $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ be the functor of Proposition 5.2.2.1. We will refer to $f_{!}$ as the functor of covariant transport along $f$.

Warning 5.2.2.3. In the situation of Construction 5.2.2.2, the covariant transport functor $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ depends not only on the cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and the morphism $f: C \rightarrow D$, but also on the system of $U$-cocartesian lifts $\{ \widetilde{f}_{X}: X \rightarrow f_{!}(X) \} _{X \in \operatorname{\mathcal{E}}_{C}}$. A different system of cocartesian lifts $\{ \widetilde{f}'_{X}: X \rightarrow f'_{!}(X) \} _{X \in \operatorname{\mathcal{E}}_{C} }$ will give rise to a different covariant transport functor $f'_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$. However, there is a canonical isomorphism of functors $\alpha : f_{!} \simeq f'_{!}$, which is uniquely determined by the requirement that for every object $X \in \operatorname{\mathcal{E}}_{C}$, the diagram

$\xymatrix@R =50pt@C=50pt{ & f_{!}(X) \ar [dr]^{ \alpha _{X} }_-{\sim } & \\ X \ar [ur]^-{ \widetilde{f}_{X} } \ar [rr]^-{ \widetilde{f}'_{X} } & & f'_{!}(X) }$

is commutative.

We now apply the results of §5.2.1 to extend Construction 5.2.2.2 to the $\infty$-categorical setting.

Definition 5.2.2.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets, let $f: C \rightarrow D$ be an edge of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}_{D} = \{ D\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ denote the corresponding fibers of $U$. We will say that a functor $F: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ is given by covariant transport along $f$ if there exists a morphism of simplicial sets $\widetilde{F}: \Delta ^1 \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}$ satisfying the following conditions:

$(1)$

The diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \Delta ^1\times \operatorname{\mathcal{E}}_{C} \ar [r]^-{ \widetilde{F} } \ar [d] & \operatorname{\mathcal{E}}\ar [d]^-{U} \\ \Delta ^1 \ar [r]^-{f} & \operatorname{\mathcal{C}}}$

commutes.

$(2)$

The restriction $\widetilde{F}|_{ \{ 0\} \times \operatorname{\mathcal{E}}_{C} }$ is the identity map $\operatorname{id}_{\operatorname{\mathcal{E}}_{C}}$, and the restriction $\widetilde{F}|_{ \{ 1\} \times \operatorname{\mathcal{E}}_{C} }$ is equal to $F$.

$(3)$

For every object $X$ of the $\infty$-category $\operatorname{\mathcal{E}}_ C$, the composite map

$\Delta ^1 \times \{ X\} \hookrightarrow \Delta ^1 \times \operatorname{\mathcal{E}}_{C} \xrightarrow { \widetilde{F} } \operatorname{\mathcal{E}}$

is a locally $U$-cocartesian edge of the simplicial set $\operatorname{\mathcal{E}}$.

If these conditions are satisfied, we say that the morphism $\widetilde{F}$ witnesses $F$ as given by covariant transport along $f$.

Example 5.2.2.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets and let $C$ be a vertex of $\operatorname{\mathcal{C}}$. Then the projection map

$\Delta ^1 \times \operatorname{\mathcal{E}}_{C} \twoheadrightarrow \operatorname{\mathcal{E}}_{C} \hookrightarrow \operatorname{\mathcal{E}}$

exhibits the identity functor $\operatorname{id}_{ \operatorname{\mathcal{E}}_{C} }$ as given by covariant transport along the degenerate edge $\operatorname{id}_{C}$. See Example 5.1.3.6.

Example 5.2.2.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left covering map of simplicial sets. Then, for every edge $f: C \rightarrow D$ in $\operatorname{\mathcal{C}}$, there is a unique functor $\operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ given by covariant transport along $f$, which can be identified with the covariant transport function given by Construction 5.2.0.1.

Example 5.2.2.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration between ordinary categories, let $f: C \rightarrow D$ be a morphism in $\operatorname{\mathcal{C}}$, and choose a collection of $U$-cocartesian morphisms $\{ \widetilde{f}_{X}: X \rightarrow f_{!}(X) \} _{X \in \operatorname{\mathcal{E}}_{C} }$ satisfying $U( \widetilde{f}_{X} ) = f$. According to Proposition 5.2.2.1, there is a unique functor $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ for which the construction $X \mapsto \widetilde{f}_{X}$ determines a natural transformation of functors $\widetilde{f}: \operatorname{id}_{ \operatorname{\mathcal{E}}_{C} } \rightarrow f_{!}$. Passing to nerves, we obtain a natural transformation $\operatorname{id}_{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{E}}_{C} )} \rightarrow \operatorname{N}_{\bullet }(f_!)$, which exhibits the functor

$\operatorname{N}_{\bullet }(f_!): \operatorname{N}_{\bullet }( \operatorname{\mathcal{E}}_ C ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{E}}_{D} )$

as given by covariant transport along $f$ (regarded as an edge of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$).

Stated more informally, the covariant transport construction for cocartesian fibrations of ordinary categories (see Construction 5.2.2.2) can be regarded as a special case Definition 5.2.2.4.

Proposition 5.2.2.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $f: C \rightarrow D$ be an edge of $\operatorname{\mathcal{C}}$. Then:

• There exists a functor $F: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ which is given by covariant transport along $f$.

• An arbitrary functor $F': \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ is given by covariant transport along $f$ if and only if it is isomorphic to $F$ (as an object of the $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )$).

Notation 5.2.2.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $f: C \rightarrow D$ be an edge of the simplicial set $\operatorname{\mathcal{C}}$. Applying Proposition 5.2.2.8, we conclude that the collection of functors $\operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ which are given by covariant transport along $f$ comprise a single isomorphism class in the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )$. We will denote this isomorphism class by $[f_{!}]$, which we regard as an element of the set $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )^{\simeq })$. We will often use the notation $f_{!}$ to denote a particular choice of representative of this isomorphism class: that is, a particular choice of functor $\operatorname{\mathcal{E}}_ C \rightarrow \operatorname{\mathcal{E}}_{D}$ which is given by covariant transport along $f$.

We now explain how to deduce Proposition 5.2.2.8 from Theorem 5.2.1.1. For this purpose, it will be convenient to introduce a bit more terminology.

Definition 5.2.2.10. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Let $K$ be another simplicial set, let $H: \Delta ^1 \times K \rightarrow \operatorname{\mathcal{E}}$ be a morphism. We will say that $H$ is a $U$-cocartesian lift of $\overline{H} = U \circ H$ if, for every vertex $x \in K$, the restriction $H|_{ \Delta ^1 \times \{ x\} }$ is a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.

Remark 5.2.2.11. In the situation of Definition 5.2.2.10, we can identify $H$ and $\overline{H}$ with edges of the simplicial sets $\operatorname{Fun}(K, \operatorname{\mathcal{E}})$ and $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$, respectively. Then $H$ is a $U$-cocartesian lift of $\overline{H}$ if and only if it is $U'$-cocartesian, where $U': \operatorname{Fun}(K,\operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}})$ is given by postcomposition with $U$. (see Theorem 5.2.1.1).

Example 5.2.2.12. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $f: C \rightarrow D$ be an edge of $\operatorname{\mathcal{C}}$, let $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}_{D} = \{ D\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ denote the corresponding fibers of $U$. Suppose we are given a commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \Delta ^1 \times \operatorname{\mathcal{E}}_{C} \ar [r]^-{\widetilde{F}} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \Delta ^1 \ar [r]^-{f} & \operatorname{\mathcal{C}}, }$

where the restriction $\widetilde{F}|_{ \{ 0\} \times \operatorname{\mathcal{E}}_{C} }$ is the identity map from $\operatorname{\mathcal{E}}_{C}$ to itself, and set $F = \widetilde{F}|_{ \{ 1\} \times \operatorname{\mathcal{E}}_{C} } \in \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )$. Then $\widetilde{F}$ witnesses $F$ as given by covariant transport along $f$ (in the sense of Definition 5.2.2.4) if and only if it is a $U$-cocartesian lift of $U \circ \widetilde{F}$ (in the sense of Definition 5.2.2.10).

Lemma 5.2.2.13. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $K$ be a simplicial set, and suppose we are given a lifting problem

5.13
$$\begin{gathered}\label{equation:existence-uniqueness-cocartesian-lift} \xymatrix@R =50pt@C=50pt{ \{ 0\} \times K \ar [r]^-{H_0} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \Delta ^1 \times K \ar [r]^-{\overline{H}} \ar@ {-->}[ur]^{H} & \operatorname{\mathcal{C}}. } \end{gathered}$$

Then:

$(1)$

The lifting problem (5.13) admits a solution $H: \Delta ^1 \times K \rightarrow \operatorname{\mathcal{E}}$ which is a $U$-cocartesian lift of $\overline{H}$.

$(2)$

Let $F$ be any object of the $\infty$-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \{ 1\} \times K, \operatorname{\mathcal{E}})$. Then $F$ is isomorphic to $H|_{ \{ 1\} \times K}$ (as an object of $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \{ 1\} \times K, \operatorname{\mathcal{E}})$) if and only if $F = H'|_{ \{ 1\} \times K }$, where $H'$ is another $U$-cocartesian lift of $\overline{H}$ which solves the lifting problem (5.13).

Proof. By virtue of Remark 5.2.2.11 (and Theorem 5.2.1.1), we can replace $U$ by the induced map $\operatorname{Fun}(K, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}})$ and thereby reduce to the case where $K = \Delta ^0$. In this case, assertion $(1)$ follows immediately from our assumption that $U$ is a cocartesian fibration, and assertion $(2)$ follows from Remark 5.1.3.8. $\square$

Proof of Proposition 5.2.2.8. Apply Lemma 5.2.2.13 in the special case where $K$ is the $\infty$-category $\operatorname{\mathcal{E}}_{C}$, $H_0: K \rightarrow \operatorname{\mathcal{E}}$ is the inclusion map, and $\overline{H}$ is the composite map $\Delta ^1 \times K \rightarrow \Delta ^1 \xrightarrow {f} \operatorname{\mathcal{C}}$ (see Example 5.2.2.12). $\square$

We also have a dual version of Definition 5.2.2.4:

Definition 5.2.2.14. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets, let $C$ and $D$ be vertices of $\operatorname{\mathcal{C}}$, and let $f: C \rightarrow D$ be an edge of $\operatorname{\mathcal{C}}$. We say that a functor $F: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ is given by contravariant transport along $f$ if there exists a morphism of simplicial sets $\widetilde{F}: \Delta ^1 \times \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}$ satisfying the following conditions:

$(1)$

The diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \Delta ^1 \times \operatorname{\mathcal{E}}_{D} \ar [r]^-{ \widetilde{F} } \ar [d] & \operatorname{\mathcal{E}}\ar [d]^-{U} \\ \Delta ^1 \ar [r]^-{f} & \operatorname{\mathcal{C}}}$

commutes.

$(2)$

The restriction $\widetilde{F}|_{ \{ 1\} \times \operatorname{\mathcal{E}}_{D}}$ is equal to the identity map $\operatorname{id}_{\operatorname{\mathcal{E}}_{D}}$, and the restriction $\widetilde{F}|_{ \{ 0\} \times \operatorname{\mathcal{E}}_{D}}$ is equal to $F$.

$(3)$

For every object $Y$ of the $\infty$-category $\operatorname{\mathcal{E}}_{D}$, the composite map

$\Delta ^1 \times \{ Y\} \hookrightarrow \Delta ^1 \times \operatorname{\mathcal{E}}_ D \xrightarrow { \widetilde{F} } \operatorname{\mathcal{E}}$

is a locally $U$-cartesian edge of the simplicial set $\operatorname{\mathcal{E}}$.

If these conditions are satisfied, we say that the morphism $\widetilde{F}$ witnesses $F$ as given by contravariant transport along $f$.

Remark 5.2.2.15. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets, let $C$ and $D$ be vertices of $\operatorname{\mathcal{C}}$, and let $f: C \rightarrow D$ be a morphism of simplicial sets. Then a functor $F: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ is given by covariant transport along $f$ if and only if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{E}}_{C}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{E}}_{D}^{\operatorname{op}}$ is given by contravariant transport along $f$ with respect to the cartesian fibration $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$.

Proposition 5.2.2.8 has a counterpart for cartesian fibrations:

Proposition 5.2.2.16. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets and let $f: C \rightarrow D$ be an edge of $\operatorname{\mathcal{C}}$. Then:

• There exists a functor $F: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ which is given by contravariant transport along $f$.

• An arbitrary functor $F': \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ is given by contravariant transport along $f$ if and only if it is isomorphic to $F$ (as an object of the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{E}}_{D}, \operatorname{\mathcal{E}}_{C} )$).

Notation 5.2.2.17. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets and let $f: C \rightarrow D$ be an edge of the simplicial set $\operatorname{\mathcal{C}}$. It follows from Proposition 5.2.2.16 that the collection of functors $\operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ which are given by contravariant transport along $f$ comprise a single isomorphism class in the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{E}}_{D}, \operatorname{\mathcal{E}}_{C} )$. We will denote this isomorphism class by $[f^{\ast }]$, which we regard as an element of the set $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{E}}_{D}, \operatorname{\mathcal{E}}_{C} )^{\simeq } )$. We will often use the notation $f^{\ast }$ to denote a particular choice of representative of this isomorphism class: that is, a particular choice of functor $\operatorname{\mathcal{E}}_ D \rightarrow \operatorname{\mathcal{E}}_{C}$ which is given by contravariant transport along $f$.

For Kan fibrations, there is a close relationship between covariant and contravariant transport:

Proposition 5.2.2.18. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a Kan fibration of simplicial sets and let $f: C \rightarrow D$ be an edge of $\operatorname{\mathcal{C}}$. Then the covariant and contravariant transport morphisms $[f_{!}] \in \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )$ and $[ f^{\ast } ] \in \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}} }( \operatorname{\mathcal{E}}_{D}, \operatorname{\mathcal{E}}_{C} )$ are inverse to one another (as morphisms in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$).

Proof. Choose morphisms of Kan complexes $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ and $f^{\ast }: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ representing the homotopy classes $[f_{!}]$ and $[ f^{\ast } ]$, respectively. We will show that $f^{\ast } \circ f_{!}$ is homotopic to the identity morphism $\operatorname{id}_{\operatorname{\mathcal{E}}_{C} }$; a similar argument will show that $f_{!} \circ f^{\ast }$ is homotopic to $\operatorname{id}_{ \operatorname{\mathcal{E}}_{D} }$. Let $\operatorname{\mathcal{D}}$ denote the fiber product $\operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$, and let $\pi : \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be the projection map onto the second factor. Since $U$ is a Kan fibration, it follows from Corollary 3.1.3.2 that $\pi$ is also a Kan fibration. Let $\widetilde{f}: \Delta ^{1} \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}$ be a morphism witnessing $f_{!}$ as given by covariant transport along $f$. Then $\widetilde{f}$ determines an edge $h$ of the simplicial set $\operatorname{\mathcal{D}}$ satisfying $\pi ( h ) = f$. Let $\widetilde{f}': \Delta ^1 \times \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}$ be a morphism which witnesses $f^{\ast }$ as given by contravariant transport along $f$, so that the composite morphism

$\Delta ^{1} \times \operatorname{\mathcal{E}}_{C} \xrightarrow { \operatorname{id}\times f_{!} } \Delta ^1 \times \operatorname{\mathcal{E}}_{D} \xrightarrow { \widetilde{f}' } \operatorname{\mathcal{E}}$

determines an edge $h'$ of the simplicial set $\operatorname{\mathcal{D}}$ satisfying $\pi (h') = f$. The edges $h$ and $h'$ have the same target (the vertex of $\operatorname{\mathcal{D}}$ corresponding to the morphism $f_{!}$). Invoking our assumption that $\pi$ is a Kan fibration, we deduce that there exists a $2$-simplex $\sigma$ of $\operatorname{\mathcal{D}}$ satisfying $d_0(\sigma ) = h'$, $d_1(\sigma ) = h$, and $\pi (\sigma ) = s_0(f)$; we can represent $\sigma$ as a diagram

$\xymatrix@R =50pt@C=50pt{ & f^{\ast } \circ f_{!} \ar [dr]^{h' } & \\ \operatorname{id}_{\operatorname{\mathcal{E}}_{C}} \ar@ {-->}[ur]^{v} \ar [rr]^-{ h } & & f_{!}. }$

We now observe that the edge $v = d_2(\sigma )$ of $\operatorname{\mathcal{D}}$ can be identified with a map of simplicial sets $V: \Delta ^1 \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C}$ which is a homotopy from $\operatorname{id}_{\operatorname{\mathcal{E}}_{C}} = V|_{ \{ 0\} \times \operatorname{\mathcal{E}}_{C} }$ to $f^{\ast } \circ f_! = V|_{ \{ 1\} \times \operatorname{\mathcal{E}}_{C} }$. $\square$

We close this section by establishing a converse to Proposition 5.2.2.18:

Theorem 5.2.2.19. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $U$ is a Kan fibration.

$(2)$

The morphism $U$ is a left fibration and, for every edge $f: C \rightarrow D$ of the simplicial set $\operatorname{\mathcal{C}}$, the covariant transport morphism $[f_{!}]: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

$(3)$

The morphism $U$ is a right fibration and, for every edge $f: C \rightarrow D$ of the simplicial set $\operatorname{\mathcal{C}}$, the contravariant transport morphism $[f^{\ast }]: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

Proof. We will show that $(1) \Leftrightarrow (2)$; the proof of the equivalence $(1) \Leftrightarrow (3)$ is similar. The implication $(1) \Rightarrow (2)$ is immediate from Proposition 5.2.2.18. For the converse, assume that $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration of simplicial sets and that, for every edge $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$, the covariant transport morphism $[f_{!}]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. We wish to show that $U$ is a Kan fibration. By virtue of Example 4.2.1.5, it will suffice to show that $U$ is a right fibration. By Proposition 4.2.6.1, this is equivalent to the assertion that the induced map

$\theta : \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{C}}) } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$

is a trivial Kan fibration. Note that our assumption that $U$ is a left fibration guarantees that $\theta$ is also a left fibration (Proposition 4.2.5.1).

Fix an edge $f: C \rightarrow D$ of the simplicial set $\operatorname{\mathcal{C}}$ and let $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}})_{f}$ denote the fiber $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) } \{ f\}$. Then evaluation at the vertex $1 \in \Delta ^1$ induces a morphism $\theta _{f}: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}})_{f} \rightarrow \operatorname{\mathcal{E}}_{D}$. Note that $\theta _{f}$ is a pullback of $\theta$, and is therefore also a left fibration. Since $\operatorname{\mathcal{E}}_{D}$ is a Kan complex (Corollary 4.4.2.3), Corollary 4.4.3.8 guarantees that $\theta _{f}$ is a Kan fibration (so $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})_{f}$ is also a Kan complex). Evaluation at the vertex $0 \in \Delta ^1$ induces another morphism of simplicial sets $u: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}})_{f} \rightarrow \operatorname{\mathcal{E}}_{C}$. Since $U$ is a left fibration, the morphism $u$ is a trivial Kan fibration. By construction, the homotopy class $[f_{!}]$ can be represented by the morphism of Kan complexes given by the composition

$\operatorname{\mathcal{E}}_{C} \xrightarrow {v} \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{E}})_{f} \xrightarrow { \theta _{f} } \operatorname{\mathcal{E}}_{D},$

where $v$ is a section of $u$ (and therefore a homotopy equivalence). Consequently, our assumption that $[f_{!}]$ is an isomorphism in $\mathrm{h} \mathit{\operatorname{Kan}}$ guarantees that $\theta _{f}$ is a homotopy equivalence of Kan complexes (Remark 3.1.6.16). Applying Corollary 3.2.7.4, we deduce that the fibers of $\theta _{f}$ are contractible Kan complexes. Since every fiber of $\theta$ can also be viewed as a fiber of $\theta _{f}$ for some edge $f$ of the simplicial set $\operatorname{\mathcal{C}}$, it follows that the fibers of $\theta$ are also contractible Kan complexes. Invoking Proposition 4.4.2.14, we conclude that $\theta$ is a trivial Kan fibration, as desired. $\square$

Corollary 5.2.2.20. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $U$ is a covering map (Definition 3.1.4.1).

$(2)$

The morphism $U$ is a left covering map (Definition 4.2.3.8) and, for every edge $f: C \rightarrow D$ of the simplicial set $\operatorname{\mathcal{C}}$, the covariant transport functor $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ is a bijection.

$(3)$

The morphism $U$ is a right covering map (Definition 4.2.3.8) and, for every edge $f: C \rightarrow D$ of the simplicial set $\operatorname{\mathcal{C}}$, the contravariant transport morphism $f^{\ast }: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ is a bijection.