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5.1.1 Covariant Transport

Let $q: X \rightarrow S$ be a left fibration of simplicial sets. Then, for every vertex $s \in S$, the fiber $X_{s} = \{ s\} \times _{S} X$ is a Kan complex (Corollary 4.4.2.2). Our goal in this section is to show that the construction $s \mapsto X_{s}$ determines a functor from the homotopy category $\mathrm{h} \mathit{S}$ (Definition 1.2.5.1) to the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$ (Construction 5.1.1.7). For a homotopy-coherent refinement of this result, we refer the reader to ยง).

Definition 5.1.1.1. Let $q: X \rightarrow S$ be a left fibration of simplicial sets, let $e: s \rightarrow s'$ be an edge of the simplicial set $S$, and let $f: X_{s} \rightarrow X_{s'}$ be a morphism of simplicial sets. We will say that $f$ is given by covariant transport along $e$ if there exists a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \times X_{s} \ar [r]^-{h} \ar [d] & X \ar [d]^{q} \\ \Delta ^{1} \ar [r]^-{e} & S } \]

satisfying $h|_{ \{ 0\} \times X_{s} } = \operatorname{id}_{ X_ s }$, $h|_{ \{ 1\} \times X_{s} } = f$, and the left vertical map is given by projection onto the first factor. In this case, we also say that the morphism $h$ witnesses $f$ as given by covariant transport along $e$.

Example 5.1.1.2. Let $q: X \rightarrow S$ be a left fibration of simplicial sets and let $s \in S$ be a vertex. Then a morphism of Kan complexes $f: X_{s} \rightarrow X_{s}$ is given by covariant transport along the degenerate edge $\operatorname{id}_{s}: s \rightarrow s$ if and only if $f$ is homotopic to the the identity morphism $\operatorname{id}_{ X_ s}$.

Remark 5.1.1.3. Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \overline{X} \ar [d]^{ \overline{q} } \ar [r] & X \ar [d]^{q} \\ \overline{S} \ar [r] & S. } \]

Let $\overline{e}: \overline{s} \rightarrow \overline{s}'$ be an edge of the simplicial set $\overline{S}$ having image $e: s \rightarrow s'$ in the simplicial set $S$. Then a morphism of Kan complexes $f: \overline{X}_{ \overline{s} } \rightarrow \overline{X}_{ \overline{s}' }$ is given by covariant transport along $\overline{e}$ if and only if, when viewed as a morphism from $X_{s}$ to $X_{s'}$, it is given by covariant transport along $e$.

Proposition 5.1.1.4. Let $q: X \rightarrow S$ be a left fibration of simplicial sets and let $e: s \rightarrow s'$ be an edge of of the simplicial set $S$. Then:

$(1)$

There exists a morphism of Kan complexes $f: X_{s} \rightarrow X_{s'}$ which is given by covariant transport along $e$.

$(2)$

Let $g: X_{s} \rightarrow X_{s'}$ be any morphism of Kan complexes. Then $g$ is given by covariant transport along $e$ if and only if it is homotopic to $f$.

Proof. We first prove $(1)$. Let $\underline{s}$, $\underline{s}'$, and $\underline{e}$ denote the images of $s$, $s'$, and $e$ under the diagonal map $S \rightarrow \operatorname{Fun}( X_{s}, S)$. Let $\overline{s}$ denote the vertex of the simplicial set $\operatorname{Fun}(X_ s, X)$ corresponding to the inclusion map $X_ s \hookrightarrow X$. Applying Corollary 4.2.3.2, we see that postcomposition with $q$ induces a left fibration $Q: \operatorname{Fun}( X_ s, X) \rightarrow \operatorname{Fun}( X_ s, S)$ satisfying $Q( \overline{s} ) = \underline{s}$. It follows that $\underline{e}$ can be lifted to an edge $\overline{e}: \overline{s} \rightarrow \overline{s}'$ of the simplicial set $\operatorname{Fun}(X_ s, X)$. Unwinding the definitions, we see that $\overline{s}'$ can be identified with a morphism of Kan complexes $f: X_{s} \rightarrow X_{s'}$ and that $\overline{e}$ can be identified with a morphism $h: \Delta ^1 \times X_{s} \rightarrow X$ which witnesses $f$ as given by covariant transport along the edge $e$.

We now prove $(2)$. Let $g: X_{s} \rightarrow X_{s'}$ be a morphism of Kan complexes. Then $g$ can be identified with a vertex $\overline{s}''$ of the simplicial set $\operatorname{Fun}( X_{s}, X)$ satisfying $Q( \overline{s}'' ) = \underline{s}'$. Unwinding the definitions, we see that $f$ is homotopic to $g$ if and only if the following condition is satisfied:

  • There exists an edge $\overline{e}': \overline{s}' \rightarrow \overline{s}''$ of the simplicial set $\operatorname{Fun}( X_ s, X)$ satisfying $Q( \overline{e}' ) = \operatorname{id}_{ \underline{s}' }$.

Similarly, $g$ is given by covariant transport along $e$ if and only if the following alternative condition is satisfied:

$(\ast ')$

There exists an edge $\overline{e}'': \overline{s} \rightarrow \overline{s}''$ of the simplicial set $\operatorname{Fun}( X_ s, X)$ satisfying $Q( \overline{e}'' ) = \underline{e}$.

To complete the proof, it suffices to show that conditions $(\ast )$ and $(\ast ')$ are equivalent to one another. The implication $(\ast ) \Rightarrow (\ast ')$ follows from the observation that the left fibration $Q: \operatorname{Fun}( X_ s, X) \rightarrow \operatorname{Fun}(X_ s, S)$ has the right lifting property with respect to the horn inclusion $\Lambda ^{2}_{0} \hookrightarrow \Delta ^2$, and the implication $(\ast ') \Rightarrow (\ast )$ follows from the observation that $Q$ has the right lifting property with respect to the horn inclusion $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$. $\square$

Notation 5.1.1.5. Let $q: X \rightarrow S$ be a left fibration of simplicial sets, and let $e: s \rightarrow s'$ be an edge of the simplicial set $S$. We will often use the symbol $e_{!}$ to denote a morphism of Kan complexes $X_{s} \rightarrow X_{s'}$ which is given by covariant transport along $e$. By virtue of Proposition 5.1.1.4, such a morphism exists and is uniquely determined up to homotopy.

Proposition 5.1.1.6 (Transitivity). Let $q: X \rightarrow S$ be a left fibration of simplicial sets and let $\sigma $ be a $2$-simplex of $S$, whose faces we display as a diagram

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

Let $f: X_{s} \rightarrow X_{s'}$ be a morphism of Kan complexes given by covariant transport along $e$, and let $f': X_{s'} \rightarrow X_{s''}$ be a morphism of Kan complexes given by covariant transport along $e'$. Then the composite map $(f' \circ f): X_{s} \rightarrow X_{s''}$ is given by covariant transport along $e''$.

Proof. For every simplex $\tau $ of the simplicial set $S$, let $\underline{\tau }$ denote its image in the simplicial set $\operatorname{Fun}(X_ s, S)$. Let $Q: \operatorname{Fun}( X_ s, X) \rightarrow \operatorname{Fun}( X_ s, S)$ be given by postcomposition with $q$. Note that the inclusion map $X_{s} \hookrightarrow X$, the morphism $f: X_{s} \rightarrow X_{s'}$, and the composite map $(f' \circ f): X_{s} \rightarrow X_{s''}$ can be identified with vertices $\overline{s}$, $\overline{s}'$, and $\overline{s}''$ of the simplicial set $\operatorname{Fun}( X_ s, X)$ satisfying $Q( \overline{s} ) = \underline{s}$, $Q( \overline{s}' ) = \underline{s}'$, and $Q( \overline{s}'') = \underline{s}''$. Let $h: \Delta ^1 \times X_{s} \rightarrow X$ witness $f$ as given by covariant transport along $e$, and let $h': \Delta ^1 \times X_{s'} \rightarrow X$ witness $f'$ as given by covariant transport along $e'$. Then $h$ can be identified with an edge $\overline{e}: \overline{s} \rightarrow \overline{s}'$ of the simplicial set $\operatorname{Fun}( X_ s, X)$ satisfying $Q( \overline{e} ) = \underline{e}$. Similarly, the composite map

\[ \Delta ^1 \times X_ s \xrightarrow { \operatorname{id}\times f} \Delta ^1 \times X_{s'} \xrightarrow {h'} X \]

can be identified with an edge $\overline{e}'$ of $\operatorname{Fun}(X_ s, X)$ satisfying $Q( \overline{e}' ) = \underline{e}'$. Since $Q$ is a left fibration (Corollary 4.2.3.2), the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{2}_{1} \ar [r]^-{ ( \overline{e}', \bullet , \overline{e} ) } \ar [d] & \operatorname{Fun}( X_ s, X) \ar [d]^{Q} \\ \Delta ^2 \ar [r]^-{ \underline{\sigma } } \ar@ {-->}[ur]^{ \overline{\sigma } } & \operatorname{Fun}(X_ s, S) } \]

admits a solution. Unwinding the definitions, we see that $d_1( \overline{\sigma } )$ can be identified with a map of simplicial sets $\Delta ^1 \times X_{s} \rightarrow X$ which witnesses $f' \circ f$ as given by covariant transport along $e''$. $\square$

Construction 5.1.1.7 (The Covariant Transport Functor). Let $q: X \rightarrow S$ be a left fibration of simplicial sets, and let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes (Construction 3.1.4.10). Combining Proposition 5.1.1.4, Proposition 5.1.1.6, and Example 5.1.1.2, we deduce that there is a unique map of simplicial sets $T: S \rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{Kan}} )$ with the following properties:

  • For each vertex $s \in S$, we have $T(s) = X_{s}$.

  • For each edge $e: s \rightarrow s'$ of the simplicial set $S$, we have $T(e) = [ e_{!} ]$. Here $e_{!}: X_{s} \rightarrow X_{s'}$ denotes a morphism of Kan complexes given by covariant transport along $e$, and $[e_{!}]$ denotes its homotopy class.

The morphism $T$ determines a functor from the homotopy category $\mathrm{h} \mathit{S}$ (Notation 1.2.5.3) to the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, which we will also denote by $T: \mathrm{h} \mathit{S} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$. We will refer to $T$ as the covariant transport functor.

We now consider a dual version of Definition 5.1.1.1.

Definition 5.1.1.8. Let $q: X \rightarrow S$ be a right fibration of simplicial sets, let $e: s \rightarrow s'$ be an edge of the simplicial set $S$, and let $g: X_{s'} \rightarrow X_{s}$ be a morphism of simplicial sets. We will say that $g$ is given by contravariant transport along $e$ if there exists a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \times X_{s'} \ar [r]^-{h} \ar [d] & X \ar [d]^{q} \\ \Delta ^{1} \ar [r]^-{e} & S } \]

satisfying $h|_{ \{ 1\} \times X_{s'} } = \operatorname{id}_{ X_{s'} }$, $h|_{ \{ 0\} \times X_{s} } = g$, and the left vertical map is given by projection onto the first factor.

Remark 5.1.1.9. In the situation of Definition 5.1.1.8, the morphism $g$ is given by contravariant transport along $e$ if and only if the opposite map $g^{\operatorname{op}}: X_{s'}^{\operatorname{op}} \rightarrow X_{s}^{\operatorname{op}}$ is given by covariant transport along $e$ with respect to the left fibration $q^{\operatorname{op}}: X^{\operatorname{op}} \rightarrow S^{\operatorname{op}}$.

Proposition 5.1.1.10. Let $q: X \rightarrow S$ be a left fibration of simplicial sets and let $e: s \rightarrow s'$ be an edge of of the simplicial set $S$. Then:

$(1)$

There exists a morphism of Kan complexes $g: X_{s'} \rightarrow X_{s}$ which is given by covariant transport along $e$.

$(2)$

Let $f: X_{s'} \rightarrow X_{s}$ be any morphism of Kan complexes. Then $f$ is given by covariant transport along $e$ if and only if it is homotopic to $g$.

Proof. Apply Proposition 5.1.1.4 to the opposite map $q^{\operatorname{op}}: X^{\operatorname{op}} \rightarrow S^{\operatorname{op}}$. $\square$

Notation 5.1.1.11. Let $q: X \rightarrow S$ be a right fibration of simplicial sets, and let $e: s \rightarrow s'$ be an edge of the simplicial set $S$. We will often use the symbol $e^{\ast }$ to denote a morphism of Kan complexes $X_{s'} \rightarrow X_{s}$ which is given by covariant transport along $e$. By virtue of Proposition 5.1.1.10, such a morphism exists and is uniquely determined up to homotopy.

Construction 5.1.1.12 (The Contravariant Transport Functor). Let $q: X \rightarrow S$ be a right fibration of simplicial sets, and let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes (Construction 3.1.4.10). Dualizing Construction 5.1.1.7, we deduce that there is a unique map of simplicial sets $T: S^{\operatorname{op}} \rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{Kan}} )$ with the following properties:

  • For each vertex $s \in S$, we have $T(s) = X_{s}$.

  • For each edge $e: s \rightarrow s'$ of the simplicial set $S$, we have $T(e) = [ e^{\ast } ]$. Here $e^{\ast }: X_{s'} \rightarrow X_{s}$ denotes a morphism of Kan complexes given by contravariant transport along $e$, and $[e^{\ast }]$ denotes its homotopy class.

The morphism $T$ determines a functor from the homotopy category $\mathrm{h} \mathit{S}^{\operatorname{op}}$ (Notation 1.2.5.3) to the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, which we will also denote by $T: \mathrm{h} \mathit{S}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$. We will refer to $T$ as the contravariant transport functor.