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5.2 Covariant Transport

Let $X \rightarrow S$ be a covering map of topological spaces. For every point $s \in S$, the fiber $X_{s} = \{ s\} \times _{S} X$ is equipped with an action of the fundamental group $\pi _{1}(S,s)$. More generally, the construction $s \mapsto X_{s}$ determines a functor from the fundamental groupoid $\pi _{\leq 1}(S)$ to the category of sets, which will refer to as the monodromy representation of the covering map $X \rightarrow S$ (see Example 5.2.0.5 below).

It will be convenient to place monodromy in a more general context. Recall that if $X \rightarrow S$ is a covering map of topological spaces, then the induced map $ \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$ is a covering map of simplicial sets (Proposition 3.1.4.9). In particular, it is a left covering map of simplicial sets (Definition 4.2.3.8).

Construction 5.2.0.1 (Covariant Transport for Left Covering Maps). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left covering map of simplicial sets. For each vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a discrete simplicial set, which we will identify with its underlying set of vertices (Remark 4.2.3.17). If $\widetilde{C}$ is a vertex of $\operatorname{\mathcal{E}}_{C}$ and $f: C \rightarrow D$ is an edge of $\operatorname{\mathcal{C}}$, then our assumption that $U$ is a left covering map guarantees that there is a unique edge $\widetilde{f}: \widetilde{C} \rightarrow f_{!}(\widetilde{C})$ of $\operatorname{\mathcal{E}}$ satisfying $U( \widetilde{f} ) = f$. The construction $\widetilde{C} \mapsto f_{!}(\widetilde{C})$ then determines a function $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$, which we will refer to as covariant transport along $f$.

Example 5.2.0.2. In the situation of Construction 5.2.0.1, suppose that $e = \operatorname{id}_{C}$ is a degenerate edge of $\operatorname{\mathcal{C}}$. Then the covariant transport function $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C}$ is the identity function.

Proposition 5.2.0.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left covering map of simplicial sets. Then there is a unique functor

\[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set} \]

with the following properties:

  • For each vertex $C \in \operatorname{\mathcal{C}}$, we have $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C) = \operatorname{\mathcal{E}}_{C}$.

  • Let $f: C \rightarrow D$ be an edge of $\operatorname{\mathcal{C}}$, and let $[f]$ denote the corresponding morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Then $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}( [f] )$ is the covariant transport function $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ of Construction 5.2.0.1.

Proof. By virtue of Example 5.2.0.2 (and the proof of Proposition 1.2.5.4), it will suffice to show that if $\sigma $ is a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & D \ar [dr]^{g} & \\ C \ar [ur]^{f} \ar [rr]^-{h} & & E, } \]

then the covariant transport function $h_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{E}$ is equal to the composition $g_{!} \circ f_{!}$. Fix a vertex $X \in \operatorname{\mathcal{E}}_{C}$. By construction, there is an edge $\widetilde{f}: X \rightarrow f_{!}(X)$ satisfying $U( \widetilde{f} ) = f$ and an edge $\widetilde{h}: X \rightarrow h_{!}(X)$ satisfying $U( \widetilde{h} ) = h$. Since $U$ is a left covering map, we can lift $\sigma $ (uniquely) to a $2$-simplex of $\operatorname{\mathcal{E}}$ with boundary indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & f_{!}(X) \ar [dr]^{\widetilde{g}} & \\ X \ar [ur]^{\widetilde{f}} \ar [rr]^-{\widetilde{h}} & & h_{!}(X). } \]

The edge $\widetilde{g}$ then satisfies $U( \widetilde{g} ) = g$, and therefore witnesses the identity $g_{!}( f_{!}(X) ) = h_{!}(X)$. $\square$

Definition 5.2.0.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left covering morphism of simplicial sets and let $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$ be the functor of Proposition 5.2.0.3. We will refer to $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the homotopy transport representation of $U$.

Example 5.2.0.5 (The Monodromy Representation). Let $X \rightarrow S$ be a covering map of topological spaces. Applying Proposition 5.2.0.3 to the induced map $\operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$, we obtain a functor from the fundamental groupoid $\pi _{\leq 1}(S)$ to the category of sets, which we will denote by $\operatorname{hTr}_{X/S}: \pi _{\leq 1}(S) \rightarrow \operatorname{Set}$ and refer to as the monodromy representation of $f$. Concretely, it is given on objects by the formula $\operatorname{hTr}_{X/S}(s) = \{ s\} \times _{S} X$.

Example 5.2.0.6. Let $\operatorname{Set}_{\ast }$ denote the category of pointed sets, so that the forgetful functor $\operatorname{Set}_{\ast } \rightarrow \operatorname{Set}$ induces a left covering morphism of simplicial sets $\operatorname{N}_{\bullet }( \operatorname{Set}_{\ast } ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set})$ (Example 4.2.3.3). Then the homotopy transport functor $\operatorname{hTr}_{ \operatorname{N}_{\bullet }( \operatorname{Set}_{\ast } ) / \operatorname{N}_{\bullet }(\operatorname{Set}) }$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{Set}}: \operatorname{Set}\rightarrow \operatorname{Set}$.

Our first goal in this section is to generalize the definition of the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ to the case where $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of simplicial sets. In §5.2.2, we associate to each edge $f: C \rightarrow D$ of the simplicial set $\operatorname{\mathcal{C}}$ a functor of $\infty $-categories $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$, which we refer to as the covariant transport functor associated to $f$ (Definition 5.2.2.4). Unlike the covariant transport function of Construction 5.2.0.1, the functor $f_{!}$ is not uniquely determined: it is well-defined only up to isomorphism (Proposition 5.2.2.8). To construct it (and to establish its uniqueness up to isomorphism), we will exploit the fact that postcomposition with $U$ induces a cocartesian fibration $\operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{C}})$, which we prove in §5.2.1 (see Theorem 5.2.1.1).

In §5.2.5, we study the behavior of covariant transport with respect to composition. Suppose we are given a $2$-simplex $\sigma $ of the simplicial set $\operatorname{\mathcal{C}}$, which we view as a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ & D \ar [dr]^{g} & \\ C \ar [ur]^-{f} \ar [rr]^-{h} & & E. } \]

In this case, we will show that there is an isomorphism of covariant transport functors $h_{!} \simeq g_{!} \circ f_{!}$ (Proposition 5.2.5.1). As a consequence, we can regard the construction $C \mapsto \operatorname{\mathcal{E}}_{C}$ as a functor from the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$ of Construction 4.5.1.1, which we denote by $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ and refer to as the homotopy transport representation of the cocartesian fibration $U$ (Construction 5.2.5.2).

The remainder of this section is devoted to the following:

Question 5.2.0.7. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\mathscr {F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ be a functor. Can $\mathscr {F}$ be obtained as the homotopy transport representation of a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$?

The answer to Question 5.2.0.7 is “no” in general. However, there are two important special cases where the answer is “yes”:

  • In §5.2.7, we show that any set-valued functor $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$ can be realized as the homotopy transport representation of a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Moreover, we can arrange that $U$ is a left covering map. In this case, the simplicial set $\operatorname{\mathcal{E}}$ is uniquely determined up to isomorphism (Corollary 5.2.7.3) and can be described explicitly using the classical category of elements construction, which we review in §5.2.6.

  • Every functor of $\infty $-categories $\operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}_1$ can be realized as the covariant transport functor associated to a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$: that is, Question 5.2.0.7 has an affirmative answer in the case $\operatorname{\mathcal{C}}= \Delta ^1$ (see Proposition 5.2.3.15). We prove this in §5.2.3 using an explicit construction which generalizes the join operation on simplicial sets (Construction 5.2.3.1). In §5.2.4, we show that the $\infty $-category $\operatorname{\mathcal{E}}$ is determined uniquely up to equivalence (see Remark 5.2.4.3).

We will eventually give a complete answer to Question 5.2.0.7: a functor between ordinary categories $\mathscr {F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ is (isomorphic to) the homotopy transport representation of a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ if and only if it can be promoted to a diagram $\widetilde{\mathscr {F}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ (Remark 5.7.5.15), where $\operatorname{\mathcal{QC}}$ denotes the $\infty $-category of small $\infty $-categories (Construction 5.6.4.1). In §5.2.8, we prove a preliminary result in this direction by showing that if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then the homotopy transport representation of any cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ can always be promoted to an enriched functor, where we regard $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ and $\operatorname{QCat}$ as enriched over the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$ (Construction 5.2.8.9).

Remark 5.2.0.8. In the preceding discussion, we have confined our attention to the case of cocartesian fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Of course, all of our results have counterparts for cartesian fibrations, which can be obtained from passing to opposite $\infty $-categories.

Structure

  • Subsection 5.2.1: Exponentiation for Cartesian Fibrations
  • Subsection 5.2.2: Covariant Transport Functors
  • Subsection 5.2.3: Example: The Relative Join
  • Subsection 5.2.4: Fibrations over the $1$-Simplex
  • Subsection 5.2.5: The Homotopy Transport Representation
  • Subsection 5.2.6: Elements of Set-Valued Functors
  • Subsection 5.2.7: Covering Space Theory
  • Subsection 5.2.8: Parametrized Covariant Transport