$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Theorem Let $\operatorname{\mathcal{C}}$ be a simplicial category having the property that, for every pair of objects $Y,Z \in \operatorname{\mathcal{C}}$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }$ is an $\infty $-category. Let $X$ be an object of $\operatorname{\mathcal{C}}$, let $\operatorname{hTr}$ denote the (enriched) homotopy transport representation associated to the cocartesian fibration $\operatorname{Pith}(U): \operatorname{Pith}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/} ) \rightarrow \operatorname{Pith}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) )$, and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the locally Kan simplicial subcategory defined above. Then the diagram of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors

\[ \xymatrix@R =70pt@C=70pt{ & \mathrm{h} \mathit{\operatorname{\mathcal{C}}'} \ar [dl]_{\Phi }^{\sim } \ar [dr]^{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, \bullet )_{\bullet }} & \\ \mathrm{h} \mathit{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})} \ar [rr]^-{ \operatorname{hTr}} & & \mathrm{h} \mathit{\operatorname{QCat}} } \]

commutes up to natural isomorphism, given explicitly by the map

\[ Y \mapsto ( [ \theta _{X,Y} ]: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X,Y) ). \]

of Construction

Proof. For every object $Y \in \operatorname{\mathcal{C}}$, the comparison functor

\[ \theta _{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X,Y) \]

is an equivalence of $\infty $-categories (Theorem, so its homotopy class $[ \theta _{X,Y} ]$ is an isomorphism when regarded as a morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$. To complete the proof, it will suffice to show that the construction $Y \mapsto [ \theta _{X,Y} ]$ determines a natural transformation of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors. Let $Y$ and $Z$ be objects of $\operatorname{\mathcal{C}}$, so that the map $\theta _{Y,Z}$ restricts to a homotopy equivalence of Kan complexes $\theta _{Y,Z}^{\simeq }: \operatorname{Hom}_{\operatorname{\mathcal{C}}'}(Y,Z)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}')}(Y,Z)$. We wish to show that the diagram of Kan complexes

\begin{equation} \begin{gathered}\label{equation:enriched-homotopy-transport-universal} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{\mathcal{C}}'}(Y,Z)_{\bullet } \ar [r] \ar [d]^-{\theta _{Y,Z}^{\simeq } } & \operatorname{Fun}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet })^{\simeq } \ar [dd]^{ \theta _{X,Z} \circ } \\ \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}')}(Y,Z) \ar [d]^-{\rho } & \\ \operatorname{Fun}( \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}} }^{\mathrm{L}}(X,Y), \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}} }^{\mathrm{L}}(X,Z) )^{\simeq } \ar [r]^-{ \circ \theta _{X,Y} } & \operatorname{Fun}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }, \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}} }^{\mathrm{L}}(X,Z) )^{\simeq } } \end{gathered} \end{equation}

commutes up to homotopy, where $\rho $ is given by parametrized covariant transport for the cocartesian fibration $\operatorname{Pith}(U): \operatorname{Pith}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/} ) \rightarrow \operatorname{Pith}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) ) \simeq \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}')$.

We will show that there exists a functor of $\infty $-categories

\[ H: \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/} \]

satisfying the following requirements:


The diagram of simplicial sets

\[ \xymatrix@R =50pt@C=25pt{ \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \ar [rr]^-{H} \ar [d] & & \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/} \ar [d]^{U} \\ \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }^{\simeq } \ar [r]^-{\theta _{Y,Z}} & \Delta ^1 \times \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(Y,Z) \ar [r] & \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) } \]



The restriction $H_0 = H|_{ \{ 0\} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }}$ is given by the composition

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \xrightarrow { \theta _{X,Y} } \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) }^{\mathrm{L} }(X,Y). \]

The restriction $H_1 = H|_{ \{ 1\} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }}$ is given by the composition

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \xrightarrow {\circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \xrightarrow { \theta _{X,Z} } \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) }^{\mathrm{L} }(X,Z). \]

For every pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$, the composite map

\[ \Delta ^1 \times \{ f\} \times \{ g\} \hookrightarrow \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \xrightarrow {H} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/} \]

is a $U$-cocartesian morphism of the $(\infty ,2)$-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/}$ (that is, it corresponds to a thin $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$; see Theorem

Assume for the moment that there exists a morphism $H$ satisfying these requirements. Note that the restriction $H|_{ \{ 1\} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( Y,Z)_{\bullet } }$ can be identified with a map of Kan complexes

\[ \lambda : \operatorname{Hom}_{\operatorname{\mathcal{C}}'}(Y,Z)_{\bullet } \rightarrow \operatorname{Fun}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }, \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}} }^{\mathrm{L}}(X,Z) )^{\simeq }. \]

It follows from requirement $(c)$ that $\lambda $ is given by clockwise composition around the diagram (5.44), and from requirements $(a)$, $(b)$, and $(d)$ that $\lambda $ is also given (up to homotopy) by counterclockwise composition around the diagram (5.44). It follows that the diagram (5.44) commutes up to homotopy, as desired.

It remains to construct the morphism $H$. Fix an auxiliary symbol $e$, let $n \geq 0$, and let $\sigma $ be an $n$-simplex of the simplicial set $\Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }$. We will identify $\sigma $ with a triple $(\alpha , f_{\sigma }, g_{\sigma } )$, where $\alpha : [n] \rightarrow [1]$ is a nondecreasing function, $f_{\sigma }$ is an $n$-simplex of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$, and $Fg_{\sigma }$ is an $n$-simplex of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }$. Let $\operatorname{Path}[ \{ e\} \star [n] ]_{\bullet }$ denote the simplicial path category of the linearly ordered set $\{ e\} \star [n] = \{ e < 0 < \cdots < n \} $ (see Notation To the $n$-simplex $\sigma $, we associate a simplicial functor $h_{\sigma }: \operatorname{Path}[ \{ e\} \star [n] ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ as follows:

  • On objects, the functor $h_{\sigma }$ is given by the formula

    \[ h_{\sigma }(i) = \begin{cases} X & \textnormal{ if $i=e$ } \\ Y & \textnormal{ if $0 \leq i \leq n$ and $\alpha (i) = 0$ } \\ Z & \textnormal{ if $0 \leq i \leq n$ and $\alpha (i) = 1$.} \end{cases} \]
  • Let $i < j$ be elements of the linearly ordered set $\{ e\} \star [n]$, so that $\operatorname{Hom}_{ \operatorname{Path}[ \{ e\} \star [n] ] }( i, j)_{\bullet }$ can be identified with the nerve $\operatorname{N}_{\bullet }(Q)$, where $Q$ is the collection of all subsets $K \subseteq \{ e \} \star [n]$ having smallest element $i$ and largest element $j$ (and we regard $Q$ as ordered by reverse inclusion). The simplicial functor $h_{\sigma }$ is given on morphisms by a map of simplicial sets $u_{i,j}: \operatorname{N}_{\bullet }(Q) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( h_{\sigma }(i), h_{\sigma }(j) )$. If $0 \leq i < j \leq n$ with $\alpha (i) = \alpha (j)$, we take $u_{i,j}$ to be the constant map taking the value $\operatorname{id}_{Y}$ (if $\alpha (i) = 0$) or $\operatorname{id}_{Z}$ (if $\alpha (i) = 1$). The remaining cases can be described as follows:

    • If $0 \leq i < j \leq n$ satisfy $\alpha (i) = 0$ and $\alpha (j) = 1$, then $u_{i,j}$ is given by the composition

      \[ \operatorname{N}_{\bullet }(Q) \xrightarrow {r_{+}} \Delta ^{n} \xrightarrow { g_{\sigma } } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }, \]

      where $r_{+}$ is given on vertices by the formula $r_{+}(K) = \min \{ k \in K: \alpha (k) = 1 \} $.

    • If $i = e$ and $\alpha (j) = 0$, then $u_{i,j}$ is given by the composition

      \[ \operatorname{N}_{\bullet }(Q) \xrightarrow { r_{-} } \Delta ^{n} \xrightarrow { f_{\sigma } } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }, \]

      where $r_{-}$ is given on vertices by the formula $r_{-}(K) = \min \{ k \in K: k > e \} $.

    • If $i = e$ and $\alpha (j) = 1$, then $u_{i,j}$ is given by the composition

      \[ \operatorname{N}_{\bullet }(Q) \xrightarrow { (r_{+}, r_{-} )} \Delta ^{n} \times \Delta ^{n} \xrightarrow { g_{\sigma } \times f_{\sigma } } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \xrightarrow {\circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }, \]

      where $r_{-}$ and $r_{+}$ are defined as above.

Note that we can identify $h_{\sigma }$ with a morphism of simplicial sets $\{ e\} \star \Delta ^ n \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ carrying $\{ e\} $ to the vertex $X$, which we can view as an $n$-simplex $H(\sigma )$ of the $(\infty ,2)$-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/}$. The construction $\sigma \mapsto H(\sigma )$ determines a morphism of simplicial sets

\[ H: \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/}. \]

Requirements $(a)$, $(b)$, and $(c)$ follow immediately from $(a')$, $(b')$, and $(c')$ (together with the definitions of the maps $\theta _{Y,Z}$, $\theta _{X,Y}$, and $\theta _{X,Z}$, respectively). Requirement $(d)$ follows from the description of the thin $2$-simplices of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ supplied by Proposition $\square$