Kerodon

Proof. Let $\operatorname{Spine}[n]$ denote the spine of the standard $n$-simplex $\Delta ^ n$ (see Example 1.5.7.7). Unwinding the definitions, we see that $\theta $ is a pullback of the restriction map

Since $q$ is an inner fibration and the inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne (Example 1.5.7.7), the morphism $\theta '$ is a trivial Kan fibration (Proposition 4.1.4.4). $\square$

Proof. Combine Corollary 4.6.9.5 with Proposition 4.6.1.10. $\square$

Proof. There is a diagram of Kan complexes

where the left diagonal arrow is induced by the map $\sigma ^0_{1}: [2] \rightarrow [1]$ of Construction 1.1.2.1 and the right diagonal arrow is induced by the map $\delta ^{1}_{2}: [1] \rightarrow [2]$ of Construction 1.1.1.4. Here the solid arrows are well-defined as morphisms of simplicial sets, while the dotted arrow is well-defined only as a morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. We now observe that the triangle on the left is strictly commutative, the triangle on the right commutes up to homotopy (by the construction of the composition law $\circ $). Assertion $(1)$ follows from the observation that the composition of the diagonal arrows is the identity on the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ (since $\sigma ^{0}_{1} \circ \delta ^{1}_{2}$ is the identity on the object $[1] \in \operatorname{{\bf \Delta }}$). Assertion $(2)$ follows by a similar argument. $\square$

Proof. By virtue of Corollary 4.6.9.5, (4.70) is isomorphic to the diagram of restriction maps

which commutes in the category of simplicial sets (and therefore also in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$). $\square$

Proof of Proposition 4.6.9.16. It will suffice to show that there exists a morphism of Kan complexes

for which the diagram

commutes (in the category of simplicial sets).

We first observe that there is a unique morphism of simplicial sets $e: \Delta ^2 \times \operatorname{\mathcal{C}}_{f/} \rightarrow \Delta ^1 \star \operatorname{\mathcal{C}}_{f/}$ with the property that $e|_{ \Delta ^1 \times \operatorname{\mathcal{C}}_{f/} }$ is given by projection onto the first factor, and $e|_{ \{ 2\} \times \operatorname{\mathcal{C}}_{f/} }$ is given by projection onto the second factor. Note that the composite map

can be identified with a morphism of simplicial sets $e': \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$. Unwinding the definition, we see that $e'$ restricts to a morphism of simplicial subsets

having the desired properties. $\square$

Proof. Using Proposition 4.6.9.16, we can replace the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \xrightarrow { \circ [f] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ with the restriction map $\theta : \operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \rightarrow \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Z\} $. The morphism $\theta $ is a left fibration (Corollary 4.3.6.12). Since the left-pinched morphism space $\operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Z\} = \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Z)$ is a Kan complex (Proposition 4.6.5.5), it follows that $\theta $ is a Kan fibration (Corollary 4.4.3.8). In particular, the homotopy fiber of the composition map $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}(Y, Z ) \xrightarrow { \circ [f] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ over the vertex $g$ can be identified with the fiber

which is homotopy equivalent to $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{X/}}(f,g)$ by virtue of Proposition 4.6.5.10. $\square$

Proof. We will show that there exists a morphism of Kan complexes

for which the diagram

is commutative.

Fix an integer $n \geq 0$. Let $\operatorname{\mathcal{E}}$ denote the simplicial category with object set $\operatorname{Ob}(\operatorname{\mathcal{E}}) = \{ x,y,z \} $ and morphism spaces given by

where the composition law $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(y,z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,z)_{\bullet }$ is an isomorphism (so that $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,z)_{\bullet }$ can be identified with the product $\Delta ^ n \times \Delta ^ n$). Note that there is a unique simplicial functor $F: \operatorname{Path}[ \Delta ^2 \times \Delta ^ n ]_{\bullet } \rightarrow \operatorname{\mathcal{E}}$ satisfying the following conditions:

• On objects, the functor $F$ is given by the formula

  \[ F(i,j) = \begin{cases} x & \text{ if $i=0$ } \\ y & \text{ if $i=1$} \\ z & \text{ if $i=2$.} \end{cases} \]

• Let $(i,j)$ and $(i',j')$ be vertices of $\Delta ^2 \times \Delta ^ n$ satisfying $i < i'$ and $j \leq j'$, so that there is a unique indecomposable morphism $u$ from $(i,j)$ to $(i', j')$ in the path category $\operatorname{Path}[ \Delta ^2 \times \Delta ^ n ]$ (given by the chain $\{ (i,j) < (i',j') \} $). If $i=0$ and $i' = 1$, then $F(u)$ is the vertex $j'$ of $\Delta ^ n = \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,y)_{\bullet }$. If $i = 1$ and $i' =2$, then $F(u)$ is the vertex $j'$ of $\Delta ^ n = \operatorname{Hom}_{\operatorname{\mathcal{E}}}(y,z)_{\bullet }$. If $i= 0$ and $i' = 2$, then $F(u)$ is the vertex $(j',j')$ of $\Delta ^ n \times \Delta ^ n = \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,z)_{\bullet }$.

Let $\sigma $ and $\tau $ be $n$-simplices of the Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y,Z)_{\bullet }$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$, respectively. Then there is a unique simplicial functor $G_{\sigma ,\tau }: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ satisfying the following conditions:

• On objects, the functor $G_{\sigma ,\tau }$ is given by $G_{\sigma ,\tau }(x) = X$, $G_{\sigma ,\tau }(y) = Y$, and $G_{\sigma ,\tau }(z) = Z$.

• The induced map $\Delta ^ n = \operatorname{Hom}_{\operatorname{\mathcal{E}}}( x,y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is the $n$-simplex $\tau $.

• The induced map $\Delta ^ n = \operatorname{Hom}_{\operatorname{\mathcal{E}}}( y,z)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }$ is the $n$-simplex $\sigma $.

The composite simplicial functor

determines a morphism from $\Delta ^2 \times \Delta ^ n$ to the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, which can be identified with an $n$-simplex $\theta _{X,Y,Z}(\sigma ,\tau )$ of the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z)_{\bullet }$. Allowing $n$ to vary, the construction $(\sigma ,\tau ) \mapsto \theta _{X,Y,Z}(\sigma ,\tau )$ determines a morphism of simplicial sets $\theta _{X,Y,Z}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}(X,Y,Z)$ having the desired properties. $\square$

Proof of Proposition 4.6.9.21. Let $\operatorname{\mathcal{C}}^{\Delta }$ denote the underlying simplicial category of the differential graded category $\operatorname{\mathcal{C}}$ (Construction 2.5.9.2). By virtue of Exercise 4.6.8.4, we can identify (4.74) with the outer rectangle of a larger diagram

where the middle horizontal map is given by the composition law of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}^{\Delta } )$. We now observe that the upper square commutes up to homotopy by virtue of Proposition 4.6.9.19, and the lower square commutes up to homotopy by the functoriality of Construction 4.6.9.9. $\square$