Kerodon

Proof of Proposition 5.6.3.1. We first prove the existence of the functor $T$ appearing in the statement of Proposition 5.6.3.1 (the uniqueness is immediate). Since the induced functor of $2$-categories $\mathrm{h} \mathit{\mathscr {F}}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ is strictly unitary, the construction $(f,u) \mapsto (f,[u])$ carries degenerate edges of the $\infty $-category $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}$ to identity morphisms in the category $\int _{\operatorname{\mathcal{C}}} \mathrm{h} \mathit{\mathscr {F}}$. It will therefore suffice to show that, for every $2$-simplex $\sigma $ of the simplicial set $\int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}$ whose boundary is indicated in the diagram

we have an identity $(h,[w]) = (g,[v]) \circ (f,[u] )$ in the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Note that the functor $\mathscr {F}$ determines a natural isomorphism $\mu : \mathscr {F}(g) \circ \mathscr {F}(f) \xrightarrow {\sim } \mathscr {F}(h)$ in the $\infty $-category $\operatorname{Fun}( \mathscr {F}(C), \mathscr {F}(E) )$. Unwinding the definitions, we see that the composition $(g,[v]) \circ (f,[u] )$ is equal to $( h, [v] \circ [ \mathscr {F}(f)(u) ]\circ [\mu (X)]^{-1} )$. We are therefore reduced to proving the commutativity of the diagram

in the homotopy category $\mathrm{h} \mathit{ \mathscr {F}(Z) }$. This commutativity is witnessed by the existence of a diagram

in the $\infty $-category $\mathscr {F}(Z)$ itself, which is supplied by the datum of the $2$-simplex $\sigma $ (see Example 5.6.2.16). This completes the construction of the functor $T$.

It follows immediately from the definitions that the functor $T$ is bijective at the level of objects and that, for every pair of objects $(C,X)$ and $(D,Y)$, the induced map

is surjective. To complete the proof, we must show that $\theta $ is also injective. Fix a pair of morphisms $(f,u): (C,X) \rightarrow (D,Y)$ and $(f',u'): (C,X) \rightarrow (D,Y)$ in the $\infty $-category $\int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}$ having the same image under $T$, so that $f = f'$ as elements of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ and the morphisms $u,u': \mathscr {F}(f)(X) \rightarrow Y$ are homotopic in the $\infty $-category $\mathscr {F}(D)$. By virtue of Corollary 1.4.3.7, there exists a morphism of simplicial sets $\theta : \operatorname{\raise {0.1ex}{\square }}^2 \rightarrow \mathscr {F}(D)$ whose restriction to the boundary $\operatorname{\partial \raise {0.1ex}{\square }}^2$ is indicated in the diagram

By virtue of Example 5.6.2.16, $\theta $ determines a $2$-simplex of the $\infty $-category $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}$ whose boundary is indicated in the diagram

which we can regard as a homotopy from $(f,u)$ to $(f',u')$. $\square$

Proof. Using Proposition 4.1.3.2, we can factor the unit map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ as a composition

where $F$ is inner anodyne and $G$ is an inner fibration of simplicial sets (so that $\operatorname{\mathcal{C}}'$ is an $\infty $-category). It follows that $\mathscr {F}$ extends uniquely to a morphism $\mathscr {F}': \operatorname{\mathcal{C}}' \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set})$. Using Remark 5.6.2.18, we obtain a pullback diagram of simplicial sets

where the vertical maps are cocartesian fibrations (Proposition 5.6.2.2). Since $F$ is inner anodyne, the map $\widetilde{F}$ is a categorical equivalence of simplicial sets (Proposition 5.3.6.1). Moreover, since $F$ is bijective at the level of vertices, $\widetilde{F}$ is also bijective at the level of vertices. It follows that $F$ and $\widetilde{F}$ induce isomorphisms of homotopy categories

Replacing $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}'$, we are reduced to proving Proposition 5.6.3.2 in the special case where $\operatorname{\mathcal{C}}$ is an $\infty $-category.

Let $\operatorname{\mathcal{D}}$ be the category of elements $\int _{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}} \mathrm{h} \mathit{\mathscr {F}}$, so that Example 5.6.2.8 supplies a pullback diagram of simplicial sets

We wish to show that $G$ exhibits $\operatorname{\mathcal{D}}$ as a homotopy category of the $\infty $-category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Note that, since $\overline{G}$ is bijective at the level of vertices, the functor $G$ has the same property. It will therefore suffice to show that, for every pair of objects $X,Y \in \int _{\operatorname{\mathcal{C}}} \mathscr {F}$, the induced map

exhibits $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(G(X), G(Y) )$ as the set of connected components of the Kan complex $\operatorname{Hom}_{\int _{\operatorname{\mathcal{C}}} \mathscr {F}}(X,Y)$. Equivalently, we wish to show that each fiber of the map $\theta _{X,Y}$ is a connected (and therefore nonempty) Kan complex. This is clear, since $\theta _{X,Y}$ is a pullback of the map

whose fibers are the connected components of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Y) )$. $\square$

Proof. The sufficiency of conditions $(1)$ and $(2)$ follows from Proposition 4.2.3.16 and Remark 4.2.3.15. To prove the converse, assume that $U$ is a left covering map. By virtue of Corollary 5.2.7.4, we may assume that $\operatorname{\mathcal{E}}= \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ for some morphism of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{Set})$. Let us abuse notation by identifying $\mathscr {F}$ with a functor from the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to the category of sets. Using Proposition 5.6.3.2, we can identify $\mathrm{h} \mathit{\operatorname{\mathcal{E}}}$ with the category of elements $\int _{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}} \mathscr {F}$ of Construction 5.2.6.1. Condition $(1)$ now follows from Remark 5.2.6.9, and condition $(2)$ by combining Example 5.6.2.8 with Remark 5.6.2.18. $\square$

Proof of Proposition 5.6.3.4. By virtue of Proposition 5.6.3.1, it will suffice to show that the simplicial set $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathscr {F} )$ is isomorphic to the nerve of a category. We will prove this by verifying the criterion of Proposition 1.3.4.1. Fix $0 < i < n$; we wish to show that every morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathscr {F} )$ can be extended uniquely to an $n$-simplex $\sigma $ of $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathscr {F} )$. Let $\overline{\sigma }_0$ denote the composition of $\sigma _0$ with the projection map $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\operatorname{D}}(\mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Proposition 1.3.4.1 then guarantees that $\overline{\sigma }_0$ extends uniquely to a morphism of simplicial sets $\overline{\sigma }: \Delta ^{n} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. It will therefore suffice to show that the lifting problem

has a unique solution.

We begin by treating the special case $n=2$ (so that $i=1$). In this case, we can identify $\sigma _0$ with a pair of composable morphisms

in the $\infty $-category $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathscr {F} )$. Set $h = g \circ f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E)$, so that the composition constraint of $\mathscr {F}$ determines an isomorphism of functors $\mu : \mathscr {F}(g) \circ \mathscr {F}(f) \xrightarrow {\sim } \mathscr {F}(h)$. Unwinding the definitions (using Example 5.6.2.16), we are reduced to proving that there is a unique morphism $w: \mathscr {F}(h)(X) \rightarrow Z$ in the category $\mathscr {F}(E)$ for which the diagram

commutes. This is clear, since $\mu (X)$ is an isomorphism in the category $\mathscr {F}(E)$.

We now treat the case $n \geq 3$. Note that the existence of a solution to the lifting problem (5.57) is automatic (since the projection map $\pi $ is a cocartesian fibration; see Proposition 5.6.2.2). It will therefore suffice to show that $\sigma $ is unique. Using Lemma 4.3.6.16 and Remark 5.5.5.7, we can rewrite (5.57) as a lifting problem

The uniqueness of its solution is now an immediate consequence of Proposition 2.3.1.9, since the horn $\Lambda ^{n+1}_{i+1}$ contains the $2$-skeleton of $\Delta ^{n+1}$. $\square$

Proof. By virtue of Corollary 4.1.5.11, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex. In this case, we wish to show that the simplicial set $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is isomorphic to the nerve of an ordinary category (see Proposition 4.1.5.10), which is a special case of Proposition 5.6.3.4. $\square$