Kerodon

Proof. It follows from Corollary 8.1.1.13 that $\lambda _{-}$ and $\lambda _{+}$ are Kan fibrations. By virtue of Proposition 3.3.7.6, it will suffice to show that the fibers of $\lambda _{-}$ and $\lambda _{+}$ are contractible Kan complexes, which is an immediate consequence of Proposition 8.1.2.1 (see Corollary 4.6.7.11). $\square$

Proof. Writing $\operatorname{\mathcal{C}}$ as the filtered colimit of its skeleta $\operatorname{sk}_{n}(\operatorname{\mathcal{C}})$ and using Proposition 6.3.6.12, we can reduce to the case where $\operatorname{\mathcal{C}}$ has dimension $\leq n$ for some integer $n \geq 0$. We proceed by induction on $n$. If $n = 0$, the morphism $\lambda _{+}$ is an isomorphism. Let us therefore assume that $n$ is positive. Let $S$ denote the collection of nondegenerate $n$-simplices of $\operatorname{\mathcal{C}}$, so that Proposition 1.1.4.12 supplies a pushout square

where the horizontal maps are monomorphisms. Combining our inductive hypothesis with Proposition 6.3.6.13, we can replace $\operatorname{\mathcal{C}}$ by $S \times \Delta ^ n$ and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is an $\infty $-category. In this case, $\lambda _{+}$ is a cocartesian fibration (Corollary 8.1.1.14) having weakly contractible fibers (Proposition 8.1.2.1 and Corollary 4.6.7.25), and is therefore universally localizing by virtue of Example 6.3.6.2. $\square$

Proof. Combine Corollary 8.1.2.4 with Remark 6.3.6.5. $\square$

Proof. We have a commutative diagram of simplicial sets

where the vertical maps are weak homotopy equivalences by virtue of Corollary 8.1.2.4. $\square$

Proof. By construction, we have a commutative diagram

where the vertical maps are left fibrations of $\infty $-categories (Propositions 4.3.6.1 and 8.1.1.11). Moreover, the $\infty $-category $\operatorname{\mathcal{C}}_{X/}$ has an initial object $\widetilde{X}$, given by the identity morphism $\operatorname{id}_{X}: X \rightarrow X$ (Proposition 4.6.7.22). Proposition 8.1.2.1 guarantees that $\iota _{X}( \widetilde{X} )$ is an initial object of the $\infty $-category $\{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$, so that $\iota _{X}$ is an equivalence of $\infty $-categories by virtue of Corollary 5.6.6.20. $\square$

Proof. Combine Proposition 8.1.2.9 with Corollary 5.1.6.4. $\square$

Proof. Since the vertical maps in the diagram (8.3) are left fibrations (Proposition 8.1.1.11), it is a categorical pullback square if and only if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map

is a homotopy equivalence of Kan complexes (Corollary 5.1.7.16). Using Notation 8.1.2.14, we see that this is equivalent to the requirement that $F$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$. $\square$

Proof. Combine Corollary 8.1.2.15 with Proposition 4.5.2.21. $\square$

Proof. Choose an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{C}}'$ is a $\kappa $-small simplicial set. It follows from Corollary 8.1.2.16 that the induced map $\operatorname{Tw}(\operatorname{\mathcal{C}}') \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}})$ is also an equivalence of $\infty $-categories. We conclude by observing that $\operatorname{Tw}(\operatorname{\mathcal{C}}')$ is also a $\kappa $-small simplicial set (Remark 8.1.1.5). $\square$

Proof. Let $H: \mathrm{h} \mathit{( \operatorname{\mathcal{C}}^{\operatorname{op}})} \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy transport representation for the left fibration $(\lambda _{-}, \lambda _{+})$, given on objects by the formula $H(X,Y) = \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} $. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, Notation 8.1.2.14 determines an isomorphism

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. We wish to show that $\alpha _{X,Y}$ depends functorially on $X$ and $Y$.

We first establish a strong form of functoriality in $Y$. Fix an object $X \in \operatorname{\mathcal{C}}$, and let $h^{X}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ denote the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor corepresented by $X$, given concretely by the formula $h^{X}(Y) = \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$. Let $H^{X}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ denote the restriction $H|_{ \{ X\} \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }$, which we also regard as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor (using Variant 5.2.8.11). Note that $h^{X}$ can be identified with the (enriched) homotopy transport representation of the left fibration $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (see Example 5.2.8.13). Corollary 4.6.4.18 and Proposition 8.1.2.9 supply equivalences

of left fibrations over $\operatorname{\mathcal{C}}$, which induce an isomorphism of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors $\alpha _{X,-}: h^{X} \xrightarrow {\sim } H^{X}$. By construction, this isomorphism carries each object $Y \in \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to the isomorphism $\alpha _{X,Y}: \underline{\operatorname{Hom}}(X,Y) \xrightarrow {\sim } H(X,Y)$ of Notation 8.1.2.14, which proves that $\alpha _{X,Y}$ depends functorially on $Y$.

We now show that $\alpha _{X,Y}$ depends functorially on $X$. Fix a morphism $f: W \rightarrow X$ in the $\infty $-category $\operatorname{\mathcal{C}}$. We then have a diagram of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors

where the vertical maps are induced by the homotopy class $[f] \in \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} }(X,W)$. To complete the proof, it will suffice to show that this diagram commutes. Using the corepresentability of the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $h^{X}$, we are reduced to showing that clockwise and counterclockwise composition around the diagram (8.4) carry $[ \operatorname{id}_{X} ] \in \pi _0( h^{X}(X) ) = \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,X)$ to the same element of $\pi _0( H^{W}(X) )$. We conclude by observing that under the identification $\pi _0( H^ W(X) ) \simeq \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(W,X)$ supplied by Corollary 8.1.2.11, both constructions carry $[ \operatorname{id}_{X} ]$ to $[f]$ (Exercise 8.1.2.13). $\square$

Proof. We will prove the first assertion; the proof of the second is similar. Construction 8.1.2.7 supplies a commutative diagram of $\infty $-categories

where the horizontal maps are equivalences of $\infty $-categories (Proposition 8.1.2.9). By virtue of Remark 7.1.5.9, it will suffice to show that $f$ is $U$-cocartesian if and only if it is a $U_{X/}$-initial object of the $\infty $-category $\operatorname{\mathcal{C}}_{X/}$, which is a special case of Example 7.1.6.9. $\square$

Proof. Apply Corollary 8.1.2.20 in the special case $\operatorname{\mathcal{D}}= \Delta ^0$ (together with Examples 7.1.5.2 and 5.1.1.4). $\square$

Proof of Proposition 8.1.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be an isomorphism in $\operatorname{\mathcal{C}}$; we wish to show that $f$ is initial when viewed as an object of the $\infty $-category $\{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$. Fix an integer $n > 0$ and a morphism $\rho _0: \operatorname{\partial \Delta }^{n} \rightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ satisfying $\rho _0( 0 ) = f$; we wish to show that $\rho _0$ can be extended to an $n$-simplex of $\{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$.

We now use a variation on the proof of Proposition 8.1.1.15. For every nonempty subset $S \subseteq [2n+1]$, let $\sigma _{S}$ denote the corresponding nondegenerate simplex of $\Delta ^{2n+1}$. Let us say that $S$ is basic if it satisfies one of the following conditions:

$(a)$

The set $S$ is contained in $\{ 0 < 1 < \cdots < n \} $.

$(b)$

There exists an integer $0 \leq i \leq n$ such that $S \cap \{ i, 2n+1-i \} = \emptyset $.

Let $K_0 \subseteq \Delta ^{2n+1}$ be the simplicial subset whose nondegenerate simplices have the form $\sigma _{S}$, where $S$ is basic. Unwinding the definitions, we can identify $\rho _0$ with a morphism of simplicial sets $\theta _0: K \rightarrow \operatorname{\mathcal{C}}$, where the composition $\Delta ^{n} \hookrightarrow K \xrightarrow {\theta _0} \operatorname{\mathcal{C}}$ is the constant map taking the value $X$ and the composition

is the morphism $f$. To complete the proof, we must show that $\theta _0$ admits an extension $\theta : \Delta ^{2n+1} \rightarrow \operatorname{\mathcal{C}}$.

Let $S$ be a nonempty subset of $[2n+1]$ which is not basic. Then there exists an integer $0 \leq i \leq n$ such that $2n+1-i$ belongs to $S$. We denote the largest such integer by $\mathrm{pr}(S)$ and refer to it as the priority of $S$. We say that $S$ is prioritized if it also contained the integer $\mathrm{pr}(S)$. Let $\{ S_1, S_2, \cdots , S_ m \} $ be an ordering of the collection of all prioritized (non-basic) subsets of $[2n+1]$ which satisfies the following conditions:

• The sequence of priorities $\mathrm{pr}(S_1), \mathrm{pr}(S_2), \cdots , \mathrm{pr}( S_ m)$ is nondecreasing. That is, if $1 \leq p \leq q \leq m$, then we have $\mathrm{pr}( S_ p ) \leq \mathrm{pr}(S_ q)$.

• If $\mathrm{pr}( S_ p ) = \mathrm{pr}( S_ q )$ for $p \leq q$, then $| S_ p | \leq | S_ q |$.

For $1 \leq q \leq m$, let $\sigma _{q} \subseteq \Delta ^{2n+1}$ denote the simplex spanned by the vertices of $S_ q$, and let $K_{q} \subseteq \Delta ^{2n+1}$ denote the union of $K_0$ with the simplices $\{ \sigma _1, \sigma _2, \cdots , \sigma _ q \} $, so that we have inclusion maps

Note that the set $S = [2n+1]$ is prioritized (with priority $n$), and is therefore equal to $S_ m$. It follows that $K_{m} = \Delta ^{2n+1}$. We will complete the proof by showing that $\theta _0$ admits a compatible sequence of extensions $\{ \theta _ q: K_ q \rightarrow \operatorname{\mathcal{C}}\} _{0 \leq q \leq m}$, so that $\theta = \theta _ m$ is an extension of $\theta _0$ to $\Delta ^{2n+1}$.

For the remainder of the proof, we fix an integer $1 \leq q \leq m$, and suppose that the morphism $\theta _{q-1}: K_{q-1} \rightarrow \operatorname{\mathcal{C}}$ has already been constructed. Let $d$ denote the dimension of the simplex $\sigma _{q}$, let us abuse notation by identifying $\sigma _{q}$ with a morphism of simplicial sets $\Delta ^{d} \rightarrow K_{q} \subseteq \Delta ^{2n+1}$, and set $L = \sigma _{q}^{-1} K_{q-1} \subseteq \Delta ^{d}$. Let $i = \mathrm{pr}(S_ q)$ denote the priority of $S_{q}$, so that $S_{q}$ contains both $i$ and $2n+1-i$. Write $S_{q} = \{ j_0 < j_1 < \cdots < j_ d \} $, so that $i = j_{k}$ for some integer $0 \leq k \leq d$. We will prove below that $L$ is equal to the horn $\Lambda ^{d}_{k} \subseteq \Delta ^{d}$, so that the diagram of simplicial sets

is a pushout square (Lemma 3.1.2.11). Let $\tau _0$ denote the composite map $L \xrightarrow { \sigma _ q } K_{q-1} \xrightarrow { \theta _{q-1} } \operatorname{\mathcal{C}}$. We will complete the proof by showing that $\tau _0$ admits an extension $\tau : \Delta ^{d} \rightarrow \operatorname{\mathcal{C}}$ (which then determines a morphism $\theta _{q}: K_{q} \rightarrow \operatorname{\mathcal{C}}$ extending $\theta _{q-1}$). The proof splits into four cases:

• Suppose that $0 < k < d$. Then $\Lambda ^{d}_{k} \subseteq \Delta ^{d}$ is an inner horn, so that $\tau _0$ admits an extension $\tau : \Delta ^{d} \rightarrow \operatorname{\mathcal{C}}$ by virtue of our assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category.

• Suppose that $k = d$. Then $S_{q}$ is contained in $\{ 0, 1, \cdots , n \} $, contradicting our assumption that $S_{q}$ is not basic.

• Suppose $k = 0$ and $i < n$, so that $i$ is the least element of $S_ q$. Our assumption $\mathrm{pr}(S_ q) = i$ guarantees that $2n-i \notin S$. Since $S_{q}$ does not satisfy $(b)$, we must also have $i+1 \in S_ q$. It follows that $d \geq 2$ (otherwise, $S_ q$ would satisfy $(a)$), and that $\tau _0: \Lambda ^{d}_{0} \rightarrow \operatorname{\mathcal{C}}$ carries the initial edge $\operatorname{N}_{\bullet }( \{ 0 < 1 \} )$ to the identity morphism $\operatorname{id}_{X}$. In this case, the existence of the extension $\tau $ follows from Theorem 4.4.2.6.

• Suppose $k = 0$ and $i = n$, so that $i = n$ is the least element of $S_ q$. Since $S_{q}$ has priority $n$, the element $n+1$ also belongs to $S_{q}$. We must then have $d \geq 2$ (otherwise, $S_ q$ would satisfy condition $(b)$). It follows that $\tau _0: \Lambda ^{d}_{0} \rightarrow \operatorname{\mathcal{C}}$ carries the initial edge $\operatorname{N}_{\bullet }( \{ 0 < 1 \} )$ to the morphism $f$, which is an isomorphism in $\operatorname{\mathcal{C}}$. In this case, the existence of the extension $\tau $ again follows from Theorem 4.4.2.6.

It remains to prove that $L = \Lambda ^{d}_{k}$, which we can formulate more concretely as follows:

$(\ast )$

Let $j$ be an element of $S_{q}$, and set $S' = S_{q} \setminus \{ j\} $. Then $\sigma _{S'}$ is contained in $K_{q-1}$ if and only if $j \neq i$.

We first treat the case $j = i$; in this case, we wish to show that $\sigma _{S'}$ is not contained in $K_{q-1}$. Note that $S'$ cannot be basic: it cannot be contained in $\{ 0, 1, \cdots , n \} $ (otherwise $S_{q} =S' \cup \{ i\} $ would have the same property) and cannot have empty intersection with a set of the form $\{ i', 2n+1- i'\} $ (otherwise $S_ q$ would have the same property; here we use the fact that $2n+1-i$ is contained in $S_ q$). Moreover, we have $\mathrm{pr}(S') = i \notin S'$, so that $S'$ is not prioritized. Assume, for a contradiction, that $\sigma _{S'}$ is contained in $K_{q-1}$. Then we must have $S' \subseteq S_{q'}$, for some $1 \leq q' < q$. Note that $2n+1-i \in S' \subseteq S_{q'}$, so that $S_{q'}$ has priority $\geq i$. Since $q' < q$, it follows that $S_{q'}$ has priority $i$ and that $|S_{q'}| \leq | S_{q} |$. Since $S_{q'}$ is prioritized, it contains the element $i$, and therefore contains $S_{q} = S' \cup \{ i\} $. It follows that $S_{q'} = S_{q}$, contradicting our assumption that $q' < q$.

We now treat the case $j \neq i$; in this case, we wish to show that $\sigma _{S'}$ is contained in $K_{q-1}$. We may assume without loss of generality that $S'$ is not basic (otherwise, the simplex $\sigma _{S'}$ is already contained in $K_0$). Let $i'= \mathrm{pr}( S' )$ denote the priority of $S'$; note that the inclusion $S' \subseteq S_{q}$ guarantees that $i' \leq i$. If $i' < i$, then $S' \cup \{ i' \} $ is a prioritized set of priority $< i$, and therefore of the form $S_{q'}$ for some $q' < q$. If $i' = i$, then $S'$ is a prioritized set of priority $i$ and cardinality $| S_{q} | - 1$, and therefore of the form $S_{q'}$ for some $q' < q$. In either case, we obtain $\sigma _{S'} \subseteq \sigma _{q'} \subseteq K_{q-1}$. $\square$