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8.1.2 Homotopy Transport for Twisted Arrows

Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\operatorname{Tw}(\operatorname{\mathcal{C}})$ denote its twisted arrow $\infty$-ategory. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, Proposition 8.1.1.10 guarantees that the fiber product

$\{ X \} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\}$

is a Kan complex, whose vertices can be identified with morphisms $f: X \rightarrow Y$. Our goal in this section is to show that this identification can be promoted to a homotopy equivalence of Kan complexes

$\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}( X,Y ) \hookrightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} ,$

where $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) = \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Y\}$ denotes the left-pinched morphism space of Construction 4.6.5.1 (see Corollary 8.1.2.6). Our starting point is the following result:

Proposition 8.1.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $f: X \rightarrow Y$ be an isomorphism in $\operatorname{\mathcal{C}}$. Then $f$ is initial when viewed as an object of the $\infty$-category $\{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$.

We will prove Proposition 8.1.2.1 at the end of this section. First, let us record some consequences.

Construction 8.1.2.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $X$ be a vertex of $\operatorname{\mathcal{C}}$. Let $\sigma$ be an $n$-simplex of the coslice simplicial set $\operatorname{\mathcal{C}}_{X/}$, which we identify with a morphism of simplicial sets $\{ x\} \star \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma (x) = X$. Then the composite map

$( \Delta ^{n} )^{\operatorname{op}} \star \Delta ^{n} \twoheadrightarrow \{ x\} \star \Delta ^{n} \xrightarrow {\sigma } \operatorname{\mathcal{C}}$

can be identified with an $n$-simplex of the twisted arrow simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$, which we will denote by $\iota _{X}( \sigma )$. The construction $\sigma \mapsto \iota _{X}( \sigma )$ is compatible with the formation of face and degeneracy maps, and therefore determines a morphism of simplicial sets $\iota _{X}: \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}})$. Moreover, the diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{X/} \ar [r]^-{ \iota _{X} } \ar [d] & \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [d]^{ \lambda _{-} } \\ \{ X\} \ar [r] & \operatorname{\mathcal{C}}^{\operatorname{op}} }$

commutes, where $\lambda _{-}$ is the projection map of Notation 8.1.1.5. It follows that $\iota _{X}$ can be regarded as a morphism of simplicial sets from $\operatorname{\mathcal{C}}_{X/}$ to the fiber $\{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$. We will refer to this morphism as the coslice inclusion.

Remark 8.1.2.4. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $X \in \operatorname{\mathcal{C}}$ be a vertex. Then an $n$-simplex $\sigma$ of $\operatorname{\mathcal{C}}_{X/}$ can be identified with an $(n+1)$-simplex of $\operatorname{\mathcal{C}}$, which we represent informally as a diagram

$X \xrightarrow {f} Y_0 \xrightarrow {v_1} Y_1 \xrightarrow {v_2} Y_2 \rightarrow \cdots \xrightarrow {v_ n} Y_ n.$

The morphism $\iota _{X}$ of Construction 8.1.2.3 carries $\sigma$ to a $(2n+1)$-simplex $\tau$ of $\operatorname{\mathcal{C}}$, which can be represented informally by the diagram

$\xymatrix@R =50pt@C=50pt{ X \ar [d]^-{f} & X \ar [l]_-{\operatorname{id}} & X \ar [l]_-{\operatorname{id}} & \cdots \ar [l]_-{\operatorname{id}} & X \ar [l]_-{\operatorname{id}} \\ Y_0 \ar [r]^-{v_1} & Y_1 \ar [r]^-{v_2} & Y_2 \ar [r] & \cdots \ar [r]^-{v_ n} & Y_ n. }$

Note that $\sigma$ can be recover from $\tau$ (by composing with the inclusion map $\Delta ^{n+1} \hookrightarrow \Delta ^{2n+1}$, given on vertices by $i \mapsto i+n$). It follows that $\iota _{X}$ is a monomorphism of simplicial sets $\operatorname{\mathcal{C}}_{X/} \hookrightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ (as suggested by our terminology).

Proposition 8.1.2.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. For every object $X \in \operatorname{\mathcal{C}}$, the coslice inclusion

$\iota _{X}: \operatorname{\mathcal{C}}_{X/} \hookrightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$

is an equivalence of $\infty$-categories.

Proof. By construction, we have a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{X/} \ar [rr]^-{ \iota _{X} } \ar [dr] & & \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [dl]^-{ \lambda _{+} } \\ & \operatorname{\mathcal{C}}, & }$

where the vertical maps are left fibrations of $\infty$-categories (Propositions 4.3.6.1 and 8.1.1.10). 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 4.6.6.23). 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 5.6.6.21. $\square$

Corollary 8.1.2.6. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the coslice inclusion $\iota _{X}$ restricts to a homotopy equivalence of Kan complexes

$\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y) \hookrightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} .$

Proof. Combine Proposition 8.1.2.5 with Corollary 5.1.5.4. $\square$

Corollary 8.1.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $f,f': X \rightarrow Y$ be morphisms of $\operatorname{\mathcal{C}}$. Then $f$ and $f'$ are homotopic (in the sense of Definition 1.3.3.1) if and only they belong to the same connected component of the Kan complex $\{ X\} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\}$. Consequently, we have a canonical isomorphism of sets

$\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) \simeq \pi _0( \{ X\} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} ).$

Exercise 8.1.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing morphisms $u: X' \rightarrow X$ and $v: Y \rightarrow Y'$, so that covariant transport for the left fibration $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ determines a morphism of Kan complexes

$T: \{ X\} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \{ X'\} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y'\} .$

Show that, under the identifications supplied by Corollary 8.1.2.7, the induced map of connected components $\pi _0(T): \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X,Y) \rightarrow \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X',Y')$ is given by the construction $[f] \mapsto [v] \circ [f] \circ [u]$.

We now apply Proposition 8.1.2.5 to describe the left fibration $(\lambda _{-}, \lambda _{+}): \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ of Proposition 8.1.1.13.

Notation 8.1.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, Proposition 4.6.5.9 and Corollary 8.1.2.6 supply homotopy equivalences of Kan complexes

$\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \hookleftarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) \hookrightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} .$

Passing to homotopy, we obtain an isomorphism

$\alpha _{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow {\sim } \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\}$

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

Corollary 8.1.2.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. Then $F$ is fully faithful if and only if the diagram

8.3
$$\begin{gathered}\label{equation:twisted-arrow-fully-faithful} \xymatrix { \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^{ \operatorname{Tw}(F) } \ar [d] & \operatorname{Tw}(\operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\ar [r]^{ F^{\operatorname{op}} \times F} & \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}} \end{gathered}$$

is a categorical pullback square.

Proof. Since the vertical maps in the diagram (8.3) are left fibrations (Proposition 8.1.1.10), it is a categorical pullback square if and only if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map

$\{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \{ F(X) \} \times _{ \operatorname{\mathcal{D}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{D}}) \times _{\operatorname{\mathcal{D}}} \{ F(Y) \}$

is a homotopy equivalence of Kan complexes (Corollary 5.1.6.15). Using Notation 8.1.2.10, we see that this is equivalent to the requirement that $F$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$. $\square$

Corollary 8.1.2.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty$-categories. Then the induced map $\operatorname{Tw}(F): \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Tw}(\operatorname{\mathcal{D}})$ is also an equivalence of $\infty$-categories.

Corollary 8.1.2.13. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ be its homotopy category, and let

$\underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad (X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$

denote the functor determined by the $\mathrm{h} \mathit{\operatorname{Kan}}$-enrichment of Construction 4.6.8.13. Then the assignment $(X,Y) \mapsto \alpha _{X,Y}$ of Notation 8.1.2.10 determines an isomorphism from $\underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}$ to the homotopy transport representation of the left fibration $(\lambda _{-}, \lambda _{+}): \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$.

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 8.1.2.10 determines an isomorphism

$\alpha _{X,Y}: \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) \xrightarrow {\sim } H(X,Y)$

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 4.6.4.17 and Proposition 8.1.2.5 supply equivalences

$\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\hookleftarrow \operatorname{\mathcal{C}}_{X/} \hookrightarrow \{ X\} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$

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 8.1.2.10, 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

8.4
$$\begin{gathered}\label{equation:homotopy-transport-twisted-arrow} \xymatrix@R =50pt@C=50pt{ h^{X} \ar [d] \ar [r]^-{\alpha _{X,-}} & H^{X} \ar [d] \\ h^{W} \ar [r]^-{ \alpha _{X,-} } & H^{W}, } \end{gathered}$$

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 8.1.2.7, both constructions carry $[ \operatorname{id}_{X} ]$ to $[f]$ (Exercise 8.1.2.9). $\square$

Warning 8.1.2.14. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Our proof of Corollary 8.1.2.13 shows that the isomorphism $\alpha _{X,Y}: \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) \rightarrow H(X,Y)$ is compatible with the $\mathrm{h} \mathit{\operatorname{Kan}}$-enrichment in the second variable. Beware that things are a bit more subtle if we wish to view $\underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$ and $H(X,Y)$ as $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors of the first variable. The functor $\underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}$ is defined using the enrichment of the category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, and can therefore be viewed an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

$(\mathrm{h} \mathit{\operatorname{\mathcal{C}}})^{\operatorname{op}} \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}.$

On the other hand, the functor $H$ is defined as the enriched homotopy transport representation of the left fibration $(\lambda _{-}, \lambda _{+}): \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$, which is an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

$\mathrm{h} \mathit{( \operatorname{\mathcal{C}}^{\operatorname{op}})} \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}.$

The $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched categories $( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )^{\operatorname{op}}$ and $\mathrm{h} \mathit{ ( \operatorname{\mathcal{C}}^{\operatorname{op}} )}$ are a priori different objects: to a pair of objects $X,Y \in \operatorname{\mathcal{C}}$, they assign morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)^{\operatorname{op}}$, respectively. It is possible to address this point (since $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)^{\operatorname{op}}$ are canonically isomorphic as objects of the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$), but we will not pursue the matter here.

We can use Proposition 8.1.2.5 to deduce a stronger form of Proposition 8.1.2.1.

Corollary 8.1.2.15. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty$-categories and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, which we regard as an object of the twisted arrow $\infty$-category $\operatorname{Tw}(\operatorname{\mathcal{C}})$. Then:

• The morphism $f$ is $U$-cocartesian if and only if it is $V$-initial, where $V$ denotes the induced map

$\{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \{ U(X) \} \times _{ \operatorname{\mathcal{D}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{D}}).$
• The morphism $f$ is $U$-cartesian if and only if it is $V'$-initial, where $V'$ denotes the induced map

$\operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \operatorname{Tw}(\operatorname{\mathcal{D}}) \times _{\operatorname{\mathcal{D}}} \{ U(Y) \} .$

Proof. We will prove the first assertion; the proof of the second is similar. Construction 8.1.2.3 supplies a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{X/} \ar [d]^{ U_{X/} } \ar [r]^-{ \iota _{X} } & \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [d]^{ V } \\ \operatorname{\mathcal{D}}_{U(X)/} \ar [r]^-{ \iota _{ U(X)} } & \{ U(X) \} \times _{ \operatorname{\mathcal{D}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{D}}), }$

where the horizontal maps are equivalences of $\infty$-categories (Proposition 8.1.2.5). By virtue of Remark 7.1.4.9, 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 7.1.5.9. $\square$

Corollary 8.1.2.16. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The morphism $f$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{C}}$.

$(2)$

The morphism $f$ is initial when regarded as an object of the $\infty$-category $\{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$.

$(3)$

The morphism $f$ is initial when regarded as an object of the $\infty$-category $\operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\}$.

Proof. Apply Corollary 8.1.2.15 in the special case $\operatorname{\mathcal{D}}= \Delta ^0$ (together with Examples 7.1.4.2 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 8.1.1.13. 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

$\Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ n < n+1 \} ) \hookrightarrow K \xrightarrow {\theta _0} \operatorname{\mathcal{C}}$

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

$K_0 \hookrightarrow K_1 \hookrightarrow K_{2} \hookrightarrow \cdots \hookrightarrow K_{m}.$

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 $L = \sigma _{q}^{-1} K_{q-1} \subseteq \Delta ^{d}$. We then have a pushout diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ L \ar [r] \ar [d] & K_{q-1} \ar [d] \\ \Delta ^{d} \ar [r]^-{ \sigma _ q} & K_{q}. }$

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 $\theta _{q-1}$).

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 $\Lambda ^{d}_{k} \subseteq \Delta ^{d}$. Assuming this, we can split the proof 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 $\sigma _{S'} \subseteq \sigma _{q'} \subseteq K_{q-1}$. $\square$