nLab twisted arrow (∞,1)-category

Idea

This is the generalization of twisted arrow category to (∞,1)-categories.

Definition

In the case of quasicategories, we can give an explicit definition. For a quasi-category $C$ we define the twisted arrow quasi-category $Tw(C)$ to be

$$Tw(C)_n = sSet((\Delta^n)^{op} \star (\Delta^n), C)$$

The insertions into the components of the join determine natural transformations $src : Tw(C) \to C^{op}$ and $tgt : Tw(C) \to C$.

By Kerodon, tag 03K9, this definition respects equivalences between quasi-categories, and thus descends to a functor $Tw : (\infty,1)Cat \to (\infty,1)Cat$.

Since $(\Delta^n)^{op} \star (\Delta^n)$ is preserved by taking opposites, this construction satisfies $Tw(C) = Tw(C^{op})$, and swaps the source and target maps.

The opposite convention often appears in the literature,

$$\overline{Tw}(C)_n = sSet((\Delta^n) \star (\Delta^n)^{op}, C)$$

and comes with natural transformations $\overline{src} : \overline{Tw}(C) \to C$ and $\overline{tgt} : \overline{Tw}(C) \to C^{op}$.

This satisfies $\overline{Tw}(C) \simeq Tw(C)^{op}$.

Properties

If $C$ is a 1-category, then $Tw(C)$ agrees with the 1-categorical version. In the quasi-category model we have an even stronger statement

Proposition

If $C$ is a 1-category, there is an isomorphism $T : N(Tw(C)) \cong Tw(N(C))$ of simplicial sets, uniquely determined by the properties that $T(f) = f$ and that the source and target maps are consistent with the obvious map $N(C^{op} \times C) \cong N(C)^{op} \times N(C)$.

Proof

This is Kerodon, tag 03JN.

For a quasi-category $C$, both $Tw(C)$ and $\overline{Tw}(C)$ are quasi-categories. This follows from the following characterization:

Proposition

• $Tw(C) \to C^{op} \times C$ is the left fibration classfied by the hom-space functor $C^{op} \times C \to \infty Gpd$.

• $\overline{Tw}(C) \to C \times C^{op}$ is the right fibration classified the hom-space functor $C^{op} \times C \to \infty Gpd$.

Proof

This description of $\overline{Tw}(C)$ is Lurie 4.2.5. The left fibration classified by a map is the opposite of the right fibration classified by the same map. Alternatively, this is Kerodon, tag 03JQ.

This means that we have the formulas $Tw(C) \simeq el(\hom_C) \simeq ({*} \downarrow \hom)$, just as in the 1-category version.

We can say more:

Proposition

If $C \to D$ is an inner fibration of simplicial sets, then the universal map

$$Tw(C) \to (C^{op} \times C) \times_{D^{\op} \times D} Tw(D)$$

is a left fibration.

Proof

This is Kerodon, tag 03JT.

Corollary

$Tw : sSet \to sSet$ is a right Quillen functor with respect to the model structure for quasicategories

Proof

Let $L$ be the left adjoint. $L$ preserves monomorphisms, so we need to show it preserves trivial cofibrations. The model structure for quasicategories is a Cisinski model structure, so a map is a trivial cofibration iff it has the left lifting property against fibrations between fibrant objects, which in this case are the categorical fibrations between quasicategories.

Suppose $i$ is a trivial cofibration, and let $f : C \to D$ be a categorical fibration between quasi-categories. Since left Kan fibrations are categorical fibrations, the previous lemma implies $Tw(f)$ is a categorical fibration between quasicategories, so $i \perp Tw(f)$, and thus $L(i) \perp f$.

Related concepts

• (∞,1)-end

• lax (∞,1)-colimit

• twisted arrow category

References

• Jacob Lurie, Formal moduli problems

• Kerodon, The Twisted Arrow Construction $[$kerodon:03JF$]$