nLab
twisted arrow (∞,1)-category

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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 CC we define the twisted arrow quasi-category Tw(C)Tw(C) to be

Tw(C) n=sSet((Δ n) op(Δ n),C) 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)C opsrc : Tw(C) \to C^{op} and tgt:Tw(C)Ctgt : Tw(C) \to C.

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

The opposite convention often appears in the literature,

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

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

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

Properties

If CC is a 1-category, then Tw(C)Tw(C) agrees with the 1-categorical version.

For a quasi-category CC, both Tw(C)Tw(C) and Tw¯(C)\overline{Tw}(C) are quasi-categories.

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

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

Proof

This description of Tw¯(C)\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.

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

References

Last revised on February 25, 2021 at 15:23:06. See the history of this page for a list of all contributions to it.