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category theory

# Contents

## 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 $tgt : Tw(C) \to C$.

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.

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

###### 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.

This means that we have the formulas $Tw(C) \simeq el(\hom_C) \simeq ({*} \downarrow \hom)$, 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.