# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeJan 31st 2021

Redirect: Barwick-Kan equivalence.

Renamed.

• CommentRowNumber2.
• CommentAuthorHurkyl
• CommentTimeFeb 26th 2021

The fact the Quillen equivalence is actually an adjoint weak equivalence of relative categories is significant enough that I reorganized things a bit so I can state it.

It also seems useful to put a little more emphasis on the fact that RelCat can model simplicial spaces.

• CommentRowNumber3.
• CommentAuthorHurkyl
• CommentTimeFeb 27th 2021

Added the compatibility with simplicial localization.

• CommentRowNumber4.
• CommentAuthorHurkyl
• CommentTimeFeb 27th 2021

Added the connection to marked simplicial sets, as well as the description of (C,W) as modeling the localization $C[W^{-1}]$.

• CommentRowNumber5.
• CommentAuthorHurkyl
• CommentTimeFeb 27th 2021

I seem to have broken the formatting system in the statement of the theorem that $RelCat \to (\infty,1)Cat$ is $(C,W) \to C[W^{-1}]$, and I can’t figure out what I’ve done wrong.

• CommentRowNumber6.
• CommentAuthorHurkyl
• CommentTimeFeb 27th 2021

By the way, does nLab have an established notation for localization of $\infty$-categories? I’m uncomfortable with $C[W^{-1}]$ due to the risk of that being interpreted as the localization of 1-categories.

• CommentRowNumber7.
• CommentAuthorDmitri Pavlov
• CommentTimeFeb 27th 2021

Fixed formatting.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeFeb 27th 2021
• (edited Feb 27th 2021)

@Hurkyl #6,

I like to use “$L_W$”, following/alluding to standard (?) notation for Dwyer-Kan simplicial localizations, such as Hammock localization “$L^H_W$”.

• CommentRowNumber9.
• CommentAuthorHurkyl
• CommentTimeFeb 28th 2021
• (edited Feb 28th 2021)

$L_W(C)$ or $L(C,W)$? I guess the difference is whether you are in a context viewing $C$ or $(C,W)$ as the primary object of interest. I’ve switched over the notation.