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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeFeb 1st 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexRedirect: Barwick-Kan equivalence. Renamed. <a href="https://ncatlab.org/nlab/revision/diff/model+structure+on+relative+categories/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/model+structure+on+relative+categories/7">v7</a>, <a href="https://ncatlab.org/nlab/show/model+structure+on+relative+categories">current</a>

   Redirect: Barwick-Kan equivalence.

   Renamed.

   diff, v7, current

2. • CommentRowNumber2.
   • CommentAuthorHurkyl
   • CommentTimeFeb 26th 2021
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexThe 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. <a href="https://ncatlab.org/nlab/revision/diff/model+structure+on+relative+categories/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/model+structure+on+relative+categories/9">v9</a>, <a href="https://ncatlab.org/nlab/show/model+structure+on+relative+categories">current</a>

   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.

   diff, v9, current

3. • CommentRowNumber3.
   • CommentAuthorHurkyl
   • CommentTimeFeb 27th 2021
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexAdded the compatibility with simplicial localization. <a href="https://ncatlab.org/nlab/revision/diff/model+structure+on+relative+categories/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/model+structure+on+relative+categories/10">v10</a>, <a href="https://ncatlab.org/nlab/show/model+structure+on+relative+categories">current</a>

   Added the compatibility with simplicial localization.

   diff, v10, current

4. • CommentRowNumber4.
   • CommentAuthorHurkyl
   • CommentTimeFeb 27th 2021
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexAdded the connection to marked simplicial sets, as well as the description of (C,W) as modeling the localization $C[W^{-1}]$. <a href="https://ncatlab.org/nlab/revision/diff/model+structure+on+relative+categories/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/model+structure+on+relative+categories/11">v11</a>, <a href="https://ncatlab.org/nlab/show/model+structure+on+relative+categories">current</a>

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

   diff, v11, current

5. • CommentRowNumber5.
   • CommentAuthorHurkyl
   • CommentTimeFeb 27th 2021
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexI 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. <a href="https://ncatlab.org/nlab/revision/diff/model+structure+on+relative+categories/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/model+structure+on+relative+categories/11">v11</a>, <a href="https://ncatlab.org/nlab/show/model+structure+on+relative+categories">current</a>

   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.

   diff, v11, current

6. • CommentRowNumber6.
   • CommentAuthorHurkyl
   • CommentTimeFeb 27th 2021
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexBy 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.

   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.

7. • CommentRowNumber7.
   • CommentAuthorDmitri Pavlov
   • CommentTimeFeb 27th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexFixed formatting. <a href="https://ncatlab.org/nlab/revision/diff/model+structure+on+relative+categories/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/model+structure+on+relative+categories/12">v12</a>, <a href="https://ncatlab.org/nlab/show/model+structure+on+relative+categories">current</a>

   Fixed formatting.

   diff, v12, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeFeb 28th 2021
   • (edited Feb 28th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItex@Hurkyl #6, I like to use "$L_W$", following/alluding to standard (?) notation for Dwyer-Kan simplicial localizations, such as Hammock localization "$L^H_W$".

   @Hurkyl #6,

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

9. • CommentRowNumber9.
   • CommentAuthorHurkyl
   • CommentTimeFeb 28th 2021
   • (edited Feb 28th 2021)
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItex$L_W(C)$ 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.

   $L_W(C)$ 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.