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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 20th 2016
   • (edited Mar 20th 2016)
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   Author: Dmitri Pavlov
   Format: MarkdownItexBoth [[relative categories]] and [[marked simplicial sets]] (over Δ^0) present the ∞-category of ∞-categories. Is the functor that sends a relative category (C,W) to the marked simplicial set (NC,NW) a right Quillen equivalence? If so, is there a reference for this fact? The closest I could find is Remark 1.3.4.2 in Higher Algebra, which states it on the level of individual objects, not model categories.

   Both relative categories and marked simplicial sets (over Δ^0) present the ∞-category of ∞-categories.

   Is the functor that sends a relative category (C,W) to the marked simplicial set (NC,NW) a right Quillen equivalence?

   If so, is there a reference for this fact? The closest I could find is Remark 1.3.4.2 in Higher Algebra, which states it on the level of individual objects, not model categories.

2. • CommentRowNumber2.
   • CommentAuthorChris SchommerPries
   • CommentTimeMar 20th 2016
   • (edited Mar 20th 2016)
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   Author: Chris SchommerPries
   Format: MarkdownItexNo. If it were then the left adjoint would be a left Quillen Equivalence and so the map $$ X \to \mathbb{R}N (L(X)) $$ would have to be an equivalence of marked simplicial sets for any cofibrant marked simplicial set $X = (X, wX)$. But it is easy to make counter examples, e.g. take $X$ to be a Kan complex with all edges marked which is not a 1-type.

   No. If it were then the left adjoint would be a left Quillen Equivalence and so the map

   $$X \to \mathbb{R}N (L(X))$$

   would have to be an equivalence of marked simplicial sets for any cofibrant marked simplicial set $X = (X, wX)$. But it is easy to make counter examples, e.g. take $X$ to be a Kan complex with all edges marked which is not a 1-type.

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 20th 2016
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   Author: Dmitri Pavlov
   Format: MarkdownItex@Chris SchommerPries: Sorry, I can't quite figure out on my own why a Kan complex, with all edges marked, that is not a 1-type, cannot be weakly equivalent to the derived nerve of a relative 1-category. According to a variant of McDuff's theorem, any ∞-groupoid X (not necessarily a 1-type) can be presented as C[C^{−1}], where C is a 1-category and ^{−1} means inverting morphisms up to a homotopy, so at least naively (before deriving) it seems to me like the relative category (C,C) should present X when we take the nerve (NC,NC).

   @Chris SchommerPries: Sorry, I can’t quite figure out on my own why a Kan complex, with all edges marked, that is not a 1-type, cannot be weakly equivalent to the derived nerve of a relative 1-category.

   According to a variant of McDuff’s theorem, any ∞-groupoid X (not necessarily a 1-type) can be presented as C[C^{−1}], where C is a 1-category and ^{−1} means inverting morphisms up to a homotopy, so at least naively (before deriving) it seems to me like the relative category (C,C) should present X when we take the nerve (NC,NC).

4. • CommentRowNumber4.
   • CommentAuthorChris SchommerPries
   • CommentTimeMar 21st 2016
   • (edited Mar 21st 2016)
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   Author: Chris SchommerPries
   Format: MarkdownItexThe problem is not whether there is some relative category which represents $X$ or with deriving. The problem is that the particular relative category L(X) doesn't do the job. If X is a Kan complex with every edge marked, then L(X) is the fundamental groupoid of X with all morphisms marked. Then $\mathbb{R}N(L(X))$ is a model for the 1-type of X, but not X itself (assuming X was not a 1-type). More generally L(X) only depends on the (marked) 2-skeleton of X.

   The problem is not whether there is some relative category which represents $X$ or with deriving. The problem is that the particular relative category L(X) doesn’t do the job.

   If X is a Kan complex with every edge marked, then L(X) is the fundamental groupoid of X with all morphisms marked. Then $\mathbb{R}N(L(X))$ is a model for the 1-type of X, but not X itself (assuming X was not a 1-type).

   More generally L(X) only depends on the (marked) 2-skeleton of X.

5. • CommentRowNumber5.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 21st 2016
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   Author: Dmitri Pavlov
   Format: MarkdownItexI see, so although the right adjoint functor (NC,NW) does compute the correct ∞-category (even without deriving, it seems), its left adjoint is badly behaved. After you pointed this out, I found out that Barwick and Kan already treat the case of nonmarked simplicial sets in §6.6 of their paper, and they recover Thomason's theorem (Theorem 6.7 in their paper). This is achieved by replacing the naive cosimplicial object (n↦(0→⋯→n)) in relative categories with its double subdvision (as a relative poset). The resulting nerve functor is weakly equivalent to the naive nerve functor, but their left adjoints are not equivalent, and it is the nonnaive left adjoint that computes the correct category that corresponds to a simplicial set. So I guess one could formulate a more refined version of the above conjecture: 1) is the subdivided relative nerve (N_ξ C,N_ξ W) weakly equivalent to the naive relative nerve (NC,NW) as a marked simplicial set? 2) is the subdivided relative nerve a right Quillen equivalence?

   I see, so although the right adjoint functor (NC,NW) does compute the correct ∞-category (even without deriving, it seems), its left adjoint is badly behaved.

   After you pointed this out, I found out that Barwick and Kan already treat the case of nonmarked simplicial sets in §6.6 of their paper, and they recover Thomason’s theorem (Theorem 6.7 in their paper). This is achieved by replacing the naive cosimplicial object (n↦(0→⋯→n)) in relative categories with its double subdvision (as a relative poset). The resulting nerve functor is weakly equivalent to the naive nerve functor, but their left adjoints are not equivalent, and it is the nonnaive left adjoint that computes the correct category that corresponds to a simplicial set.

   So I guess one could formulate a more refined version of the above conjecture: 1) is the subdivided relative nerve (N_ξ C,N_ξ W) weakly equivalent to the naive relative nerve (NC,NW) as a marked simplicial set? 2) is the subdivided relative nerve a right Quillen equivalence?

6. • CommentRowNumber6.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 8th 2016
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   Author: Dmitri Pavlov
   Format: MarkdownItexI asked the improved question here: <http://mathoverflow.net/questions/235696/from-relative-categories-to-marked-simplicial-sets>

   I asked the improved question here: http://mathoverflow.net/questions/235696/from-relative-categories-to-marked-simplicial-sets