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1. • CommentRowNumber1.
   • CommentAuthorUlrik
   • CommentTimeOct 30th 2018
   • PermaLink
   Author: Ulrik
   Format: MarkdownItexPage created, but author did not leave any comments. <a href="https://ncatlab.org/nlab/revision/%28%E2%88%9E%2C1%29-comparison+lemma/1">v1</a>, <a href="https://ncatlab.org/nlab/show/%28%E2%88%9E%2C1%29-comparison+lemma">current</a>

   Page created, but author did not leave any comments.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUlrik
   • CommentTimeOct 30th 2018
   • PermaLink
   Author: Ulrik
   Format: MarkdownItexI added a stubby version of [[(∞,1)-comparison lemma]] with a version by Hoyois. We should also have pages for the (∞,1)-versions of [[comorphism of sites]] and [[morphism of sites]]. I was wondering if there's any useful generalization not requiring a lot of pullbacks to exist in the [[(∞,1)-sites]]? Of course, we have to be careful as [this discussion](https://nforum.ncatlab.org/discussion/4457/cech-model-structure-on-simplicial-sheaves/) shows. Is there some property of an (∞,1)-site that guarantees that the sheaf topos has finite homotopy dimension, and could we combine such a property with the usual [[comparison lemma]] to get a different, general (∞,1)-comparison lemma?

   I added a stubby version of (∞,1)-comparison lemma with a version by Hoyois. We should also have pages for the (∞,1)-versions of comorphism of sites and morphism of sites.

   I was wondering if there’s any useful generalization not requiring a lot of pullbacks to exist in the (∞,1)-sites? Of course, we have to be careful as this discussion shows.

   Is there some property of an (∞,1)-site that guarantees that the sheaf topos has finite homotopy dimension, and could we combine such a property with the usual comparison lemma to get a different, general (∞,1)-comparison lemma?

3. • CommentRowNumber3.
   • CommentAuthorMarc Hoyois
   • CommentTimeOct 30th 2018
   • PermaLink
   Author: Marc Hoyois
   Format: MarkdownItexI made a few adjustments (in particular a locally small assumption on $D$ is missing from the published version of my paper). I never found a useful generalization that does not assume existence of some pullbacks. For example there are obvious formulations of a and c that do not require pullbacks (let's assume $D$ small for simplicity): a. for every $\tau$-sieve $U\to X$, $a_\rho u_!(U\to X)$ is an equivalence; c. for every $X\in D$, its image in $Shv_\rho(D)$ belongs to the smallest subcategory generated by the image of $C$ under colimits. Then a+b+c are necessary and sufficient for the conclusion of the lemma. Condition a is also what the definition of "continuous functor" should be in the (∞,1)-context (b is of course the definition of "cocontinuous functor").

   I made a few adjustments (in particular a locally small assumption on $D$ is missing from the published version of my paper).

   I never found a useful generalization that does not assume existence of some pullbacks. For example there are obvious formulations of a and c that do not require pullbacks (let’s assume $D$ small for simplicity): a. for every $\tau$-sieve $U\to X$, $a_\rho u_!(U\to X)$ is an equivalence; c. for every $X\in D$, its image in $Shv_\rho(D)$ belongs to the smallest subcategory generated by the image of $C$ under colimits. Then a+b+c are necessary and sufficient for the conclusion of the lemma. Condition a is also what the definition of “continuous functor” should be in the (∞,1)-context (b is of course the definition of “cocontinuous functor”).

4. • CommentRowNumber4.
   • CommentAuthorUlrik
   • CommentTimeOct 30th 2018
   • (edited Oct 30th 2018)
   • PermaLink
   Author: Ulrik
   Format: MarkdownItexThanks for your comments and adjustments! <a href="https://ncatlab.org/nlab/revision/diff/%28%E2%88%9E%2C1%29-comparison+lemma/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/%28%E2%88%9E%2C1%29-comparison+lemma/3">v3</a>, <a href="https://ncatlab.org/nlab/show/%28%E2%88%9E%2C1%29-comparison+lemma">current</a>

   Thanks for your comments and adjustments!

   diff, v3, current

5. • CommentRowNumber5.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 4th 2019
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCross-linked to [[dense subsite]]. <a href="https://ncatlab.org/nlab/revision/diff/%28%E2%88%9E%2C1%29-comparison+lemma/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/%28%E2%88%9E%2C1%29-comparison+lemma/4">v4</a>, <a href="https://ncatlab.org/nlab/show/%28%E2%88%9E%2C1%29-comparison+lemma">current</a>

   Cross-linked to dense subsite.

   diff, v4, current