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1. • CommentRowNumber1.
   • CommentAuthoradeelkh
   • CommentTimeNov 19th 2014
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   Author: adeelkh
   Format: MarkdownItexWhat is the definition of an [[infinity-stack]] on an [[infinity-site]] with values in an arbitrary infinity-category (bicomplete, say)? The definition I know works only for stacks of infinity-groupoids.

   What is the definition of an infinity-stack on an infinity-site with values in an arbitrary infinity-category (bicomplete, say)? The definition I know works only for stacks of infinity-groupoids.

2. • CommentRowNumber2.
   • CommentAuthoradeelkh
   • CommentTimeNov 19th 2014
   • (edited Nov 19th 2014)
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   Author: adeelkh
   Format: MarkdownItexMaybe this: $F : C^{op} \to D$ is a stack/sheaf iff $Hom(d, F(-)) : C^{op} \to Spc$ is a sheaf for all objects $d \in D$?

   Maybe this: is a stack/sheaf iff is a sheaf for all objects ?

3. • CommentRowNumber3.
   • CommentAuthorZhen Lin
   • CommentTimeNov 19th 2014
   • (edited Nov 19th 2014)
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   Author: Zhen Lin
   Format: MarkdownItexThat's reasonable, at least for categories of structures "defined by finite (homotopy) limits".

   That’s reasonable, at least for categories of structures “defined by finite (homotopy) limits”.

4. • CommentRowNumber4.
   • CommentAuthoradeelkh
   • CommentTimeNov 19th 2014
   • (edited Nov 19th 2014)
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   Author: adeelkh
   Format: MarkdownItexZhen Lin: what does that mean? (At the moment I'm interested in stacks of infinity-categories.)

   Zhen Lin: what does that mean? (At the moment I’m interested in stacks of infinity-categories.)

5. • CommentRowNumber5.
   • CommentAuthorDylan Wilson
   • CommentTimeNov 19th 2014
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   Author: Dylan Wilson
   Format: MarkdownItexDAGV says, for X a topos, a C-valued sheaf is a limit preserving functor X^op ---> C, for any arbitrary C.

   DAGV says, for X a topos, a C-valued sheaf is a limit preserving functor X^op —> C, for any arbitrary C.

6. • CommentRowNumber6.
   • CommentAuthoradeelkh
   • CommentTimeNov 19th 2014
   • (edited Nov 19th 2014)
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   Author: adeelkh
   Format: MarkdownItexDylan: that definition doesn't take into account the topology on the site though. Oh wait, X is already a topos. I'm confused now...

   Dylan: that definition doesn’t take into account the topology on the site though. Oh wait, X is already a topos. I’m confused now…

7. • CommentRowNumber7.
   • CommentAuthorZhen Lin
   • CommentTimeNov 19th 2014
   • (edited Nov 19th 2014)
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   Author: Zhen Lin
   Format: MarkdownItexWell, going back to ordinary examples, we know that a "sheaf of local rings" is not actually defined to be a sheaf taking values in local rings in this sense. But otherwise the obvious thing works well for many examples: groups, rings, etc. These are structures "defined by finite limits". See [[sketch]]. I expect the same holds true for "homotopical" structures. > Dylan: that definition doesn't take into account the topology on the site though. Oh wait, X is already a topos. I'm confused now... It's secretly the same as your definition. The idea is that every limit preserving functor $\mathcal{X}^{op} \to \mathcal{S}$ is representable.

   Well, going back to ordinary examples, we know that a “sheaf of local rings” is not actually defined to be a sheaf taking values in local rings in this sense. But otherwise the obvious thing works well for many examples: groups, rings, etc. These are structures “defined by finite limits”. See sketch. I expect the same holds true for “homotopical” structures.

   Dylan: that definition doesn’t take into account the topology on the site though. Oh wait, X is already a topos. I’m confused now…

   It’s secretly the same as your definition. The idea is that every limit preserving functor is representable.

8. • CommentRowNumber8.
   • CommentAuthoradeelkh
   • CommentTimeNov 19th 2014
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   Author: adeelkh
   Format: MarkdownItexRight, of course, thanks to both of you. Perhaps this should go on the page [[infinity-stack]].

   Right, of course, thanks to both of you. Perhaps this should go on the page infinity-stack.

9. • CommentRowNumber9.
   • CommentAuthorMarc Hoyois
   • CommentTimeNov 19th 2014
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   Author: Marc Hoyois
   Format: MarkdownItexEach definition has an advantage over the other though. On the one hand, one can speak of $D$-valued sheaves on (∞,1)-topoi that don't arise from a site. On the other hand, one can have $D$-valued sheaves on $C$ that do not lift to $Shv(C)$ (if $D$ doesn't have enough limits).

   Each definition has an advantage over the other though. On the one hand, one can speak of -valued sheaves on (∞,1)-topoi that don’t arise from a site. On the other hand, one can have -valued sheaves on that do not lift to (if doesn’t have enough limits).

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 19th 2014
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    Author: Urs
    Format: MarkdownItexThe definition recalled in #5 is to be thought of as giving sheaves (stacks) on the topos regarded as a big site equipped with the [[canonical topology]].

    The definition recalled in #5 is to be thought of as giving sheaves (stacks) on the topos regarded as a big site equipped with the canonical topology.

11. • CommentRowNumber11.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 20th 2014
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    Author: Marc Hoyois
    Format: MarkdownItex@Urs: one might think about it this way, but it doesn't actually work for a general (∞,1)-topos.

    @Urs: one might think about it this way, but it doesn’t actually work for a general (∞,1)-topos.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeNov 20th 2014
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    Author: Urs
    Format: MarkdownItexSorry, say again: what doesn't work?

    Sorry, say again: what doesn’t work?

13. • CommentRowNumber13.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 20th 2014
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    Author: Marc Hoyois
    Format: MarkdownItexThe Yoneda embedding $\mathbf H\to Shv_{can}(\mathbf H)$ is only an equivalence if $\mathbf H$ is a sheaf (∞,1)-topos.

    The Yoneda embedding is only an equivalence if is a sheaf (∞,1)-topos.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeNov 20th 2014
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    Author: Urs
    Format: MarkdownItexWhere by "sheaf $\infty$-topos" you mean topological localization of presheaves, I suppose. I'd still think that for #6 this $\mathbf{H} \simeq Sh_{can}(\mathbf{H})$ clarifies what's going on, conceptually. Hypercompletion may probably be brought into the picture nicely somehow?

    Where by “sheaf -topos” you mean topological localization of presheaves, I suppose.

    I’d still think that for #6 this clarifies what’s going on, conceptually. Hypercompletion may probably be brought into the picture nicely somehow?

15. • CommentRowNumber15.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 20th 2014
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    Author: Marc Hoyois
    Format: MarkdownItexI was wondering about this also, and I'm not sure. That's what the notion of ``hypertopology'' on a category $C$ would accomplish, I guess.

    I was wondering about this also, and I’m not sure. That’s what the notion of “hypertopology” on a category would accomplish, I guess.

16. • CommentRowNumber16.
    • CommentAuthorZhen Lin
    • CommentTimeDec 11th 2014
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    Author: Zhen Lin
    Format: MarkdownItexSo, what's the conclusion? Is it true that a functor $\mathcal{X}^{op} \to \mathcal{S}$ is representable if and only if it is a sheaf for the canonical topology? This [MO question](http://mathoverflow.net/q/95172/11640) remains unresolved.

    So, what’s the conclusion? Is it true that a functor is representable if and only if it is a sheaf for the canonical topology? This MO question remains unresolved.

17. • CommentRowNumber17.
    • CommentAuthorMarc Hoyois
    • CommentTimeDec 12th 2014
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    Author: Marc Hoyois
    Format: MarkdownItexIs this a correct argument? An (∞,1)-topos $\mathbf{H}$ is the union of its subcategories of $\kappa$-compact objects for all cardinals $\kappa$ such that $\mathbf{H}$ is $\kappa$-accessible and closed under finite limits: $$ \mathbf{H} = colim_\kappa \mathbf{H}^\kappa. $$ So $$ Fun(\mathbf{H}, \infty Grpd^{op}) = lim_\kappa Fun(\mathbf{H}^\kappa, \infty Grpd^{op}). $$ A presheaf on $\mathbf{H}$ is a sheaf iff its restrictions to all $\mathbf{H}^{\kappa}$'s are sheaves, since every Cech nerve in $\mathbf{H}$ is in $\mathbf{H}^\kappa$ for some $\kappa$. Hence $$ Sh(\mathbf{H}) = lim_\kappa Sh(\mathbf{H}^\kappa) = \mathbf{H}. $$

    Is this a correct argument? An (∞,1)-topos is the union of its subcategories of -compact objects for all cardinals such that is -accessible and closed under finite limits:

    So

    A presheaf on is a sheaf iff its restrictions to all ’s are sheaves, since every Cech nerve in is in for some . Hence

18. • CommentRowNumber18.
    • CommentAuthorZhen Lin
    • CommentTimeDec 12th 2014
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    Author: Zhen Lin
    Format: MarkdownItexThat seems correct, modulo the question of whether a sheaf for the canonical topology on $\mathbf{H}^\kappa$ necessarily satisfies descent for Čech nerves in $\mathbf{H}^\kappa$. But that's just a question of what the definition of canonical topology is, I guess.

    That seems correct, modulo the question of whether a sheaf for the canonical topology on necessarily satisfies descent for Čech nerves in . But that’s just a question of what the definition of canonical topology is, I guess.

19. • CommentRowNumber19.
    • CommentAuthorMarc Hoyois
    • CommentTimeDec 12th 2014
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    Author: Marc Hoyois
    Format: MarkdownItexRight, I was thinking of the topology on $\mathbf H^\kappa$ induced by epimorphic covers (in the topos $\mathbf H$), so the canonical topology in the sense of HTT. Actually, do we even know that $Sh(\mathbf H^\kappa)=\mathbf H$?

    Right, I was thinking of the topology on induced by epimorphic covers (in the topos ), so the canonical topology in the sense of HTT. Actually, do we even know that ?

20. • CommentRowNumber20.
    • CommentAuthorZhen Lin
    • CommentTimeDec 12th 2014
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    Author: Zhen Lin
    Format: MarkdownItexOh, sorry. I think I had some wires crossed. Yes, it would suffice to check that $Sh (\mathbf{H}^\kappa) \simeq \mathbf{H}$. In view of the fact that every limit-preserving functor $\mathbf{H}^{op} \to \infty Grpd$ is representable, it suffices to verify that the right Kan extension of a sheaf $(\mathbf{H}^\kappa)^{op} \to \infty Grpd$ along the inclusion $\mathbf{H}^\kappa \to \mathbf{H}$ preserves limits. By accessibility, it should be enough to verify that a sheaf $(\mathbf{H}^\kappa)^{op} \to \infty Grpd$ preserves limits for $\kappa$-small diagrams. But is this really any easier than the question we started with?

    Oh, sorry. I think I had some wires crossed. Yes, it would suffice to check that . In view of the fact that every limit-preserving functor is representable, it suffices to verify that the right Kan extension of a sheaf along the inclusion preserves limits. By accessibility, it should be enough to verify that a sheaf preserves limits for -small diagrams. But is this really any easier than the question we started with?

21. • CommentRowNumber21.
    • CommentAuthorMarc Hoyois
    • CommentTimeDec 13th 2014
    • (edited Dec 13th 2014)
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    Author: Marc Hoyois
    Format: MarkdownItexIt doesn't seem any easier, no. Let me list some facts which I think we know for sure: - if you factor the inclusion $\mathbf H\subset PSh(\mathbf H^\kappa)$ as a cotopological localization followed by a topological one, then the topological one is exactly $Sh(\mathbf H^\kappa)$, by definition of the canonical topology. Hence $\mathbf H$ is a cotopological localization of $Sh(\mathbf H^\kappa)$. - By HTT 6.2.4.6, that the inclusion $\mathbf H\subset Sh(\mathbf H^\kappa)$ preserves effective epimorphisms. - If $\mathbf H=Sh(C)$ where $C$ has finite limits, the restriction along $C\to \mathbf H^\kappa$ is a geometric morphism $Sh(\mathbf H^\kappa)\to\mathbf H$, and the composition $$ \mathbf H \subset Sh(\mathbf H^\kappa)\to\mathbf H $$ is the identity. So $\mathbf H$ is even a retract of $Sh(\mathbf H^\kappa)$.

    It doesn’t seem any easier, no. Let me list some facts which I think we know for sure:

    • if you factor the inclusion as a cotopological localization followed by a topological one, then the topological one is exactly , by definition of the canonical topology. Hence is a cotopological localization of .
    • By HTT 6.2.4.6, that the inclusion preserves effective epimorphisms.
    • If where has finite limits, the restriction along is a geometric morphism , and the composition

    is the identity. So is even a retract of .