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nLab > nLab General Discussions: infinity-stack valued in an arbitrary infinity-category

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    • CommentRowNumber1.
    • CommentAuthoradeelkh
    • CommentTimeNov 19th 2014

    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.

    • CommentRowNumber2.
    • CommentAuthoradeelkh
    • CommentTimeNov 19th 2014
    • (edited Nov 19th 2014)

    Maybe this: F:C op→DF : C^{op} \to D is a stack/sheaf iff Hom(d,F(−)):C op→SpcHom(d, F(-)) : C^{op} \to Spc is a sheaf for all objects d∈Dd \in D?

    • CommentRowNumber3.
    • CommentAuthorZhen Lin
    • CommentTimeNov 19th 2014
    • (edited Nov 19th 2014)

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

    • CommentRowNumber4.
    • CommentAuthoradeelkh
    • CommentTimeNov 19th 2014
    • (edited Nov 19th 2014)

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

    • CommentRowNumber5.
    • CommentAuthorDylan Wilson
    • CommentTimeNov 19th 2014

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

    • CommentRowNumber6.
    • CommentAuthoradeelkh
    • CommentTimeNov 19th 2014
    • (edited Nov 19th 2014)

    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…

    • CommentRowNumber7.
    • CommentAuthorZhen Lin
    • CommentTimeNov 19th 2014
    • (edited Nov 19th 2014)

    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 𝒳 op→𝒮\mathcal{X}^{op} \to \mathcal{S} is representable.

    • CommentRowNumber8.
    • CommentAuthoradeelkh
    • CommentTimeNov 19th 2014

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

    • CommentRowNumber9.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 19th 2014

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

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 19th 2014

    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.

    • CommentRowNumber11.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 20th 2014

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

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeNov 20th 2014

    Sorry, say again: what doesn’t work?

    • CommentRowNumber13.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 20th 2014

    The Yoneda embedding H→Shv can(H)\mathbf H\to Shv_{can}(\mathbf H) is only an equivalence if H\mathbf H is a sheaf (∞,1)-topos.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeNov 20th 2014

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

    I’d still think that for #6 this H≃Sh can(H)\mathbf{H} \simeq Sh_{can}(\mathbf{H}) clarifies what’s going on, conceptually. Hypercompletion may probably be brought into the picture nicely somehow?

    • CommentRowNumber15.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 20th 2014

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

    • CommentRowNumber16.
    • CommentAuthorZhen Lin
    • CommentTimeDec 11th 2014

    So, what’s the conclusion? Is it true that a functor 𝒳 op→𝒮\mathcal{X}^{op} \to \mathcal{S} is representable if and only if it is a sheaf for the canonical topology? This MO question remains unresolved.

    • CommentRowNumber17.
    • CommentAuthorMarc Hoyois
    • CommentTimeDec 12th 2014

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

    H=colim κH κ. \mathbf{H} = colim_\kappa \mathbf{H}^\kappa.

    So

    Fun(H,∞Grpd op)=lim κFun(H κ,∞Grpd op). Fun(\mathbf{H}, \infty Grpd^{op}) = lim_\kappa Fun(\mathbf{H}^\kappa, \infty Grpd^{op}).

    A presheaf on H\mathbf{H} is a sheaf iff its restrictions to all H κ\mathbf{H}^{\kappa}’s are sheaves, since every Cech nerve in H\mathbf{H} is in H κ\mathbf{H}^\kappa for some κ\kappa. Hence

    Sh(H)=lim κSh(H κ)=H. Sh(\mathbf{H}) = lim_\kappa Sh(\mathbf{H}^\kappa) = \mathbf{H}.
    • CommentRowNumber18.
    • CommentAuthorZhen Lin
    • CommentTimeDec 12th 2014

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

    • CommentRowNumber19.
    • CommentAuthorMarc Hoyois
    • CommentTimeDec 12th 2014

    Right, I was thinking of the topology on H κ\mathbf H^\kappa induced by epimorphic covers (in the topos H\mathbf H), so the canonical topology in the sense of HTT. Actually, do we even know that Sh(H κ)=HSh(\mathbf H^\kappa)=\mathbf H?

    • CommentRowNumber20.
    • CommentAuthorZhen Lin
    • CommentTimeDec 12th 2014

    Oh, sorry. I think I had some wires crossed. Yes, it would suffice to check that Sh(H κ)≃HSh (\mathbf{H}^\kappa) \simeq \mathbf{H}. In view of the fact that every limit-preserving functor H op→∞Grpd\mathbf{H}^{op} \to \infty Grpd is representable, it suffices to verify that the right Kan extension of a sheaf (H κ) op→∞Grpd(\mathbf{H}^\kappa)^{op} \to \infty Grpd along the inclusion H κ→H\mathbf{H}^\kappa \to \mathbf{H} preserves limits. By accessibility, it should be enough to verify that a sheaf (H κ) op→∞Grpd(\mathbf{H}^\kappa)^{op} \to \infty Grpd preserves limits for κ\kappa-small diagrams. But is this really any easier than the question we started with?

    • CommentRowNumber21.
    • CommentAuthorMarc Hoyois
    • CommentTimeDec 13th 2014
    • (edited Dec 13th 2014)

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

    • if you factor the inclusion H⊂PSh(H κ)\mathbf H\subset PSh(\mathbf H^\kappa) as a cotopological localization followed by a topological one, then the topological one is exactly Sh(H κ)Sh(\mathbf H^\kappa), by definition of the canonical topology. Hence H\mathbf H is a cotopological localization of Sh(H κ)Sh(\mathbf H^\kappa).
    • By HTT 6.2.4.6, that the inclusion H⊂Sh(H κ)\mathbf H\subset Sh(\mathbf H^\kappa) preserves effective epimorphisms.
    • If H=Sh(C)\mathbf H=Sh(C) where CC has finite limits, the restriction along C→H κC\to \mathbf H^\kappa is a geometric morphism Sh(H κ)→HSh(\mathbf H^\kappa)\to\mathbf H, and the composition
    H⊂Sh(H κ)→H \mathbf H \subset Sh(\mathbf H^\kappa)\to\mathbf H

    is the identity. So H\mathbf H is even a retract of Sh(H κ)Sh(\mathbf H^\kappa).

1 to 21 of 21

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