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Stream: deprecated: topos theory

Topic: stacks and other things

James Deikun (May 26 2022 at 13:10):

I've gathered from the literature, especially Caramello, the following analogies:

• stack : internal category in a sheaf topos :: fibration : split fibration :: indexed category : strict indexed category
• stack on $\mathcal{E}$ : geometric morphism into $\mathcal{E}$ :: small category : site :: presheaf topos : Grothendieck topos

First off, it feels like these analogies should come with some size conditions on the stack, but they seem to get glossed over. What are they?

Second, the first set of analogies seems like it's not specific to sites/toposes, but the definition of stack is. Is there a more general way to define stacks so they generalize internal categories, or does that just reduce to a fibration/indexed category?

Third, it seems like the second set of analogies would associate stacks with a particularly nice kind of model of a geometric theory, like presheaf toposes are particularly nice toposes. Is there a good characterization of what makes these models nicer?

Zhen Lin Low (May 26 2022 at 14:20):

There's more subtlety than just that. Not every internal category in a sheaf topos gives rise to a stack. (More precisely, every internal category in a category with finite limits can be "externalised" to obtain a strict indexed category, but that does not automatically satisfy the effective descent condition.)

Zhen Lin Low (May 26 2022 at 14:21):

Stacks are only defined for sites for the same reason sheaves are only defined for sites: you need to know which descent conditions to impose.

Zhen Lin Low (May 26 2022 at 14:29):

I would prefer to push the analogy that a stack is the correct notion of a Cat-valued sheaf, where we understand Cat to be the 2-category of categories, whereas an internal category in a sheaf topos is basically the same thing as a sheaf taking values in the 1-category of "categories with equality between objects". (Put it another way, the notion of internal category in a 1-category is really an internalisation of the notion of "category with equality between objects".)

James Deikun (May 26 2022 at 14:39):

I understand there is a 'stackification' of any fibration much like there is a 'sheafification' of any presheaf; what distinctions between internal categories are lost by passing to the stackification of the externalization?

James Deikun (May 26 2022 at 14:41):

I get the idea of whatever might make stacks 'more special' than internal categories isn't enough to prevent there being a relative Giraud's theorem using stacks instead of internal categories; is this correct?

Zhen Lin Low (May 26 2022 at 14:43):

In the case of the terminal site, you lose the distinction between non-isomorphic but equivalent categories. But much more happens in general – it's where the first sheaf cohomology group comes from, in some sense.

Zhen Lin Low (May 26 2022 at 14:46):

Take an internal group G and consider it as an internal category. Its externalisation will usually fail to be a stack. There is a natural identification between the set of connected components of the fibre over 1 of the stackification and the sheaf cohomology group H^1 (G).

James Deikun (May 26 2022 at 14:49):

(Whereas the fiber over 1 of the plain externalization is always connected, IIUC.)

Zhen Lin Low (May 26 2022 at 14:51):

Yes. So by passing to the stackification you lose the distinction between internal categories that are weakly equivalent but not strictly equivalent (let alone isomorphic).

James Deikun (May 26 2022 at 14:54):

Oh, there's a specific notion of weak equivalence of internal categories in play here?

Zhen Lin Low (May 26 2022 at 14:55):

It is more or less the internalisation of the usual notion: a functor is a weak equivalence if it is fully faithful and every object in the codomain category is isomorphic to an object in the image. The existential quantifier is doing the work of making this interesting in a sheaf topos.

James Deikun (May 26 2022 at 15:03):

Ah, so it's just internal anaequivalence! But I guess in a general sheaf topos you get some interesting homotopy from anaequivalence, and that's what the stacks 'see' by forming a bicategory that is not (I'm guessing) globally equivalent to the bicategory of internal categories?

Zhen Lin Low (May 26 2022 at 15:04):

I believe so.

James Deikun (May 26 2022 at 15:08):

Thanks, this really clears up the first set of analogies; and in particular its limitations and why stacks might be nicer to work with than internal categories to an even greater degree than non-split fibrations in general.

James Deikun (May 26 2022 at 15:18):

(I guess it also clears up why you want $\infty$-stacks; in a similar way, they can see all of the latent homotopy behind sheaf cohomology, basically, not just the lowest levels?)

Zhen Lin Low (May 26 2022 at 15:38):

Sure. At least it clarifies why ∞-stacks of ∞-groupoids are significant. IIRC this is explained as early as Ken Brown's paper introducing categories of fibrant objects. (He doesn't use these words though...)

James Deikun (May 27 2022 at 12:38):

Okay, summarizing my progress from the discussion yesterday and my own thoughts:

I guess I can refine the analogies to:

• stack : internal category in a sheaf topos :: anaequivelance class of fibrations : split fibration :: anaequivalence class of indexed categories : strict indexed category
• stacks on $\mathcal{S}$ with stack maps : $\mathcal{S}$-toposes with $\mathcal{S}$-essential geometric morphisms over $\mathcal{S}$ :: small categories with functors : sites with ??? :: presheaf toposes with essential geometric morphisms : $\bold{EssTopos}/\bold{Set}$

I still have no idea on the size conditions.

I think the second question is mostly answered; stacks correspond to something like fibrant objects in a bicategorical CFO or fibrant-cofibrant objects in a 2-categorical model structure on fibered/indexed categories with internal anaequivalences as weak equivalences, and if you figure out a way to construct such a thing outside of topos theory you could consider them "generalized stacks" if you want to.

The third question, on the other hand, seems to have grown another question: if an $\mathcal{S}$-topos $e$ is also a $\mathbb{Th}(\mathcal{S})$-model in $|e|$ then what is the model-theoretic significance of geometric morphisms over $\mathcal{S}$ in general, or $\mathcal{S}$-essential ones in particular? OTOH my estimate of the likelihood of a meaningful answer for the question in general has fallen due to realizing the most natural maps for $\mathcal{S}$-toposes as models and as generalized stacks are different...

James Deikun (May 27 2022 at 12:56):

I guess a geometric morphism from $e$ to $f$ over $\mathcal{S}$ corresponds to a way of seeing $e$ as an internalization of $f$ in $|e|$ ... and a slice morphism of $\bold{Topos}/\mathcal{S}$ in general with a nontrivial transformation in the triangle corresponds to a model homomorphism from $e$ to an internalization of $f$ in $|e|$ as seen in https://arxiv.org/abs/2104.05650 ...

James Deikun (May 27 2022 at 15:20):

I guess then the difference when you have an $\mathcal{S}$-essential geometric morphism is that aside from the homomorphism from $e$ to $g^{*}f$ in $|e|$ you also get a corresponding model $g_{!}e$ and a homomorphism from it to $f$ in $|f|$ ...

James Deikun (May 27 2022 at 15:29):

Basically the homomorphism in $|e|$ arises from an "underlying" homomorphism that already exists in $|f|$?