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Stream: learning: questions

Topic: On the classifying topos of an object

James Deikun (Apr 02 2025 at 15:42):

Let $\mathcal S$ be an elementary topos with natural numbers object, $\mathcal S[X]$ the classifying topos over $\mathcal S$ of an object, and $\mathcal S[X^\bullet]$ the classifying topos over $\mathcal S$ of a pointed object. Are any of these true/well-known facts of topos theory?

1. The morphism from $\mathcal S[X^\bullet]$ to $\mathcal S[X]$ picking out the object part of the generic pointed object is an [[etale geometric morphism]] which is equivalent to $\mathcal S[X]/X$ where X is the generic object.
2. Any [[etale geometric morphism]] $\mathcal E/E \to \mathcal E$ over $\mathcal S$ is a pullback of the above morphism along the geometric morphism classifying the object $E$.
3. The global sections morphism of $\mathcal S[X]$ represents a stack on $\mathcal S$ whenever $\mathcal S$ is such that it makes sense to ask if it represents a stack.
4. Geometric morphisms that represent stacks are closed under precomposition by etale geometric morphisms, and whenever this makes sense, under precomposition by geometric morphisms that represent stacks.

Also are any of the equivalents in $\infty$-topos theory known?

James Deikun (Apr 02 2025 at 19:18):

Evidence for the propositions above:

1. A morphism into $\mathcal S[X^\bullet]$ or $\mathcal S[X]/X$ is an object equipped with a global section, so they classify the same theory.
2. A morphism into $\mathcal E/E$ is a whatever-$\mathcal E$-classifies with a constant of type $E$. This is equivalent to a pointed object and a whatever-$\mathcal E$-classifies with the pointed object happening to be the same object that represents the sort $E$.
3. This one is just a feeling and I'm genuinely curious about it.
4. Local homomorphisms of locales are closed under composition and geometric morphisms representing stacks are similar in nature.

James Deikun (Apr 03 2025 at 09:42):

Although it's not outright stated, I found enough proof in https://arxiv.org/abs/2107.04417 that if (3) is true then the represented stack is the canonical stack on $\mathcal S$.

Morgan Rogers (he/him) (Apr 03 2025 at 20:34):

1 follows from the discussion in the other topic. One way of seeing 2 is actually as a special case of Diaconescu's theorem(!) The slice over X is equivalent the category of internal presheaves over X viewed as a discrete category, and Diaconescu's theorem says that this construction is pullback-stable in the way that you state.
For the stack-based questions I don't even know the definition of stack in enough detail to figure out if they're plausible so I'll let someone else answer :)

Fernando Yamauti (Apr 04 2025 at 01:13):

What is a geometric morphism that represents a stack? Every geometric morphism canonically defines a topos-valued stack with respect to the canonical topology on the base. Is that what you're talking about?

James Deikun (Apr 04 2025 at 02:01):

In the paper I linked above they form a "fundamental adjunction" between stacks on a site $(\mathcal C,J)$ and essential, locally connected geometric morphisms over the topos $\mathbf{Sh}(\mathcal C,J)$, specifically given by continuous comorphisms over $(\mathcal C,J)$. The left adjoint is given by Cartesian lifts in the fibration of the category of sites and comorphisms over the category of categories and functors. The right adjoint is given by homming the image of $\mathcal C$ in $\mathbf{EssTopos}/\mathbf{Sh}(\mathcal C,J)$ given by sheafifying the representables and slicing over them, the indexed version of what's called the "canonical stack", fiberwise into the argument, and taking the Grothendieck construction. I think when your site is the canonical topology on a topos the canonical stack is the self-indexing. This stack is not topos-valued; it's a different construction meant to represent all the small stacks (and it still gets some of the large ones like the canonical stack).

James Deikun (Apr 04 2025 at 02:16):

I'm saying a geometric morphism "represents a stack" if it is the image of a stack under the left adjoint of the fundamental adjunction.

Fernando Yamauti (Apr 04 2025 at 02:19):

What's the site in question 3?

James Deikun (Apr 04 2025 at 02:20):

If I'm right it should work with any site that represents $\mathcal S$.

Fernando Yamauti (Apr 04 2025 at 02:25):

So your base is not an elementary topos with NNO anymore?

James Deikun (Apr 04 2025 at 02:27):

I think it probably would work with an internal site, like the trivial internal site, too, but the groundwork hasn't been done except for Grothendieck toposes and external sites, so it's probably best to stick with that for now.

James Deikun (Apr 04 2025 at 02:51):

There's an obvious geometric embedding of $\mathcal S^\to$, as the classifying topos over $\mathcal S$ of a subterminal object, into $\mathcal S[X]$. The Yoneda embedding of $\mathcal S$ should factor into this and a geometric embedding of $\mathcal S[X]$ into presheaves on $\mathcal S^\to$.

And the Grothendieck topology should be describable as "a sieve covers iff it contains Cartesian lifts of all arrows in a jointly effective-epimorphic sieve".

Fernando Yamauti (Apr 04 2025 at 03:11):

I see now what they mean. Their notation is a little nonstandard, but it seems they are talking about relative (or internal) presheaf topoi. So what you need to prove is that $\mathcal{S}[X]$ is a presheaf topos relative to $\mathcal{S}$. The obvious candidate would be taking $\mathcal{S}$-presheaves over the $\mathcal{S}$-cat given by $\mathcal{S}_{fin}/(-)$, but I don't know if that works.

It would be nice to have an external characterisation of what it means to be an internal presheaf topos without all the obscure blabbing about internal language.

James Deikun (Apr 04 2025 at 03:21):

Wait, so is it that the image of the stacks under the left adjoint is the internal presheaf topoi as seen in Diaconescu's theorem? I guess that would be true up to equivalence since every (small) stack is equivalent to (the externalization of) an internal category ...

James Deikun (Apr 04 2025 at 03:26):

But wait, in that case how would it be possible to represent a large stack like the canonical stack?

James Deikun (Apr 04 2025 at 03:33):

And if $\mathcal S[X]$ represents any stack it's the canonical stack, since $\mathbf{Topos}/\mathcal S(\mathcal S/Y,\mathcal S[X]) \simeq \mathcal S/Y$ ...

James Deikun (Apr 04 2025 at 03:49):

In any case $\mathbf{Set}[X] \simeq \mathbf{Set}^\mathbf{FinSet}$ but the stack on $1$ it represents (if any) is $\mathbf{Set}$ itself, not $\mathbf{FinSet}^\mathsf{op}$.

James Deikun (Apr 04 2025 at 04:08):

Maybe this adjunction just isn't idempotent, unlike the other "fundamental adjunction"? That would be more than a little bothersome.

Fernando Yamauti (Apr 04 2025 at 04:13):

You're right. My suggestion can't possibly work. I was thinking, instead, about finite objects as an internal category, so my formula above is wrong. I confess I just skimmed through the paper quickly, but they seem to be repeating Giraud's construction on "Classifying Topos".

Fernando Yamauti (Apr 04 2025 at 05:21):

James Deikun said:

But wait, in that case how would it be possible to represent a large stack like the canonical stack?

I don't think that change much. We can still consider the indexed version of presheaf topoi (see 2.2 in Giraud's paper). The left adjoint will still send the identity fibration to the identity geometric morphism.

Looking back again, I think my suggestion was not wrong (apart from the lack of taking the opposite cat fiberwise).

James Deikun (Apr 04 2025 at 10:30):

Okay, I think I figured it out. $\mathbf{Set}[X]$ actually does represent the $\mathbf{Set}$-stack $\mathbf{FinSet}^\mathsf{op}$. The text had led me to think otherwise, but they really do mean the slice category $\mathbf{EssTopos}^\mathsf{co}/\mathcal S$, with the morphisms also being essential. You get away without this for sheaves because geometric morphisms are basically something similar to the Ind-completion of essential geometric morphisms (exactly the Ind-completion in the case of morphisms from stacks into $\mathbf{Set}[X]$) and Ind-completion acts trivially on discrete categories. This way everything makes sense, and the adjunction is idempotent, and the representing site for the canonical stack might be too big to present a topos but there are lots of smaller classifying objects for the small-enough sheaves. I think (3) is still true and the process to prove it is what you said.

Morgan Rogers (he/him) (Apr 04 2025 at 12:53):

I look forward to understanding this discussion. In the mean time, out of curiosity, what motivated this question @James Deikun ?

James Deikun (Apr 04 2025 at 14:20):

I was interested in the relationship between the classifying theory of an object and the discrete opfibration classifiers in a 2-topos.