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Stream: theory: category theory

Topic: Exponential of presheaf topos by locale

Fernando Yamauti (Dec 31 2024 at 19:13):

Hi. Does anyone know a way of computing explicitly exponentials of the form $\widehat{C}^{\widetilde{X}}$ for $C$ small and $X$ a locally compact topological space?

Morgan Rogers (he/him) (Dec 31 2024 at 19:23):

Which constructions do the accents refer to, and in which category are you working?

Fernando Yamauti (Dec 31 2024 at 19:32):

Morgan Rogers (he/him) said:

working

I'm working inside the cat of topoi. The hat means presheaves and tilde means sheaves.

Morgan Rogers (he/him) (Jan 01 2025 at 13:04):

Gotcha. One way to present it (which is not especially satisfying) is to extract an explicit presentation for the theory it classifies: since a geometric morphism $\mathcal{F} \to \widehat{C}^{\tilde{X}}$ corresponds to a geometric morphism $\mathcal{F} \times \tilde{X}\to \widehat{C}$, it should be the theory of "X-indexed C-torsors".
I wonder if you could derive a more explicit description from that for the exponential of the object classifier somehow.

Fernando Yamauti (Jan 01 2025 at 18:44):

Morgan Rogers (he/him) said:

Gotcha. One way to present it (which is not especially satisfying) is to extract an explicit presentation for the theory it classifies: since a geometric morphism $\mathcal{F} \to \widehat{C}^{\tilde{X}}$ corresponds to a geometric morphism $\mathcal{F} \times \tilde{X}\to \widehat{C}$, it should be the theory of "X-indexed C-torsors".
I wonder if you could derive a more explicit description from that for the exponential of the object classifier somehow.

If $C$ is the lex completion of say $D$, then one can just take the lax limit with shape $D$ of $\mathcal{S}[O]^{\widetilde{X}}$. But that still seems like a complete hell to compute.

Perhaps a more conceptual syntactic description would be better. How can one describe what's a $X$-indexed $C$-torsor in an arbitrary topos $\mathcal{X}$?

First, one would have to describe what's $X$ as an internal locale by presenting the internal site corresponding to the geometric morphism $\mathcal{X} \otimes \widetilde{X} \rightarrow \mathcal{X}$ and, then, describe what it means for an internal locale to be a $C$-torsor. Maybe at least one of those steps was already explicitly written down somewhere?

Graham Manuell (Jan 02 2025 at 13:45):

This exponential doesn't need to exist. Not all exponentiable locales are exponentiable as toposes.

Adrian Clough (Jan 02 2025 at 14:04):

Do you know a concrete example of an exponentiable locale that is not exponentiable as a topos?

Fernando Yamauti (Jan 02 2025 at 14:31):

Graham Manuell said:

This exponential doesn't need to exist. Not all exponentiable locales are exponentiable as toposes.

Thanks for the comment. I didn't know about that. I thought a continuous poset was continuous as a category iff it was continuous as a poset. I guess the failure has to do with the difference between proper morphisms of topoi and locales.

Still, in my case, I'm mostly interested in compact Haudorff spaces (I think locally quasi-compact and quasi-separated is enough to imply exponentiability of the respective topos). Indeed, any explicit description of things like $\widehat{C}^{\widetilde{[0, 1]}}$ or $\widehat{C}^{\widetilde{\beta S}}$ for $S$ an infinite set would already be very illuminating.

Graham Manuell (Jan 02 2025 at 19:24):

Adrian Clough said:

Do you know a concrete example of an exponentiable locale that is not exponentiable as a topos?

No, but I believe this is discussed in the Elephant. The key phrase is metastably locally compact.

Morgan Rogers (he/him) (Jan 03 2025 at 11:47):

Fernando Yamauti said:

First, one would have to describe what's $X$ as an internal locale by presenting the internal site corresponding to the geometric morphism $\mathcal{X} \otimes \widetilde{X} \rightarrow \mathcal{X}$ and, then, describe what it means for an internal locale to be a $C$-torsor. Maybe at least one of those steps was already explicitly written down somewhere?

The internal locale is the "constant sheaf" valued in the opens of X, which is to say the corresponding frame is that obtained by applying the inverse image functor of the global sections morphism. I expect the theory you want is constructed by extending a theory with a sort for each open of X, although I would need to scribble on a whiteboard for a while to figure out a more precise description.

Fernando Yamauti (Jan 03 2025 at 17:45):

Morgan Rogers (he/him) said:

The internal locale is the "constant sheaf" valued in the opens of X, which is to say the corresponding frame is that obtained by applying the inverse image functor of the global sections morphism. I expect the theory you want is constructed by extending a theory with a sort for each open of X, although I would need to scribble on a whiteboard for a while to figure out a more precise description.

Thanks. I see now. In general, the pullback of a localic topoi is the inverse image applied to the frame as an internal category. So as an indexed category, it's just the constant sheaf on the frame of $X$. But going back and forth between bounded geometric morphisms and internal sites is something I have a lot of difficult with.

Then, in the second step, we are left with internalising into $\mathcal{X}$ the geometric $\mathcal{X}$-morphism $\mathcal{X} \otimes \widetilde{X} \rightarrow \mathcal{X} \otimes \widehat{C}$. But I'm not comfortable enough with indexed/internal topoi to be sure if the obvious site presentation of $\mathcal{X} \otimes \widehat{C}$ inside $\mathcal{X}$, namely the inverse image applied to the category $C$, works.

Is it true, in general, that for an arbitrary geometric morphism $f \colon \mathcal{X} \rightarrow \mathcal{Y}$ and $C \in \mathbf{Cat} (\mathcal{Y})$, $f^* \textnormal{Fun}_{\mathcal{Y}} (C^{o}, \mathcal{Y}) \cong \textnormal{Fun}_{\mathcal{X}} ((f^*C)^{o}, \mathcal{X})$ ? (the outermost $f^*$ is the pullback of internal/indexed topoi as opposed to the one of internal/indexed categories).

That seems like asking for preservations of exponentials. But, for objects, as opposed to categories, that is only true in specific cases (like loc conn geometric morphisms).

Morgan Rogers (he/him) (Jan 03 2025 at 18:02):

Agreed, that isomorphism needn't hold in general. My original guess was that the theory would abstract a description of what a C-torsor in Sh(X) looks like "externally".

Fernando Yamauti (Jan 03 2025 at 18:58):

Morgan Rogers (he/him) said:

Agreed, that isomorphism needn't hold in general. My original guess was that the theory would abstract a description of what a C-torsor in Sh(X) looks like "externally".

Yep, but that requires saying what's the correct version of $C$ inside an arbitrary topos $\mathcal{X}$ (i.e., "the abstract description").

I will stop being lazy and I will try to start writing down everything. $\mathcal{X} \otimes \widetilde{X} \rightarrow \widehat{C}$ over the terminal topos is the same thing as $\mathcal{X} \otimes \widetilde{X} \rightarrow \mathcal{X} \otimes \widehat{C}$ over $\mathcal{X}$ (using the universal property of the pullback). So the "correct abstract notion" of a $C$-torsor is a flat morphism from an $\mathcal{X}$-site that presents $\mathcal{X} \otimes \widehat{C}$. But I don't know if that $\mathcal{X}$-site is just $\Gamma^* (C)$ where $\Gamma$ is the global section morphism of $\mathcal{X}$ and $\Gamma^* (C) = \Gamma^* (C_1) \rightrightarrows \Gamma^* (C_0)$ (or, in the indexed language, it's just post-composition by $\Gamma^*$ applied to the associated sheaf of categories).

I'm starting to think that formula is actually true and, then, the correct abstract $C$ will be the naive one. That would be a variant of Lemma 2.5.3 in C2.5, p.596 of the Elephant, I think.

John Baez (Jan 03 2025 at 19:03):

\matthcal should be \mathcal