Amplifying what Urs said in #53, from their new article

- Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal,
*Left-exact Localizations of ∞-Topoi III: The Acyclic Product*[arXiv:2308.15573]

they write

The structural analogy between the theories of topoi and rings is folkloric but underexploited. One goal of this work is to deepen and extend it.

and

]]>imagine a version of algebraic geometry, where a ring $A$ is called a scheme, where the word ‘ring’ is never used, and where a morphism of schemes is denoted $A \to B$ to mean a ring morphism $B\to A$. This is the current state of topos theory.

I don’t think it’s useful to think of “logos theory” as a new theory that is waiting to be applied now where we only had topos theory before. (It’s defined just as the opposite category of topoi!)

Instead it’s a suggestive perspective on the latter in some circumstances, and as such not quite new if maybe more pronounced now.

(A move reminiscent of advertising “toposes as bridges”, if you have heard about that. After all, there is a good reason for Johnstone’s analogy between toposes and the proverbial elephant which is so huge and varied that blind men inspecting it from different ends may think they are dealing with different animals altogether.)

]]>Could be interesting to see what logos theory could be used for in e.g. 2307.15106, since up until when they take op to construct stacks, they’re basically using this (as in section 31 of Joyal’s notes). But some kind soul should first elaborate on the more basic things in logos.

]]>What they refer to as topos/logos duality is not to do with physics much. Also it’s not all that new (the ambition towards evocative terminology is):

The dual logical/algebraic perspective on spatial/geometric toposes dates all the way back to the conception of “elementary topoi” in the 1960s, and has been much expanded on at least in the case of (0,1)-toposes, where its the formal duality between frames (logical) and locales (spatial).

For instance, there is an old notion of *logical functor* which conceptually, apart from some extra technical fine-print, is the kind of opposite map to a geometric morphism (“spatial functor”) that the *Topo-logie* is about.

Thanks for adding, wasn’t aware of the notion of a logos. Is this what ultimately should account for the Heisenberg/Schrodinger duality? I know in Quantum Certification via Linear Homotopy Types you mention this is supposed to be accounted for by using Bohr topoi but this sounds more natural.

]]>added pointer to:

- Mathieu Anel, André Joyal, §4.2.3 in:
*Topo-logie*, in*New Spaces for Mathematics and Physics*, Cambridge University Press (2021) 155-257 [doi:10.1017/9781108854429.007, pdf]

Added some clarification on the relationship to what Ching et al. are calling ’tangent ∞-categories’.

]]>I removed the note

As of August 2022, this paper is withdrawn due to an error; a corrected version appeared on 25 Feb 2023,

since there appear to be no questions about the current version.

]]>Added a note that Tangent infinity-categories and Goodwillie calculus was withdrawn due to an error.

]]>added publication data for:

- Vincent Braunack-Mayer,
*Combinatorial parametrised spectra*Algebr. Geom. Topol.**21**(2021) 801-891 [arXiv:1907.08496, doi:10.2140/agt.2021.21.801]

[Administrative note: I have merged this thread with one named ’tangent (oo,1)-category’, where most of the previous discussion had taken place; the present thread is the one which is picked up by the edit announcer. I took the liberty of deleting a comment of Urs’ suggesting that the present thread be used instead of the one named ’tangent (oo,1)-category’, since it would now be a bit confusing (and didn’t have other content that needed preserving).]

]]>Was the construction of taking the tangent $(\infty, 1)$-topos for the whole $(\infty, 2)$-category (∞,1)Topos ever considered around these parts?

]]>The gulf between devising things and having them be taken up is vast, and without constantly being on the case of promoting them, it seems that credit often goes missing. I await 2050 when modal HoTT becomes everyday in analytic philosophy.

]]>An $n$Lab entry stating their basic idea (here) exists since 2013 (rev 1).

I had tried to advertize formalizing this in HoTT in Paris 2014. Back then the chairman (vv) shut down my talk after I mentioned Prop. 2.5, which he claimed was false. While that was silly and abusive of him, I can see how it was pointless to try to give that talk to that audience at that time. Maybe in 10 years from now I’ll try again.

I think stable Cohomotopy – whose role in the scheme of things I didn’t appreciate back then – will lend itself to constructive formalization. So that might be a topic for 2031.

]]>Re #25, and now the article has appeared:

- Mitchell Riley, Eric Finster, Daniel R. Licata,
*Synthetic Spectra via a Monadic and Comonadic Modality*, (arXiv:2102.04099)

Hmm, that’s pretty outrageous not to have mentioned your work - dcct, Quantization via Linear homotopy types, etc.

]]>Redirect: tangent ∞-category, tangent ∞-topos.

]]>Redirect: tangent ∞-category.

]]>Some rewording to emphasise what’s different in

- {#BauerBurkeChing21} Kristine Bauer, Matthew Burke, Michael Ching,
*Tangent $\infty$-categories and Goodwillie calculus*(arXiv:2101.07819)

I was just quoting from the paper

The goal of this note is to introduce two further examples of tangent ∞-categories…

So maybe better to choose the second option. But certainly no time at the moment.

]]>Thanks for adding. But let me suggest that we need to change the wording:

If I understand well (have only skimmed the articles) these recent articles talk about a notion of “tangent structure” on $\infty$-categories which subsumes the notion discussed on our page here (which they maybe call “Goodwillie tangent structure”), but has other examples, too.

It is only in this sense that it makes sense to write articles on new examples for “tangent structures”, I suppose.

So I think the line

Two further examples of tangent $(\infty,1)$-categories

needs to be changed to something like

Two further examples of tangent structures on infinity-categories.

That, or we need to change the title and content of this page here, generalizing it all appropriately. I guess the first option is less tedious.

[edit: so I made that change in wording. But I don’t have the time now to do this any justice at the moment. Please feel invited to adjust this edit.]

]]>Added

]]>Two further examples of tangent $(\infty,1)$-categories on (∞,1)-Topos and its opposite:

- Michael Ching,
Dual tangent structures for infinity-toposes, (arXiv:2101.08805)

Added

]]>Two further examples of tangent $(\infty,1)$-categories on (∞,1)-Topos and its opposite:

- Michael Ching,
Dual tangent structures for infinity-toposes, (https://arxiv.org/abs/2101.08805)

It would be good to get to see how all this work connects.

]]>Thanks for the pointer, I had not seen that.

On the other hand, the observation that the tangent $\infty$-category of parametrized spectra is infinitesimally cohesive (dcct, Prop. 4.1.9) is used by Riley-Licata-Finster, around their slide 18. (Without any attribution, but then it’s not such a deep observation…)

]]>