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I rewrote the few sentences at tangent (infinity,1)-category in an attempt to make it run more smoothly.
In any case, there is not much there yet...
rephrased the text still a bit more
added the statement about the characterization of $\infty$-(co-)limits in an tangent $\infty$-category.
added an explicit remark (with pointers to the proof) that that tangent category fibration is classified under $\infty$-Grothendieck construction by
$c \mapsto Stab(\mathcal{C}_{/c}) \,.$As announced in another thread, I started typing some remark on the tangent $\infty$-category of an $\infty$-topos here.
I had planned to do more, but then the $n$Lab was down all the time. Now I need to quit, so it remains a bit unsatisfactory. But maybe it’s still a start for anyone interested to join in and play with.
So what are the tangent $\infty$-categories to ETop$\infty$Grpd, Smooth$\infty$Grpd, SynthDiff$\infty$Grpd, Super$\infty$Grpd, and SmoothSuper$\infty$Grpd? Are there smooth spectra?
The most rapid of searches yields a paper by Bunke – Differential cohomology
These course note first provide an introduction to secondary characteristic classes and differential cohomology. They continue with a presentation of a stable homotopy theoretic approach to the theory of differential extensions of generalized cohomology theories including products and Umkehr maps.
The ’differential function spectra’ there reappear in Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory.
See at differential function complexes on the $n$Lab. This is the method by which Hopkins-Singer in Quadratic Functions in Geometry, Topology, and M-Theory produced differential refinements of generalized cohomology theories.
In cohesion there is a canonical way to produce these “homotopy fibers of Chern-characters” and hence differential cohomology, whence the name “differential cohomology in cohesive $\infty$-toposes”.
Without an explicit notion of spectra of course, in cohesion one can only take one abelian infinity-group $G$ (infinite loop space) at a time and produce its differential cohomology $\mathbf{B}G_{conn}$ (actually a braided infinity-group is sufficient for the canonical construction).
When we get stable objects into the game, for instance by showing that the tangent $\infty$-category $T(Smooth\infty Grpd)$ is again an $\infty$-topos and hence (by the above note) a cohesive $\infty$-topos, then the full range of stable homotopy constructions would be available more intrinsically, not just piecewise by each of its underlying infinite loop spaces/abelian $\infty$-groups
So when is the tangent to an $\infty$-topos again an $\infty$-topos? Do you think it likely that $T(Smooth\infty Grpd)$ is an $\infty$-topos?
So when is the tangent to an ∞-topos again an ∞-topos?
Yes, that’s the big question here.
André Joyal told me he has a theory of what he calls “loci”, I think, that can answer this question for a given $\infty$-topos. But I don’t know yet. I should check.
Do you have a hunch about the “big question”? Is it more likely to hold for a cohesive $\infinity$-topos?
I really don’t know. But I’ll try to find out.
My understanding of the theory of ’loci’ was that it addresses the dual question, namely when is the category of families of objects of some category a topos.
I have added pointers to the alternative discussion in terms of excisive functors here. Should eventually be further expanded.
At Stable extension ofcohesion, where it says
$\Omega^{\infty} \circ tot:T$ H $\to$ H assigns the total space of a spectrum bundle;
its left adjoint is the tangent complex functor;
shouldn’t ’tot’ be ’dom’ and ’tangent’ be ’cotangent’, or is this some other construction?
Is there a ’co-jet’ complex?
This is the same construction that Jacob Lurie considers in the article on deformation theory.
But here we do need to call it differently: because in Lurie’s article the category $\mathcal{C}$ of which one considers the tangent $\infty$-category $T \mathcal{C}$ is thought of as a category of algebras, whereas we are now speaking about taking the tangent $\infty$-category of a category of spaces. In the present context $\mathcal{C} = \mathbf{H}^{op}$.
Therefore “tangent complex” instead of “co-tangent complex”.
Secondly, concerning “dom” and “tot”: this is the same functor, but I find it more descriptive to speak of the functor that assigns the total space of a bundle (which is a description of its meaning) than to speak of the domain functor (which is a description only of one specific way of constructing this functor, and in fact not the way which is used in the subsection where the term appears)-
If there’s differential cohesion available, is it possible to form the tangent (∞,1)-category by means of the modalities $\Re \dashv \Im \dashv \&$?
Recall that a tangent $\infty$-topos is internally witnessed as being infinitesimally cohesive over the base, see here.
I’m not sure I know what “internally witnessed” means there. I was just having a look at Quantization via Linear homotopy types and reading the definition:
For $X \in \mathbf{H}$ then $Mod(X)$ are the ∗/-modal types in $\mathbf{H}_{/X}$ which are left orthogonal to $\Im$-modal types,
and wondering if becomes easier to speak of the tangent at an object $X \in \mathbf{H}$ with modalities such as $\Im$ available, rather like I can give an easier account of the tangent space of a manifold $M$ in SDG as $M^D$, since I have infinitesimals available.
A different sense of ’tangent ∞-category’?
Tangent ∞-categories and Goodwillie calculus, Michael Ching, talk
Goodwillie calculus is a set of tools in homotopy theory developed, to some extent, by analogy with ordinary differential calculus. The goal of this talk is to make that analogy precise by describing a common category-theoretic framework that includes both the calculus of smooth maps between manifolds, and Goodwillie calculus of functors, as examples. This framework is based on the notion of “tangent category” introduced first by Rosicky and recently developed by Cockett and Cruttwell in connection with models of differential calculus in logic, with the category of smooth manifolds as the motivating example. In joint work with Kristine Bauer and Matthew Burke (both at Calgary) we generalize to tangent structures on an (∞,2)-category and show that the (∞,2)-category of presentable ∞-categories possesses such a structure. This allows us to make precise, for example, the intuition that the ∞-category of spectra plays the role of the real line in Goodwillie calculus. As an application we show that Goodwillie’s definition of n-excisive functor can be recovered purely from the tangent structure in the same way that n-jets of smooth maps are in ordinary calculus. If time permits, I will suggest how other concepts from differential geometry, such as connections, may play out into the context of functor calculus.
Thanks for the alert. I have forwarded this to Vincent.
Perhaps a few talks of interest there at Homotopy harnessing higher structures. I think you were looking at graph complexes and configuration spaces before Christmas:
Configuration spaces of points and real Goodwillie-Weiss calculus, Thomas Willwacher, talk
The manifold calculus of Goodwillie and Weiss proposes to reduce questions about embedding spaces of manifolds to questions about mapping spaces of the (little-disks modules of) configuration spaces of points on those manifolds. We will discuss real models for these configuration spaces. Furthermore, we will see that a real version of the aforementioned mapping spaces is computable in terms of graph complexes. In particular, this yields a new tool to study diffeomorphism groups and moduli spaces.
added pointer to
added pointer to
Maybe a tangent category $T(C)$ is an infinitesimal path space object $C^D$, just like one considers the path space object $\text{Type}^I$ in type theory to get homotopies.
One statement roughly in this direction is: A tangent $\infty$-topos is infinitesimally cohesive over its base (scroll down to the third example in the list there).
That’s the property that was recently used by somebody in the HoTT community to axiomatize spectra via modal HoTT (around slide 18 here. (Of course not attributing it this way.)
I also have the goal of making sense of a “map $X \rightarrow \text{Scheme}$ for a scheme $X$, which sends infinitesimal intervals to infinitesimal intervals”. Such a map will glue via dependent sum to a smooth map, I think. This would also formalize the intuition of “continuously varying fibers”.
$x : X \vdash Y(x) : \text{Scheme}$ goes to $\sum_{x : X} Y(x) : \text{Scheme}$, with the canonical map $\sum_{x : X} Y(x) \rightarrow X$ being a smooth map; I intend the smooth maps to be the fibrations. The reason I expect this is that they satisfy the infinitesimal lifting property, which is exactly analogous to the path lifting property for fibrations.
Anyhow, I understand you must be pretty busy. Thanks for all the time you’ve spent helping me with these “geometric type theory” ideas.
Now I’m convinced that the right lifting property is against $\text{Red}(X) \rightarrow X$.
For each $X$, and each commutative triangle $\text{Red}(X) \rightarrow Y \rightarrow X = \text{Red}(X) \rightarrow X$ with $Y \rightarrow X$ a fibration (smooth map), there is a retract $X \rightarrow Y$. That’s the same as being formally smooth.
I just need to make sense of context extension. $x : X \vdash Y(x) : \text{Scheme}$ should be a certain kind of sheaf that works well with reduction and glues to a smooth cover of $X$. The property I want to require should also be analogous to univalence.
I think I found the condition: it’s that $(U_{red} \times_X Y)_{red} \rightarrow (U \times_X Y)_{red}$ is an isomorphism.
Take $U : X \vdash F(U) : \text{Scheme}$. Infinitesimally close shapes/points $U \sim U_{red}$ are sent to infinitesimally close schemes $F(U_{red}) = U_{red} \times_X Y \sim U \tmes_X Y = F(U)$ (their reductions are the same). This is gives us what the “univalence axiom” should be.
Infinitesimal univalence says that infinitesimally close objects in $\text{Type}$ are types whose reductions are the same.
Something to follow up when I have a moment. A paper from the talk at #19
Thanks. I have added it here.
we might speculate on how the Goodwillie tangent structure fits into the much bigger programme of ‘higher differential geometry’ developed by Schreiber [Sch13, 4.1], or into the framework of homotopy type theory [Pro13],though we don’t have anything concrete to say about these possible connections.
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…)
It would be good to get to see how all this work connects.
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)
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)
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.]
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.
Some rewording to emphasise what’s different in
Re #25, and now the article has appeared:
Hmm, that’s pretty outrageous not to have mentioned your work - dcct, Quantization via Linear homotopy types, etc.
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.
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.
Was the construction of taking the tangent $(\infty, 1)$-topos for the whole $(\infty, 2)$-category (∞,1)Topos ever considered around these parts?
[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).]
added publication data for:
Added a note that Tangent infinity-categories and Goodwillie calculus was withdrawn due to an error.
added pointer to:
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.
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.
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.
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.)
Amplifying what Urs said in #53, from their new article
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.
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