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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.
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