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With Igor Khavkine we finally have a polished version of what is now “Part I” of a theory of variational calculus in a differentially cohesive $\infty$-topos. It’s now called:
Synthetic geometry of differential equations
We keep our latest version of the file here.
Comments are most welcome.
Abstract:
We give an abstract (synthetic) formulation of the formal theory of partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as super-geometry, higher (stacky) differential geometry, or even a combination of both. A motivation for such a level of generality is the eventual goal of solving the open problem of covariant geometric pre-quantization of locally variational field theories, which may include fermions and (higher) gauge fields.
A remarkable observation of Marvan 86 is that the jet bundle construction in ordinary differential geometry has the structure of a comonad, whose (Eilenberg-Moore) category of coalgebras is equivalent to Vinogradov’s category of PDEs. We give a synthetic generalization of the jet bundle construction and exhibit it as the base change comonad along the unit of the “infinitesimal shape” functor, the differential geometric analog of Simpson’s “de Rham shape” operation in algebraic geometry. This comonad structure coincides with Marvan’s on ordinary manifolds. This suggests to consider PDE theory in the more general context of any topos equipped with an “infinitesimal shape” monad (a “differentially cohesive” topos).
We give a new natural definition of a category of formally integrable PDEs at this level of generality and prove that it is always equivalent to the Eilenberg-Moore category over the synthetic jet comonad. When restricted to ordinary manifolds, Marvan’s result shows that our definition of the category of PDEs coincides with Vinogradov’s, meaning that it is a sensible generalization in the synthetic context.
Finally we observe that whenever the unit of the “infinitesimal shape” ℑ\Im operation is epimorphic, which it is in examples of interest, the category of formally integrable PDEs with independent variables ranging in Σ is also equivalent simply to the slice category over ℑΣ. This yields in particular a convenient site presentation of the categories of PDEs in general contexts.
This looks very interesting, Urs; thanks for this.
Here is a somewhat idle or open-ended question: is there any chance that your work on cohesive $\infty$-toposes makes contact with the h-principle?
Our own h-principle could do with beefing up. But that does sound as though there could be some abstract general treatment. I wonder what the ’range’ of the h-principle is. I see the Oka principle is consider by Gromov an example, and there’s a higher complex analytic geometric account of it given there.
Small afterthought: from MathOverflow answer, “h-principles work best for open conditions”. Terry Tao once distinguished open conditions as those more amenable to analysis (and closed conditions, such as equalities, more amenable to algebra). I extracted some of that discussion here.
Just a quick remark. The h-principle has to do with finding/constructing local solutions (on a neighborhood of a point) to differential equations/inequalities. In this work we essentially stop at considering formal solutions (power series at a point). In smooth geometry, formal solutions are a strictly weaker notion than local solutions, while in analytic geometry they are the same.
Generally speaking, there is as much contact with the h-principle within the formal theory of PDEs in the topos setting (our formalism) as without the topos setting (what already existed before us). I would mention though that the notion of PDE that we use is general enough to admit both differential equations (aka closed conditions) as well as differential inequalities (aka open conditions).
Sorry, Igor, I didn’t mean to exclude you where I said “[Urs’s] work on cohesive $\infty$-toposes…”. Thanks to you and David for responses.
That’s a good point about the h-principle. While we don’t say anything about it at the moment ( “part 1” talks just about 1-toposes), I suppose Todd’s point is that the combination of PDE theory with the concept of homotopy in the h-principle makes it a natural concept to consider in a context of PDEs inside a cohesive $\infty$-topos.
I suppose one would form the internal space $\mathbf{\Gamma}_\Sigma(\mathcal{E})$ of sections of a sub-bundle $\mathcal{E} \hookrightarrow J^\infty_\Sigma E$ of some jet bundle over some $\Sigma$ (hence a space of non-holonomic solutions to the PDE embodied by $\mathcal{E}$), then form its shape $ʃ \mathbf{\Gamma}_{\Sigma}(\mathcal{E}) \in \infty Grpd$ and then ask for the connected components of holonomic solutions in this shape. The original PDE $\mathcal{E}$ would be said to satisfy the h-principle if $\tau_0 ʃ \mathbf{\Gamma}_{\Sigma}(\mathcal{E}) \simeq \tau_0 ʃ \mathbf{\Gamma}^{holonomic}_{\Sigma}(\mathcal{E})$.
I don’t know at the moment if there is anything new to be said about the h-principle this way, as in Igor’s #4, but it does seem like something worth exploring.
Unfortunately, both Igor and myself will have to let the synthetic PDE project rest during summer, due to other duties. Then we need to come back and polish the writeup of the formulation of variational caclulus via synthetic Euler-Lagrange complexes in differentially cohesive $\infty$-toposes. But then, maybe somebody else picks up a cohesive analysis of the h-principle in the meantime?!
I suppose one would form the internal space $\mathbf{\Gamma}_\Sigma(\mathcal{E})$ of sections of a sub-bundle $\mathcal{E} \hookrightarrow J^\infty_\Sigma E$ of some jet bundle over some $\Sigma$ (hence a space of non-holonomic solutions to the PDE embodied by $\mathcal{E}$), then form its shape $ʃ \mathbf{\Gamma}_{\Sigma}(\mathcal{E}) \in \infty Grpd$ and then ask for the connected components of holonomic solutions in this shape. The original PDE $\mathcal{E}$ would be said to satisfy the h-principle if $\tau_0 ʃ \mathbf{\Gamma}_{\Sigma}(\mathcal{E}) \simeq \tau_0 ʃ \mathbf{\Gamma}^{holonomic}_{\Sigma}(\mathcal{E})$.
Urs, I like how you encapsulated the potential for studying the h-principle in this context. There is a slightly refined way to phrase a putative h-principle. In general there will be an inclusion $\mathbf{\Gamma}^{holonomic}_{\Sigma}(\mathcal{E}) \hookrightarrow \mathbf{\Gamma}_{\Sigma}(\mathcal{E})$. The main question is then how close is this map to a homotopy equivalence (which includes information about the higher homotopy groups)?
Yes. We could speak of a “higher h-principle” if this inclusion is an iso on geometric homotopy groups also in positive degree. This would mean that not only every non-holonomic solution is homotopic to a holonomic one, but that moreover the space of choices of homotopies by which it is so, is a contractible space.
John Francis in these nice brief notes takes the h-principle to involve full weak homotopy equivalence. The final section suggests a more abstract approach:
Those inclined to greater generalization, topologists less analytically inclined, or those who have an example of interest that doesn’t quite fit in the rubric of differential relations, might ask whether a version of this h-principle exists without the trappings of analysis at all.
They’re part of a longer course.
True, that terminology is the opposite. Instead of calling the strengthening a “strong h-principle” it calls the usual version the “0-parametric h-principle”.
I changed the URLs for the notes by John Francis.
Looking at the 5 page intro again, there are suggestions that a cohesive $(\infty, 1)$-topos treatment might be interesting:
The presheaf satisfying the h-principle is a type of homotopy sheaf condition. For instance, if $\mathcal{F}$ is a sheaf of spaces, and every restriction map $\mathcal{F}(V) \to \mathcal{F}(U)$ is a fibration, then $\mathcal{F}$ adheres to the h-principle. (Those familiar with model categories may recognize this as close to the fibrancy condition for the Joyal or Jardine model structure on presheaves of spaces.)
In the special case of Oka-Grauert-Gromov principle there is a model category structure behind, as studied by Larusson and Forstnerič. In general, h-principle involves deformations; can it be said in general within an abstract homotopy theory setup ? It is not clear to me if your discussion above assumes a generalization of the model category structure of Larusson or not.
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