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Stream: community: our work

Topic: Emilio Minichiello

Emilio Minichiello (Apr 21 2024 at 15:25):

I have finally defended my PhD thesis! It consists of two papers: Diffeological Principal Bundles and Infinity Bundles and The Diffeological Cech deRham Obstruction. Comments and critiques are appreciated.

So here I'll try and say a few words about the idea of the thesis. First of all, the main objects of study are diffeological spaces. These are a large class of smooth spaces that include finite dimensional smooth manifolds. However, the category of finite dimensional smooth manifolds is quite bad. It has very few limits (though it does have finite products) or colimits, and is not cartesian closed. Thus many constructions and modern day spaces of interest to differential geometers leave this category.

Thus a possible motivation for diffeological spaces is the following: 1) find a category $D$ such that there is a fully faithful functor $\mathsf{Man} \hookrightarrow D$ and 2) extend as many constructions and theorems from $\mathsf{Man}$ to all of $D$. Now there are many candidates for such a $D$. The real hard part is (2). Diffeological spaces are a good solution for both (1) and (2). You can check out the textbook Diffeology by Patrick Iglesias-Zemmour to see how much differential geometry has been extended to all diffeological spaces. Some examples include: differential forms, smooth homotopy groups, singular cohomology, deRham cohomology, bundle theory, etc. However, since $\mathsf{Diff}$ the category of diffeological spaces is so much larger than $\mathsf{Man}$, we'd expect many theorems that hold for $\mathsf{Man}$ to not hold for $$\mathsf{Diff}$. One easy example is that for finite dimensional smooth manifolds, there are two equivalent ways of representing the tangent space at a point. For diffeological spaces, these two constructions are no longer equivalent! See this paper for more on that.

It turns out that the category of diffeological spaces is equivalent to the category of concrete sheaves over the site $\mathsf{Cart}$ whose objects are spaces diffeomorphic to $R^n$ for some $n \geq 0$ and whose morphisms are smooth maps. This was proved by Baez and Hoffnung in 2008 (little wrinkle: Baez-Hoffnung prove it for a different underlying site, I prove they are equivalent). Thus this shows that the category of diffeological spaces is a Grothendieck quasi-topos! This is a significantly better category than $\mathsf{Man}$ and is a good candidate for a categorical framework for differential geometry.

However, there is another categorical framework for differential geometry using Higher Topos Theory (perhaps one should say model topos theory here, I don't need or use infinity categories in my papers), which has become very refined and powerful lately, see the introductions to my papers for more references. Here we embed diffeological spaces, thought of certain kinds of sheaves of sets, into presheaves of spaces (simplicial presheaves). Now we are in a world with a really nice homotopy theory, and lots of tools available to us. One can cofibrantly replace any diffeological space in a certain model category on simplicial presheaves over $\mathsf{Cart}$ and then map this into various coefficient $\infty$-sheaves. Taking $\pi_0$ of the resulting mapping space gives a notion of $\infty$-sheaf cohomology of the diffeological space.

In my thesis, I use this $\infty$-sheaf (I call it $\infty$-stack, because both sheaves of sets and stacks of groupoids are examples of $\infty$-sheaves) cohomology to prove several results.

• I show that diffeological principal $G$-bundles for a diffeological group $G$ are classified by $\infty$-stack cohomology with coefficients in the nerve of the presheaf of groupoids $\mathbf{B}G : U \mapsto [C^\infty(U,G) \rightrightarrows *]$, and furthermore show that diffeological principal bundles are examples of principal $\infty$-bundles,
• I show that there are interesting maps between $\infty$-stack cohomology of a diffeological space and some other examples of diffeological space cohomology put forward in the literature, and conjecture that these maps are isomorphisms in all dimensions.
• I prove that if $K$ is a diffeologiclly discrete subgroup of $\mathbb{R}$, then the $\infty$-stack cohomology of the irrational torus $\mathbb{R}/K$ (more on this fascinating class of diffeological spaces here) is the group cohomology of $K$ with values in $\mathbb{R}$. This was proved first by Patrick Iglesias-Zemmour here, however his proof is computational and difficult, whereas mine uses well-understood tools from higher differential geometry, resulting in a proof that is short and conceptual.
• I show that there is an obstruction to the Cech deRham isomorphism holding for diffeological spaces in all degrees. This was discovered by Patrick Iglesias-Zemmour in the paper above for degree 1. The obstruction to this isomorphism holding is contained in the group of isomorphism classes of diffeological principal $\mathbb{R}$-bundles that admit a connection. I extend this to higher dimensions, showing the obstruction in higher degrees lives in the group of isomorphism classes of $\mathbb{R}$-bundle $k$-gerbes with a chosen deRham representative for the curvature of a connection. Funnily enough I discovered during my writing that David Jaz Myers had discovered something very similar in the context of homotopy type theory. Thus we end up obtaining two interrelated exact sequences in every degree.
• I show that the canonical notion for connections on diffeological principal bundles put forward by higher topos theory is equivalent to a previously defined notion in the literature by Waldorf.

I think this is really interesting, because it shows the power of abstract thinking and methods. It shows that using category theory one can gain a really nice vantage point on a subject, and by using the correct tools in the right places, interesting theorems can just fall right out of the machinery.

John Baez (Apr 21 2024 at 15:57):

Really interesting stuff! Congratulations!

Matteo Capucci (he/him) (Apr 22 2024 at 10:07):

Cool stuff, congrats!

Matteo Capucci (he/him) (Apr 22 2024 at 10:08):

Can you say something more about the obstructions of Cech-de Rahm isos? Does it mean for diffeological spaces the two cohomology theories differ? What's the intuition of this failure being parametrized by principal R-bundles with a connection?

Emilio Minichiello (Apr 22 2024 at 11:40):

Yes, exactly. Let me give a little more background. So given a diffeological space $X$, we can define differential forms on it. The fact that $X$ is a sheaf on $\mathsf{Cart}$ makes this really easy to define. A differential $k$-form on $X$ is simply a map of sheaves $X \to \Omega^k$. Then from here you can define deRham cohomology of $X$ in the usual way. Now originally (in the 80s) Patrick Iglesias-Zemmour crafted his own version of Cech cohomology by hand and showed that the the deRham cohomology and his version of Cech cohomology differed on the irrational torus. He wrote up a preprint in French about this and never published it. He finally rewrote the paper in English and it recently was published..

Now for me, I have a competing definition of Cech cohomology for diffeological spaces that I call $\infty$-stack cohomology. This is just the cohomology you are forced into once you think of diffeological spaces as objects in an $\infty$-topos (more on this idea here). I showed in my thesis that with regards to the irrational torus you get the same exact results as Patrick with my cohomology, and you get the same results regarding the Cech deRham obstruction but now extended to all degrees rather than just degree 1.

Now for intuition, that's a bit hard, let me see what I can say. The main idea is the following. Suppose you have a manifold $M$, and a good open cover $U$ of $M$. You can then set up the Cech deRham bicomplex, and because you have partitions of unity, it turns out that every row and column is exact. So the spectral sequence of the bicomplex degenerates and gives an isomorphism between classical deRham cohomology and Cech cohomology. The idea one should really have here is that one can take a closed form and "unravel it" to build a bundle $\mathbb{R}$-gerbe with flat connection. Let us look at degree 1 as an example. Given a closed 1-form A on $M$, on every open subset $U_i$ of the good cover, you get an exact form $A_i$, and therefore there exist maps $\lambda_i: U_i \to \mathbb{R}$ such that $d \lambda_i = A_i$. It turns out these maps assemble into a cocycle for a principal $\mathbb{R}$-bundle, and $A$ gives a flat connection on the bundle. You obtain a map $\Omega^1_{\text{cl}} \to B \mathbb{R}^\delta$ where the latter $\infty$-stack is Dold-Kan of the presheaf of chain complexes $[\mathbb{R} \xrightarrow{d} \Omega^1_{\text{cl}}].$ There is an objectwise quasi-isomorphism $[\mathbb{R} \xrightarrow{d} \Omega^1_{\text{cl}}] \simeq [\mathbb{R}^\delta \to 0]$, where $\mathbb{R}^\delta$ is the sheaf of locally constant maps to $\mathbb{R}$, equivalently $\mathbb{R}$ equipped with the discrete diffeology.

Now you can include bundles with flat connections into bundles with any connection, $B_\nabla \mathbb{R}$ classifies these. Then you get a map $B_\nabla \mathbb{R} \to \Omega^2_{\text{cl}}$ by taking curvature of the connection. Then you can unravel this closed form to obtain a map $\Omega^2_{\text{cl}} \to B^2 \mathbb{R}^\delta$. It turns out that in every degree you obtain a homotopy fiber sequence $B^k_\nabla \mathbb{R} \to \Omega^{k+1}_{\text{cl}} \to B^{k+1} \mathbb{R}^\delta$ (this is how David Jaz Myers defines $B^k_\nabla \mathbb{R}$ in modal homotopy type theory), and this is how you obtain the exact sequence in every degree.

So in some sense, the obstruction you are seeing is coming from the inability to reverse this "unravelling procedure". The bundles with connection are inherent in the isomorphism above between Cech cohomology with values in $\mathbb{R}^\delta$ and principal $\mathbb{R}$-bundles with flat connection. For manifolds you don't have this problem, thanks to partitions of unity. Apologies if this was kind of sketchy.

Tim Hosgood (Apr 22 2024 at 14:14):

if you don't have a partition of unity, then when you try to "unravel" a one form will you end up not with a flat connection but instead an entire twisting cochain/Maurer–Cartan element of some Čech–de Rham bialgebra? (just trying to imagine what might happen in the holomorphic world)

Emilio Minichiello (Apr 22 2024 at 14:25):

Apologies, my above explanation could probably be improved (perhaps I should write a blog post), given a diffeological space $X$ and a closed $k$-form $A$ on $X$, one can always unravel $A$ to obtain a $k$-bundle $\mathbb{R}$-gerbe with flat connection on $X$. This process is explained nicely in Patrick's paper on page 20. It is the inverse construction, gluing together the pieces to get a global closed form, I think, that requires partitions of unity.

Emilio Minichiello (Apr 22 2024 at 14:30):

I also wonder about the holomorphic case. There is probably a ton to be said in that domain. For instance, complex diffeological spaces immediately exist: take concrete sheaves on your favorite complex manifold site, but I've never seen them defined or used explicitly in the literature.

Tim Hosgood (Apr 22 2024 at 18:24):

Emilio Minichiello said:

I also wonder about the holomorphic case. There is probably a ton to be said in that domain. For instance, complex diffeological spaces immediately exist: take concrete sheaves on your favorite complex manifold site, but I've never seen them defined or used explicitly in the literature.

mmm same, and I'm interested as to whether or not there's a reason beyond just a historical "falling out of fashion" of holomorphic stuff at key points in recent mathematical history

Tim Hosgood (Apr 22 2024 at 18:24):

it would be very interesting to study this though!

Emilio Minichiello (Apr 22 2024 at 18:38):

Ah yes, I have noticed this as well, but don't know the reason for it. There are a lot of people at the CUNY Graduate Center who know a lot of complex differential geometry, but unfortunately I am not one of them.

Tim Hosgood (Apr 22 2024 at 19:42):

yeah, Mahmoud Zeinalian would be exactly the first person that I would ask! it would be interesting to see what he has to say about this..