am working on putting some genuine detailed content into *smooth groupoid*. So far there is now discussion of the groupoid-enriched category of groupoid-valued presheaves, Cech nerves, and the stack condition.

Then it breaks off and some rough old material kicks in which needs to be harmonized. Will continue later, need to go offline now for a little.

]]>I have started creating a hyperlinked index at

- Theodore Frankel,
*The Geometry of Physics - An Introduction*

A colleague may use this for a course and maybe we get a chance to polish and/or write up some more related material in relevant $n$Lab entries.

]]>I noticed that we had no entry *density*, so I very briefly created one. While cross-linking it, I noticed that at *volume form* there is related discussion re “pseudo-volume forms”. Maybe somebody here would enjoy to add a bit more glue? (I won’t at the moment.)

While working at *geometry of physics* on the next chapter *Differentiation* I am naturally led back to think again about how to best expose/introduce infinitesimal cohesion. To the reader but also, eventually, to Coq.

First the trivial bit, concerning terminology: I am now tending to want to call it *differential cohesion*, and *differential cohesive homotopy type theory*. What do you think?

Secondly, I have come to think that the extra right adjoint in an infinitesimally cohesive neighbourhood need not be part of the axioms (although it happens to be there for $Sh_\infty(CartSp) \hookrightarrow Sh_\infty(CartSp_{th})$ ).

So I am now tending to say

**Definition.** A *differential structure* on a cohesive topos is an ∞-connected and locally ∞-connected geometric embedding into another cohesive topos.

And that’s it. This induces a homotopy cofiber sequence

$\array{ CohesiveType &\hookrightarrow& InfThickenedCohesiveType &\to& InfinitesimalType \\ & \searrow & \downarrow & \swarrow \\ && DiscreteType }$Certainly that alone is enough axioms to say in the model of smooth cohesion all of the following:

- reduced type, infinitesimal path ∞-groupoid, de Rham space, jet bundle, D-geometry, ∞-Lie algebra (synthetically), Lie differentiation, hence “Formal Moduli Problems and DG-Lie Algebras” , formally etale morphism, formally smooth morphism, formally unramified morphism, smooth etale ∞-groupoid, hence ∞-orbifold etc.

So that seems to be plenty of justification for these axioms.

We should, I think, decide which name is best (“differential cohesion”?, “infinitesimal cohesion”?) and then get serious about the “differential cohesive homotopy type theory” or “infinitesimal cohesive homotopy type theory” or maybe just “differential homotopy type theory” respectively.

]]>started something at *Reeb sphere theorem*

created an entry for *Lorentzian geometry*, prompted by this Physics.SE question

quick entry for *pullback of differential forms*, to be further expanded

started a bare minimum at *Poisson-Lie T-duality*, for the moment just so as to have a place to record the two original references

I finally gave this statement its own entry, in order to be able to conveniently point to it:

*embedding of smooth manifolds into formal duals of R-algebras*

created *equivariant de Rham cohomology* with a brief note on the Cartan model.

(I seem to remember that we had discussion of this in the general context of Lie algebroids elsewhere already, several years back. But now I cannot find it….)

]]>created *Einstein manifold*

(for the moment only to record the example of weak $G_2$-manifolds…)

]]>I have expanded, streamlined and re-organized a little at *differential forms on simplices*.

gave the statement that *derivations of smooth functions are vector fields* a dedicated entry of its own, in order to be able to convieniently point to it

spelled out the definition at *formal adjoint differential operator*

started *M-theory on G2-manifolds*

started something at *propagation of singularities theorem*

For ease of linking to from various entries, and in order to have all the relevant material in one place, I am creating an entry

Presently this contains

an Idea-section,

some preliminaries to set the scene,

the statement and proof for the case of compactly supported distributions, taken from what I had just writted into the entry

*compactly supported distribution*,the informal statement for general distributions, so far just with a pointer to Kock-Reyes 04,

a section “Applications”, so far with

some comments on the relevance in pQFT;

some vague pointer to Lawvere-Kock’s generalization to a more general theory of “extensive quantity”

both of which deserve to be expanded.

Eventually I want to have more details on the page, but I’ll leave it at that for the time being. Please feel invited to join in.

I’ll go now and add pointers to this page from “distribution” and from other pages that mention the fact.

]]>Today I was asked for what I know about the development of the theory of Kan-fibrant simplicial manifolds. I realized that the nLab does not discuss this, so I have started a page now with the facts that come to mind right away. (Likely I forgot various things that should still be added.)

]]>Does anyone know if there is a definition/construction of spinor bundles which starts with a plain manifold, rather than with a manifold equipped with a metric?

Or to ask an equivalent question: given two Riemannian manifolds with spin structures and a diffeomorphism which preserves the spin structures, is it possible to pull back sections of the spinor bundle from one manifold to the other? And prior to that, what “should” it even mean for a diffeomorphism to preserve the spin structure?

I’d appreciate any pointers to the literature or other sort of input that people here might have.

]]>I have split off spin^c from spin^c structure

]]>I have polished a little at *geometry of physics – smooth sets*, in reaction to feedback that Arnold Neumaier provided over on PF here.

I finally gave the *Connes-Lott-Chamseddine-Barrett model* its own entry. So far it contains just a minimum of an Idea-section and a minimum of references.

This was prompted by an exposition on *PhysicsForums Insights* that I wrote: *Spectral standard model and String compactifications*

I have re-written the content at *differentiable manifold*, trying to make it look a little nicer. Also gave *topological manifold* some minimum of content.

created Stokes theorem

]]>I have been added a first approximation to an Idea-section to *torsion of a G-structure* -

Have also added a pointer to Lott 90 and started a stub *torsion constraints in supergravity*, for the moment only to record some references.

Have also further touched related entries such as *torsion of a Cartan connection*.