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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• I am running into the following simple question and am wondering if there is anything useful to be said.

Let

$\mathcal{A} \in dgcAlg_\mathbb{Q}$

be a differential graded-commutative algebra in characteristic zero, whose underlying graded algebra is free graded-commutative on some graded vector space $V$:

$\mathcal{A} = (Sym(V), d) \,.$

$c \in \mathcal{A}_{odd} \,,$

and write $(c)$ for the ideal it generates.

In this situation I’d like to determine whether it is true that

1. there is an inclusion $\mathcal{A}/(c) \hookrightarrow \mathcal{A}$;

2. for every element $\omega \in \mathcal{A}$ there is a decomposition

$\omega = \omega_0 + c \omega_1$

for unique $\omega_0, \omega_1 \in \mathcal{A}/(c) \hookrightarrow \mathcal{A}$.

For example if $c \neq 0 \in V_{odd} \hookrightarrow \mathcal{A}_{odd} \hookrightarrow \mathcal{A}$ is a generator, then these conditions are trivially true.

On the other extreme, if $c$ is the product of an odd number $\gt 1$ of odd generators, then it is not true. For example if $c = c_1 c_2 c_3$, with $c_1, c_2, c_3 \in V_{odd} \hookrightarrow \mathcal{A}_{odd}$, then for instance $c (1 + c_1) = c (1 + c_2) = c$ and so the coefficient $\omega_1$ is not unique.

Is there anything useful that one can say in general?

• Took a stab at a general formulation of Poisson summation formula, although the class of functions to which it is supposed to apply wasn’t nailed down (yet).

(Some of the ingredients of Tate’s thesis are currently on my mind.)

• I have spent some minutes starting to put some actual expository content into the Idea-section on higher gauge theory. Needs to be much expanded, still, but that’s it for the moment.

• added a brief historical comment to Higgs field and added the historical references

• at surjective geometric morphism I have spelled out in detail most of the proof of the various equivalent characterizations, and all of the proof of the statement that geometric surjections are comonadic.

• I have added the Fierz identities that give the $S^2$-valued supercocycle in 5d here.

Added this briefly also at Fierz identity: here

• I have added to orthogonal factorization system

1. in the Definition-section three equivalent explicit formulations of the definition;

2. in the Properties-section the statement of the cancellability property.

Wanted to add more (and to add the proofs). But have to quit now. Maybe later.

• I have touched formal group a bit, but don’t have time to do anything substantial.

I need to adjust some of the terminology that I had been setting up at cohesive (infinity,1)-topos related to infinitesimal cohesion : the abstract notion currently called “$\infty$-Lie algebroid” there should be called “formal cohesive $\infty$-groupoid”. The actual L-infinity algebroids are (just) the first order formal smooth $\infty$-groupoids.

While on the train I started expanding some other entries on this point, but I need to quit now and continue after a little interruption.

• added to S-matrix a useful historical comment by Ron Maimon (see there for citation)

• The observation that the conditional expectation enjoys a universal property inspired me to write some ''random text''.
• At field (physics) I am beginning to write an actual introduction to the topic, now in a new section titled “A first idea of quantum fields”.

This means to introduce the concept with precise detail, but in a simple context (trivial and bosonic field bundles over Minkowski spacetime, perturbatively quantized) that allows to get a quick idea of the idea of the concept of (quantum) fields as such, without being distracted by other details.

So far I made it up to the derivation of the EOMs. Discussion of (deformation) quantization is to follow (maybe by tonight, depending on how much trouble I have with the trains) and I plan to sprinkle in the detailed example from scalar field in parallel with the abstract discussion.

• at KLT relations I have expanded the list of references. I added also references for the generalization of these relations that is known these days as “gravity is Yang-Mills squared” or similar (eventually this might want to be a separate entry).

In this course I also expanded the list of references at quantum gravityAs a perturbative quantum field theory

• Have added to HowTo a description for how to label equations

In the course of this I restructured the section “How to make links to subsections of a page” by giving it a few descriptively-titled subsections.

• there is an old article (Berends-Gastman 75) that computes the 1-loop corrections due to perturbative quantum gravity to the anomalous magnetic moment of the electron and the muon. The result turns out to be independent of the choice of (“re”-)normalization (hence what they call “finite”).

I have added a remark on this in the $(g-2)$-entry here and also at quantum gravity here.

• The definition of braided monoidal category was wrong or at least nonstandard, because it left out one of the hexagon axioms and included a 'compatibility with the unit object' law which follows from the usual definition.

I changed it to the usual definition.

If the nonstandard definition is equivalent to the usual one, I'd love to know why! But I don't see how you get two hexagons from one, even given compatibility with the unit object.

(Of course for a symmetric monoidal category we just need one hexagon.)

I also beefed up the definition at symmetric monoidal category so the poor reader doesn't need to run back to braided monoidal, then monoidal.
• At dichotomy between nice objects and nice categories I added a quote from Deligne about allowing awful schemes gave a nice category of schemes. I can’t find the page I was thinking of where this dichotomy is also mentioned along with attribution of the general idea to Grothendieck. I wanted to add it there as well.