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

• Added the proof of the fundamental theorem of covering spaces in HoTT

• tried to edit Ext a bit. But this needs to be expanded, eventually.

• I added to covering space a section In terms of homotopy fibers that explains the universal covering space as the homotopy fiber/principal oo-bundle classified by the cocycle that is the constant path inclusion $X \to \Pi_1(X)$ of topological groupoids.

To fit this into the entry, I added some new sections and restructured slightly. Todd and David should please have a look.

What I just added is essentially what David Roberts says in various query boxes, notably in what is currently the last query box. Back then we talked about the "Roberts-Schreiber construction" or whatnot, but really what this is is just the standard way to compute homotopy fibers in the oo-category of oo-groupoids.

I suspect that Todd's bar construction described there can similarly be understood as being nothing but another way to compute the more abstractly defined homotopy pullback in concrete terms. I'll have to think about this, though. But probably Tim Porter or Mike Shulman will immediately recognize this as the relevant bar construction of homotopy pullbacks in homotopy coherent category theory.

• following discussion here I am starting an entry with a bare list of references (sub-sectioned), to be !include-ed into the References sections of relevant entries (mainly at homotopy theory and at algebraic topology) for ease of updating and syncing these lists.

The organization of the subsections and their items here needs work, this is just a start. Let’s work on it.

I’ll just check now that I have all items copied, and then I will !include this entry here into homotopy theory and algebraic topology. It may best be viewed withing these entries, because there – but not here – will there be a table of contents showing the subsections here.

• corrected a typo

Anonymous

• added to derivation at the very end in the exampls section a discussion of derivations on smooth functions (and how they are vector fields) and f derivations on continuous functions (and how they are trivial).

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• I began to add a definition of conformal field theory using the Wightman resp. Osterwalder-Schrader axiomatic approach. My intention is to define and explain the most common concepts that appear again and again in the physics literature, but are rarely defined, like “primary field” or “operator product expansion”.

(I remember that I asked myself, when I first saw an operator product expansion, if the existence of one is an axiom or a theorem, I don’t remember reading or hearing an answer of that until I looked in the book by Schottenloher).

• Not sure about the original intention but sure enough that having literally “phi” (I mean, not the greek letter but this combination of three latin letters) is inappropriate

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• a stub entry, for the moment just to record some references and to satisfy links at defect brane

• starting something. Not done yet, but need to save

• brief category:people-entry for hyperlinking references

• Corrected a link. Before the word “derivation” linked to the page for derivations in differential algebra.

Sam Winnick

• Created:

\tableofcontents

## Definition

(Definition 2.1 in Bhatt–Scholze.)

Fix a prime $p$. A δ-ring is a pair $(R,δ)$, where $R$ is a commutative ring and $\delta\colon R\to R$ is a map of underlying sets such that $\delta(0)=0$, $\delta(1)=0$,

$\delta(xy)=x^p \delta(y)+y^p \delta(x) + p\delta(x)\delta(y),$

and

$\delta(x+y)=\delta(x)+\delta(y)+(x^p+y^p-(x+y)^p)/p.$

## Properties

If $(R,\delta)$ is a δ-ring, then the map $\phi\colon R\to R$ given by $\phi(x)=x^p + p\delta(x)$ is a ring homomorphism that lifts the Frobenius endomorphism on $R/p$.

For $p$-torsionfree rings, the above correspondence between δ-structures and lifts of the Frobenius endomorphism on $R/p$ to $R$ is bijective. This motivates the identities in the definition of a δ-structure.

## References

• stub entry for the last remaining item in the brane scan, for the moment just to record references

• Am finally giving this neglected entry (in contrast to M2-brane) a tad more text, a little history, disambiguation and pointers to more specific entries and more references.

• stub entry for one more item in the brane scan, for the moment just to record references

• I am trying to give more of the entries of the brane scan in low ambient dimension their proper names.

Next to the little string in $D = 6$ the brane scan says that there is a Green-Schwarz action functional for a 3-brane $\sigma$-model in $D = 6$. This has been first written down in

• James Hughes, Jun Liu, Joseph Polchinski, Supermembranes, Physics Letters B Volume 180, Issue 4, 20 November 1986, Pages 370–374

but it seems to go by no specific name apart from “the 3-brane in 6d”. So I created a stub entry with that title, 3-brane in 6d.

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• starting something, with a hat-tip to Charles Rezk

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• expanded the discussion at equivariant homotopy theory

• expanded the statement of the classical Elmendorf theorem

• added the statement of the general Elmendorf theorem in general model categories

• added remarks on G-equivariant oo-stacks, as special cases of this

• brief category:people-entry for hyperlinking references

• for the moment just for completeness (and to record references)

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• started a Properties-section, with material copied from other entries:

1. the rational cohomology of iterated loop spaces of spheres

2. the relation to configuration spaces of points

• Added in the usual group presentation of the dihedral group $D_{2n}$ plus a warning that this group is also denoted $D_n$ by some authors (including myself!!!)

• added to equivariant K-theory comments on the relation to the operator K-theory of crossed product algebras and to the ordinary K-theory of homotopy quotient spaces (Borel constructions). Also added a bunch of references.

(Also finally added references to Green and Julg at Green-Julg theorem).

This all deserves to be prettified further, but I have to quit now.

• i polished the definition in bundle gerbe and then reorganized the former material on “Interpretations” in a new section

that first shows how to get a shifted central extension of groupoids form the bundle gerbe, and then demonstrates that this is the total space of a principal 2-bundle

• Inspired by a discussion with Martin Escardo, I created taboo.

• Page created, but author did not leave any comments.

Anonymous

• needed to point to ring of integers of a number field. The term used to redirect just to integers. I have split it off now with a minimum of content. Have to rush off now.

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• added the Hurewicz model structure as an example of 2-trivial model structure

Daniel Teixeira

• am splitting this off from diffeological space as a stand-alone entry, for ease of hyperlinking

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